Produced by Marius Masi, Don Kretz and the Online
Distributed Proofreading Team at http://www.pgdp.net









Transcriber's notes:

(1) Numbers following letters (without space) like C2 were originally
      printed in subscript. Letter subscripts are preceded by an
      underscore, like C_n.

(2) Characters following a carat (^) were printed in superscript.

(3) Side-notes were relocated to function as titles of their respective
      paragraphs.

(4) Macrons and breves above letters and dots below letters were not
      inserted.

(5) [root] stands for the root symbol; [alpha], [beta], etc. for greek
      letters.

(6) The following typographical errors have been corrected:

    ARTICLE HUSS: "This appointment had a deep influence on the already
      vigorous religious life of Huss himself ..." 'appointment' amended
      from 'appoinment'.

    ARTICLE HYACINTH: "... the wild hyacinth of western North America,
      Camassia esculenta." 'America' amended from 'Amercia'.

    ARTICLE HYDRAULICS: "Fig. 74 shows an arrangement designed for the
      Manchester water works. The water enters from the reservoir at
      chamber A, the object of which is to still the irregular motion of
      the water." 'at' amended from 'a'.

    ARTICLE HYDRAULICS: "But the velocity at this point was probably
      from Howden's statements 16.58 × 40/26 = 25.5 ft. per second, an
      agreement as close as the approximate character of the data would
      lead us to expect." Added 'per second'.

    ARTICLE HYDRAULICS: "... as the velocity and area of cross section
      are different in different states of the river." 'different'
      amended from 'differest'.

    ARTICLE HYDROGEN: "... for example, formic, glycollic, lactic,
      tartaric, malic, benzoic and other organic acids are readily
      oxidized in the presence of ferrous sulphate ..." 'glycollic'
      amended from 'glygollic'.



  THE

  ENCYCLOPÆDIA BRITANNICA

  ELEVENTH EDITION




  FIRST  edition, published in three    volumes, 1768-1771.
  SECOND    "        "        ten          "     1777-1784.
  THIRD     "        "        eighteen     "     1788-1797.
  FOURTH    "        "        twenty       "     1801-1810.
  FIFTH     "        "        twenty       "     1815-1817.
  SIXTH     "        "        twenty       "     1823-1824.
  SEVENTH   "        "        twenty-one   "     1830-1842.
  EIGHTH    "        "        twenty-two   "     1853-1860.
  NINTH     "        "        twenty-five  "     1875-1889.
  TENTH     "   ninth edition and eleven
                  supplementary volumes,         1902-1903.
  ELEVENTH  "  published in twenty-nine volumes, 1910-1911.


  COPYRIGHT

  in all countries subscribing to the Bern Convention

  by

  THE CHANCELLOR, MASTERS AND SCHOLARS
  of the
  UNIVERSITY OF CAMBRIDGE

  _All rights reserved_




  THE

  ENCYCLOPÆDIA BRITANNICA

  A DICTIONARY OF
  ARTS, SCIENCES, LITERATURE AND GENERAL INFORMATION

  ELEVENTH EDITION

  VOLUME XIV
  HUSBAND to ITALIC

  New York

  Encyclopædia Britannica, Inc.
  342 Madison Avenue


  Copyright, in the United States of America, 1910,
  by
  The Encyclopædia Britannica Company.


     VOLUME XIV, SLICE I

    Husband to Hydrolysis




ARTICLES IN THIS SLICE:


  HUSBAND                               HYADES
  HUSBAND AND WIFE                      HYATT, ALPHEUS
  HUSHI                                 HYBLA
  HUSKISSON, WILLIAM                    HYBRIDISM
  HUSS                                  HYDANTOIN
  HUSSAR                                HYDE (17th century English family)
  HUSSITES                              HYDE, THOMAS
  HUSTING                               HYDE (market town)
  HUSUM                                 HYDE DE NEUVILLE, JEAN GUILLAUME
  HUTCHESON, FRANCIS                    HYDE PARK
  HUTCHINSON, ANNE                      HYDERABAD (city of India)
  HUTCHINSON, JOHN (puritan soldier)    HYDERABAD (state of India)
  HUTCHINSON, JOHN (theological writer) HYDERABAD (capital of Hyderabad)
  HUTCHINSON, SIR JONATHAN              HYDER ALI
  HUTCHINSON, THOMAS                    HYDRA (island of Greece)
  HUTCHINSON (Kansas, U.S.A.)           HYDRA (legendary monster)
  HUTTEN, PHILIPP VON                   HYDRA (constellation)
  HUTTEN, ULRICH VON                    HYDRACRYLIC ACID
  HUTTER, LEONHARD                      HYDRANGEA
  HUTTON, CHARLES                       HYDRASTINE
  HUTTON, JAMES                         HYDRATE
  HUTTON, RICHARD HOLT                  HYDRAULICS
  HUXLEY, THOMAS HENRY                  HYDRAZINE
  HUY                                   HYDRAZONE
  HUYGENS, CHRISTIAAN                   HYDROCARBON
  HUYGENS, SIR CONSTANTIJN              HYDROCELE
  HUYSMANS (Flemish painters)           HYDROCEPHALUS
  HUYSMANS, JORIS KARL                  HYDROCHARIDEAE
  HUYSUM, JAN VAN                       HYDROCHLORIC ACID
  HWANG HO                              HYDRODYNAMICS
  HWICCE                                HYDROGEN
  HYACINTH (flower)                     HYDROGRAPHY
  HYACINTH (gem-stone)                  HYDROLYSIS
  HYACINTHUS




INITIALS USED IN VOLUME XI. TO IDENTIFY INDIVIDUAL CONTRIBUTORS,[1] WITH
THE HEADINGS OF THE ARTICLES IN THIS VOLUME SO SIGNED.




  A. Ba.
    ADOLFO BARTOLI (1833-1894).

      Formerly Professor of Literature at the Istituto di studi
      superiori at Florence. Author of Storia della letteratura
      Italiana; &c.

    Italian Literature (_in part_).

  A. Bo.*
    AUGUSTE BOUDINHON, D.D., D.C.L.

      Professor of Canon Law at the Catholic University of Paris.
      Honorary Canon of Paris. Editor of the _Canoniste contemporain_.

    Index Librorum Prohibitorum;
    Infallibility.

  A. Cy.
    ARTHUR ERNEST COWLEY, M.A., LITT.D.

      Sub-Librarian of the Bodleian Library, Oxford. Fellow of Magdalen
      College.

    Ibn Gabirol;
    Inscriptions: _Semitic_.

  A. C. G.
    ALBERT CHARLES LEWIS GOTTHILF GÜNTHER, M.A., M.D., PH.D., F.R.S.

      Keeper of Zoological Department, British Museum, 1875-1895. Gold
      Medallist, Royal Society, 1878. Author of _Catalogues of Colubrine
      Snakes, Batrachia Salientia, and Fishes in the British Museum_;
      _Reptiles of British India_; _Fishes of Zanzibar_; _Reports on the
      "Challenger" Fishes_; &c.

    Ichthyology (_in part_).

  A. E. G.*
    REV. ALFRED ERNEST GARVIE, M.A., D.D.

      Principal of New College, Hampstead. Member of the Board of
      Theology and the Board of Philosophy, London University. Author of
      _Studies in the inner Life of Jesus_; &c.

    Immortality;
    Inspiration.

  A. E. H. L.
    AUGUSTUS EDWARD HOUGH LOVE, M.A., D.SC., F.R.S.

      Sedleian Professor of Natural Philosophy in the University of
      Oxford. Hon. Fellow of Queen's College, Oxford; formerly Fellow of
      St John's College, Cambridge. Secretary to the London Mathematical
      Society.

    Infinitesimal Calculus.

  A. F. C.
    ALEXANDER FRANCIS CHAMBERLAIN, A.M., PH.D.

      Assistant Professor of Anthropology, Clark University, Worcester,
      Massachusetts. Member of American Antiquarian Society; Hon. Member
      of American Folk-lore Society. Author of _The Child and Childhood
      in Folk Thought_.

    Indians, North American.

  A. G.
    MAJOR ARTHUR GEORGE FREDERICK GRIFFITHS (d. 1908).

      H.M. Inspector of Prisons, 1878-1896. Author of _The Chronicles of
      Newgate_; _Secrets of the Prison House_; &c.

    Identification.

  A. Ge.
    SIR ARCHIBALD GEIKIE, LL.D.

      See the biographical article, GEIKIE, SIR A.

    Hutton, James.

  A. Go.*
    REV. ALEXANDER GORDON, M.A.

      Lecturer on Church History in the University of Manchester.

    Illuminati.

  A. G. G.
    SIR ALFRED GEORGE GREENHILL, M.A., F.R.S.

      Formerly Professor of Mathematics in the Ordnance College,
      Woolwich. Author of _Differential and Integral Calculus with
      Applications_; _Hydrostatics_; _Notes on Dynamics_; &c.

    Hydromechanics.

  A. H.-S.
    SIR A. HOUTUM-SCHINDLER, C.I.E.

      General in the Persian Army. Author of _Eastern Persian Irak_.

    Isfahan (_in part_).

  A. M. C.
    AGNES MARY CLERKE.

      See the biographical article, CLERKE, A. M.

    Huygens, Christiaan.

  A. N.
    ALFRED NEWTON, F.R.S.

      See the biographical article, NEWTON, ALFRED.

    Ibis;
    Icterus.

  A. So.
    ALBRECHT SOCIN, PH.D. (1844-1899).

      Formerly Professor of Semitic Philology in the Universities of
      Leipzig and Tübingen. Author of _Arabische Grammatik_; &c.

    Irak-Arabi (_in part_).

  A. S. Wo.
    ARTHUR SMITH WOODWARD, LL.D., F.R.S.

      Keeper of Geology, Natural History Museum, South Kensington.
      Secretary of the Geological Society, London.

    Ichthyosaurus;
    Iguanodon.

  A. W. H.*
    ARTHUR WILLIAM HOLLAND.

      Formerly Scholar of St John's College, Oxford. Bacon Scholar of
      Gray's Inn, 1900.

    Imperial Cities;
    Instrument of Government.

  A. W. Po.
    ALFRED WILLIAM POLLARD, M.A.

      Assistant Keeper of Printed Books, British Museum. Fellow of
      King's College, London. Hon. Secretary Bibliographical Society.
      Editor of _Books about Books_ and _Bibliographica_. Joint-editor
      of The Library. Chief Editor of the "Globe" _Chaucer_.

    Incunabula.

  A. W. R.
    ALEXANDER WOOD RENTON, M.A., LL.B.

      Puisne judge of the Supreme Court of Ceylon. Editor of
      _Encyclopaedia of the Laws of England_.

    Inebriety, Law of;
    Insanity: _Law_.

  C. F. A.
    CHARLES FRANCIS ATKINSON.

      Formerly Scholar of Queen's College, Oxford. Captain, 1st City of
      London (Royal Fusiliers). Author of _The Wilderness and Cold
      Harbour_.

    Infantry;
    Italian Wars.

  C. G.
    COLONEL CHARLES GRANT.

      Formerly Inspector of Military Education in India.

    India: _Costume_.

  C. H. Ha.
    CARLTON HUNTLEY HAYES, A.M., PH.D.

      Assistant Professor of History at Columbia University, New York
      City. Member of the American Historical Association.

    Innocent V., VIII.

  C. Ll. M.
    CONWAY LLOYD MORGAN, LL.D., F.R.S.

      Professor of Psychology at the University of Bristol. Principal of
      University College, Bristol, 1887-1909. Author of _Animal Life and
      Intelligence_; _Habit and Instinct_.

    Instinct;
    Intelligence in Animals.

  C. R. B.
    CHARLES RAYMOND BEAZLEY, M.A., D.LITT., F.R.G.S., F.R.HIST.S.

      Professor of Modern History in the University of Birmingham.
      Formerly Fellow of Merton College, Oxford; and University Lecturer
      in the History of Geography. Lothian Prizeman, Oxford, 1889.
      Lowell Lecturer, Boston, 1908. Author of _Henry the Navigator_;
      _The Dawn of Modern Geography_; &c.

    Ibn Batuta (_in part_);
    Idrisi.

  C. S.*
    CARLO SALVIONI.

      Professor of Classical and Romance Languages, University of Milan.

    Italian Language (_in part_).

  C. T. L.
    CHARLTON THOMAS LEWIS, PH.D. (1834-1904).

      Formerly Lecturer on Life Insurance, Harvard and Columbia
      Universities, and on Principles of Insurance, Cornell University.
      Author of _History of Germany_; _Essays_; _Addresses_; &c.

    Insurance (_in part_).

  C. We.
    CECIL WEATHERLY.

      Formerly Scholar of Queen's College, Oxford. Barrister-at-Law,
      Inner Temple.

    Infant Schools.

  D. B. Ma.
    DUNCAN BLACK MACDONALD, M.A., D.D.

      Professor of Semitic Languages, Hartford Theological Seminary,
      U.S.A. Author of _Development of Muslim Theology, Jurisprudence
      and Constitutional Theory_; _Selection from Ibn Khaldum_;
      _Religious Attitude and Life in Islam_; &c.

    Imam.

  D. G. H.
    DAVID GEORGE HOGARTH, M.A.

      Keeper of the Ashmolean Museum, Oxford. Fellow of Magdalen
      College, Oxford. Fellow of the British Academy. Excavated at
      Paphos, 1888; Naucratis, 1899 and 1903; Ephesus, 1904-1905;
      Assiut, 1906-1907; Director, British School at Athens, 1897-1900;
      Director, Cretan Exploration Fund, 1899.

    Ionia (_in part_);
    Isauria.

  D. H.
    DAVID HANNAY.

      Formerly British Vice-Consul at Barcelona. Author of _Short
      History of Royal Navy, 1217-1688_; _Life of Emilio Castelar_; &c.

    Impressment.

  D. F. T.
    DONALD FRANCIS TOVEY.

      Author of _Essays in Musical Analysis_; comprising _The Classical
      Concerto_, _The Goldberg Variations_, and analyses of many other
      classical works.

    Instrumentation.

  D. S. M.
    DUGALD SUTHERLAND MACCOLL, M.A., LL.D.

      Keeper of the National Gallery of British Art (Tate Gallery).
      Lecturer on the History of Art, University College, London; Fellow
      of University College, London. Author of Nineteenth Century Art;
      &c.

    Impressionism.

  E. A. M.
    EDWARD ALFRED MINCHIN, M.A., F.Z.S.

      Professor of Protozoology in the University of London. Formerly
      Fellow of Merton College, Oxford; and Lecturer on Comparative
      Anatomy in the University of Oxford. Author of "Sponges and
      Sporozoa" in Lankester's _Treatise on Zoology_; &c.

    Hydromedusae;
    Hydrozoa.

  E. Br.
    ERNEST BARKER, M.A.

      Fellow and Lecturer in Modern History, St John's College, Oxford.
      Formerly Fellow and Tutor of Merton College. Craven Scholar, 1895.

    Imperial Chamber.

  E. Bra.
    EDWIN BRAMWELL, M.B., F.R.C.P., F.R.S. (Edin.).

      Assistant Physician, Royal Infirmary, Edinburgh.

    Hysteria (_in part_).

  E. C. B.
    RIGHT REV. EDWARD CUTHBERT BUTLER, O.S.B., D.LITT.

      Abbot of Downside Abbey, Bath. Author of "The Lausiac History of
      Palladius" in _Cambridge Texts and Studies_.

    Imitation of Christ.

  E. C. Q.
    EDMUND CROSBY QUIGGIN, M.A.

      Fellow, Lecturer in Modern History, and Monro Lecturer in Celtic,
      Gonville and Caius College, Cambridge.

    Ireland: _Early History_.

  E. F. S.
    EDWARD FAIRBROTHER STRANGE.

      Assistant Keeper, Victoria and Albert Museum, South Kensington.
      Member of Council, Japan Society. Author of numerous works on art
      subjects. Joint-editor of Bell's "Cathedral" Series.

    Illustration: _Technical Developments_.

  E. F. S. D.
    LADY DILKE.

      See the biographical article: DILKE, SIR C. W., BART.

    Ingres.

  E. G.
    EDMUND GOSSE, LL.D.

      See the biographical article, GOSSE, EDMUND.

    Huygens, Sir Constantijn;
    Ibsen;
    Idyl.

  E. Hü.
    EMIL HÜBNER.

      See the biographical article, HÜBNER, EMIL.

    Inscriptions: _Latin_ (_in part_).

  E. H. B.
    SIR EDWARD HERBERT BUNBURY, BART., M.A., F.R.G.S. (d. 1895).

      M.P. for Bury St Edmunds, 1847-1852. Author of a _History of
      Ancient Geography_; &c.

    Ionia (_in part_).

  E. H. M.
    ELLIS HOVELL MINNS, M.A.

      Lecturer and Assistant Librarian, and formerly Fellow, Pembroke
      College, Cambridge University Lecturer in Palaeography.

    Iazyges;
    Issedones.

  E. H. P.
    EDWARD HENRY PALMER, M.A.

      See the biographical article, PALMER, E. H.

    Ibn Khaldun (_in part_).

  E. K.
    EDMUND KNECHT, PH.D., M.SC.TECH.(Manchester), F.I.C.

      Professor of Technological Chemistry, Manchester University. Head
      of Chemical Department, Municipal School of Technology,
      Manchester. Examiner in Dyeing, City and Guilds of London
      Institute. Author of _A Manual of Dyeing_; &c. Editor of J_ournal
      of the Society of Dyers and Colourists_.

    Indigo.

  E. L. H.
    THE RIGHT REV. THE BISHOP OF LINCOLN (EDWARD LEE HICKS).

      Honorary Fellow of Corpus Christi College, Oxford. Formerly Canon
      Residentiary of Manchester. Fellow and Tutor of Corpus Christi
      College. Author of _Manual of Greek Historical Inscriptions_; &c.

    Inscriptions: Greek (_in part_).

  Ed. M.
    EDUARD MEYER, PH.D., D.LITT.(Oxon.), LL.D.

      Professor of Ancient History in the University of Berlin. Author
      of _Geschichte des Alterthums_; _Geschichte des alten Aegyptens_;
      _Die Israeliten und ihre Nachbarstämme_.

    Hystaspes;
    Iran.

  E. M. T.
    SIR EDWARD MAUNDE THOMPSON, G.C.B., I.S.O., D.C.L., LITT.D., LL.D.

      Director and Principal Librarian, British Museum, 1898-1909.
      Sandars Reader in Bibliography, Cambridge, 1895-1896. Hon. Fellow
      of University College, Oxford. Correspondent of the Institute of
      France and of the Royal Prussian Academy of Sciences. Author of
      _Handbook of Greek and Latin Palaeography_. Editor of _Chronicon
      Angliae_. Joint-editor of publications of the Palaeographical
      Society, the New Palaeographical Society, and of the Facsimile of
      the Laurentian Sophocles.

    Illuminated MSS.

  E. O.*
    EDMUND OWEN, M.B., F.R.C.S., LL.D., D.SC.

      Consulting Surgeon to St Mary's Hospital, London, and to the
      Children's Hospital, Great Ormond Street; late Examiner in Surgery
      at the Universities of Cambridge, Durham and London. Author of _A
      Manual of Anatomy for Senior Students_.

    Hydrocephalus.

  F. A. F.
    FRANK ALBERT FETTER, PH.D.

      Professor of Political Economy and Finance, Cornell University.
      Member of the State Board of Charities. Author of _The Principles
      of Economics_; &c.

    Interstate Commerce.

  F. C. C.
    FREDERICK CORNWALLIS CONYBEARE, M.A., D.TH.(Giessen).

      Fellow of the British Academy. Formerly Fellow of University
      College, Oxford. Author of _The Ancient Armenian Texts of
      Aristotle_; _Myth, Magic and Morals_; &c.

    Iconoclasts;
    Image Worship.

  F. G. M. B.
    FREDERICK GEORGE MEESON BECK, M.A.

      Fellow and Lecturer in Classics, Clare College, Cambridge.

    Hwicce.

  F. J. H.
    FRANCIS JOHN HAVERFIELD, M.A., LL.D., F.S.A.

      Camden Professor of Ancient History in the University of Oxford.
      Fellow of Brasenose College. Fellow of the British Academy.
      Formerly Censor, Student, Tutor and Librarian of Christ Church,
      Oxford. Ford's Lecturer, 1906-1907. Author of Monographs on Roman
      History, especially Roman Britain; &c.

    Icknield Street.

  F. Ll. G.
    FRANCIS LLEWELLYN GRIFFITH, M.A., PH.D., F.S.A.

      Reader in Egyptology, Oxford University. Editor of the
      Archaeological Survey and Archaeological Reports of the Egypt
      Exploration Fund. Fellow of Imperial German Archaeological
      Institute.

    Hyksos;
    Isis.

  F. P.*
    FREDERICK PETERSON, M.D., PH.D.

      Professor of Psychiatry, Columbia University. President of New
      York State Commission in Lunacy, 1902-1906. Author of _Mental
      Diseases_; &c.

    Insanity: _Hospital Treatment._

  F. S. P.
    FRANCIS SAMUEL PHILBRICK, A.M., PH.D.

      Formerly Fellow of Nebraska State University, and Scholar and
      Resident Fellow of Harvard University. Member of American
      Historical Association.

    Independence, Declaration of.

  F. Wa.
    FRANCIS WATT, M.A.

      Barrister-at-Law, Middle Temple. Author of _Law's Lumber Room_.

    Inn and Innkeeper.

  F. W. R.*
    FREDERICK WILLIAM RUDLER, I.S.O., F.G.S.

      Curator and Librarian of the Museum of Practical Geology, London,
      1879-1902. President of the Geologists' Association, 1887-1889.

    Hyacinth
    Iolite.

  F. Y. P.
    FREDERICK YORK POWELL, D.C.L., LL.D.

      See the biographical article, POWELL, FREDERICK YORK.

    Iceland: _History_, and _Ancient Literature_.

  G. A. B.
    GEORGE A. BOULENGER, F.R.S., D.SC., PH.D.

      In charge of the collections of Reptiles and Fishes, Department of
      Zoology, British Museum. Vice-President of the Zoological Society
      of London.

    Ichthyology (_in part_).

  G. A. Gr.
    GEORGE ABRAHAM GRIERSON, C.I.E., PH.D., D.LITT.(Dublin).

      Member of the Indian Civil Service, 1873-1903. In charge of
      Linguistic Survey of India, 1898-1902. Gold Medallist, Royal
      Asiatic Society, 1909. Vice-President of the Royal Asiatic
      Society. Formerly Fellow of Calcutta University. Author of _The
      Languages of India_; &c.

    Indo-Aryan Languages.

  G. A. J. C.
    GRENVILLE ARTHUR JAMES COLE.

      Director of the Geological Survey of Ireland. Professor of
      Geology, Royal College of Science for Ireland, Dublin. Author of
      _Aids in Practical Geology_; &c.

    Ireland: _Geology_.

  G. B.
    SIR GEORGE CHRISTOPHER MOLESWORTH BIRDWOOD, K.C.I.E.

      See the biographical article, BIRDWOOD, SIR G. C. M.

    Incense.

  G. F. H.*
    GEORGE FRANCIS HILL, M.A.

      Assistant in Department of Coins and Medals, British Museum.
      Author of _Sources for Greek History 478-431_ B.C.; _Handbook of
      Greek and Roman Coins_; &c.

    Inscriptions: Greek (_in part_).

  G. G. Co.
    GEORGE GORDON COULTON, M.A.

      Birkbeck Lecturer in Ecclesiastical History, Trinity College,
      Cambridge. Author of _Medieval Studies_; _Chaucer and his
      England_; &c.

    Indulgence.

  G. H. C.
    GEORGE HERBERT CARPENTER, B.SC. (Lond.).

      Professor of Zoology in the Royal College of Science, Dublin.
      Author of _Insects: their Structure and Life_.

    Hymenoptera;
    Ichneumon-Fly;
    Insect.

  G. I. A.
    GRAZIADIO I. ASCOLI.

      Senator of the Kingdom of Italy. Professor of Comparative Grammar
      at the University of Milan. Author of _Codice Islandese_; &c.

    Italian Language (_in part_).

  G. J.
    GEORGE JAMIESON, C.M.G., M.A.

      Formerly Consul-General at Shanghai, and Consul and Judge of the
      Supreme Court, Shanghai.

    Hwang Ho.

  G. K.
    GUSTAV KRÜGER, PH.D.

      Professor of Church History in the University of Giessen. Author
      of _Das Papstthum_; &c.

    Irenaeus.

  G. P. M.
    GEORGE PERCIVAL MUDGE, A.R.C.S., F.Z.S.

      Lecturer on Biology, London Hospital Medical College, and London
      School of Medicine for Women, University of London. Author of _A
      Text Book of Zoology_; &c.

    Incubation and Incubators.

  G. W. K.
    VERY REV. GEORGE WILLIAM KITCHIN, M.A., D.D., F.S.A.

      Dean of Durham, and Warden of the University of Durham. Hon.
      Student of Christ Church, Oxford. Fellow of King's College,
      London. Dean of Winchester, 1883-1894. Author of _A History of
      France_; &c.

    Hutten, Ulrich von.

  G. W. T.
    REV. GRIFFITHES WHEELER THATCHER, M.A., B.D.

      Warden of Camden College, Sydney, N.S.W. Formerly Tutor in Hebrew
      and Old Testament History at Mansfield College, Oxford. Author of
      a _Commentary on Judges_; _An Arabic Grammar_; &c.

    Ibn 'Abd Rabbihi;
    Ibn 'Arabi;
    Ibn Athir;
    Ibn Duraid;
    Ibn Faradi;
    Ibn Farid;
    Ibn Hazm;
    Ibn Hisham;
    Ibn Ishaq;
    Ibn Jubair;
    Ibn Khaldun (_in part_);
    Ibn Khallikan;
    Ibn Qutaiba;
    Ibn Sa'd;
    Ibn Tufail;
    Ibn Usaibi'a;
    Ibrahim Al-Mausili.

  H. Ch.
    HUGH CHISHOLM, M.A.

      Formerly Scholar of Corpus Christi College, Oxford. Editor the
      11th edition of the _Encyclopaedia Britannica_; Co-editor of the
      10th edition.

    Iron Mask;
    Ismail.

  H. C. R.
    SIR HENRY CRESWICKE RAWLINSON, BART., K.C.B.

      See the biographical article, RAWLINSON, SIR HENRY CRESWICKE.

    Isfahan: _History_.

  H. L. H.
    HARRIET L. HENNESSY, M.D., (Brux.) L.R.C.P.I., L.R.C.S.I.

    Infancy;
    Intestinal Obstruction.

  H. M. H.
    HENRY MARION HOWE, A.M., LL.D.

      Professor of Metallurgy, Columbia University. Author of
      _Metallurgy of Steel_; &c.

    Iron and Steel.

  H. N. D.
    HENRY NEWTON DICKSON, M.A., D.SC., F.R.G.S.

      Professor of Geography, University College, Reading. Author of
      _Elementary Meteorology_; _Papers on Oceanography_; &c.

    Indian Ocean.

  H. O.
    HERMANN OELSNER, M.A., PH.D.

      Taylorian Professor of the Romance Languages in University of
      Oxford. Member of Council of the Philological Society. Author of
      _A History of Provencal Literature_; &c.

    Italian Literature (_in part_).

  H. St.
    HENRY STURT, M.A.

      Author of _Idola Theatri_; _The Idea of a Free Church_; and
      _Personal Idealism_.

    Induction.

  H. T. A.
    REV. HERBERT THOMAS ANDREWS.

      Professor of New Testament Exegesis, New College, London. Author
      of the "Commentary on Acts" in the _Westminster New Testament_;
      _Handbook on the Apocryphal Books_ in the "Century Bible."

    Ignatius.

  H. Y.
    SIR HENRY YULE, K.C.S.I., C.B.

      See the biographical article, YULE, SIR HENRY.

    Ibn Batuta (_in part_).

  I. A.
    ISRAEL ABRAHAMS, M.A.

      Reader in Talmudic and Rabbinic Literature in the University of
      Cambridge. Formerly President, Jewish Historical Society in
      England. Author of _A Short History of Jewish Literature_; _Jewish
      Life in the Middle Ages_; &c.

    Ibn Tibbon;
    Immanuel Ben Solomon.

  J. A. F.
    JOHN AMBROSE FLEMING, M.A., F.R.S., D.SC.

      Pender Professor of Electrical Engineering in the University of
      London. Fellow of University College, London. Formerly Fellow of
      St John's College, Cambridge, and Lecturer on Applied Mechanics in
      the University. Author of _Magnets and Electric Currents_.

    Induction Coil.

  J. Bs.
    JAMES BURGESS, C.I.E., LL.D., F.R.S.(Edin.), F.R.G.S.,
    HON.A.R.I.B.A.

      Formerly Director General of Archaeological Survey of India.
      Author of _Archaeological Survey of Western India_. Editor of
      Fergusson's _History of Indian Architecture_.

    Indian Architecture.

  J. B. T.
    SIR JOHN BATTY TUKE, KT., M.D., F.R.S.(Edin.), D.SC., LL.D.

      President of the Neurological Society of the United Kingdom.
      Medical Director of New Saughton Hall Asylum, Edinburgh. M.P. for
      the Universities of Edinburgh and St Andrews, 1900-1910.

    Hysteria (_in part_);
    Insanity: _Medical._

  J. C. H.
    RIGHT REV. JOHN CUTHBERT HEDLEY, O.S.B., D.D.

      R.C. Bishop of Newport. Author of _The Holy Eucharist_; &c.

    Immaculate Conception.

  J. C. Van D.
    JOHN CHARLES VAN DYKE.

      Professor of the History of Art, Rutgers College, New Brunswick,
      N.J. Formerly Editor of _The Studio and Art Review_. Author of
      _Art for Art's Sake_; _History of Painting_; _Old English
      Masters_; &c.

    Inness, George.

  J. C. W.
    JAMES CLAUDE WEBSTER.

      Barrister-at-Law, Middle Temple.

    Inns of Court.

  J. D. B.
    JAMES DAVID BOURCHIER, M.A., F.R.G.S.

      King's College, Cambridge. Correspondent of _The Times_ in
      South-Eastern Europe. Commander of the Orders of Prince Danilo of
      Montenegro and of the Saviour of Greece, and Officer of the Order
      of St Alexander of Bulgaria.

    Ionian Islands.

  J. F. F.
    JOHN FAITHFULL FLEET, C.I.E., PH.D.

      Commissioner of Central and Southern Divisions of Bombay,
      1891-1897. Author of _Inscriptions of the Early Gupta Kings_; &c.

    Inscriptions: _Indian_.

  J. F.-K.
    JAMES FITZMAURICE-KELLY, LITT.D., F.R.HIST.S.

      Gilmour Professor of Spanish Language and Literature, Liverpool
      University. Norman McColl Lecturer, Cambridge University. Fellow
      of the British Academy. Member of the Royal Spanish Academy.
      Knight Commander of the Order of Alphonso XII. Author of A History
      of Spanish Literature; &c.

    Isla, J. F. de.

  J. G. K.
    JOHN GRAHAM KERR, M.A., F.R.S.

      Regius Professor of Zoology in the University of Glasgow. Formerly
      Demonstrator in Animal Morphology in the University of Cambridge.
      Fellow of Christ's College, Cambridge, 1898-1904. Walsingham
      Medallist, 1898. Neill Prizeman, Royal Society of Edinburgh, 1904.

    Ichthyology (_in part_).

  J. G. Sc.
    SIR JAMES GEORGE SCOTT, K.C.I.E.

      Superintendent and Political Officer, Southern Shan States. Author
      of _Burma, a Handbook_; _The Upper Burma Gazetteer_; &c.

    Irrawaddy.

  J. H. A. H.
    JOHN HENRY ARTHUR HART, M.A.

      Fellow, Theological Lecturer and Librarian, St John's College,
      Cambridge.

    Hyrcanus.

  J. H. Mu.
    JOHN HENRY MUIRHEAD, M.A., LL.D.

      Professor of Philosophy in the University of Birmingham. Author of
      _Elements of Ethics_; _Philosophy and Life_; &c. Editor of
      _Library of Philosophy_.

    Idealism.

  J. H. Be.
    VERY REV. JOHN HENRY BERNARD, M.A., D.D., D.C.L.

      Dean of St Patrick's Cathedral, Dublin. Archbishop King's
      Professor of Divinity and formerly Fellow of Trinity College,
      Dublin. Joint-editor of the Irish _Liber Hymnorum_; &c.

    Ireland, Church of.

  J. H. van't H.
    JACOBUS HENRICUS VAN'T HOFF, LL.D., D.SC., D.M.

      See the biographical article VAN'T HOFF, JACOBUS HENRICUS.

    Isomerism.

  J. L. M.
    JOHN LYNTON MYRES, M.A., F.S.A., F.R.G.S.

      Wykeham Professor of Ancient History in the University of Oxford.
      Formerly Gladstone Professor of Greek and Lecturer in Ancient
      Geography, University of Liverpool. Lecturer in Classical
      Archaeology in University of Oxford.

    Iberians;
    Ionians.

  J. Mn.
    JOHN MACPHERSON, M.D.

      Formerly Inspector-General of Hospitals, Bengal.

    Insanity: _Medical_ (_in part_).

  J. M. A. de L.
    JEAN MARIE ANTOINE DE LANESSAN.

      See the biographical article, LANESSAN, J. M. A. DE.

    Indo-China, French (_in part_).

  J. M. M.
    JOHN MALCOLM MITCHELL.

      Sometime Scholar of Queen's College, Oxford. Lecturer in Classics,
      East London College (University of London). Joint-editor of
      Grote's _History of Greece_.

    Hyacinthus.

  J. P. E.
    JEAN PAUL HIPPOLYTE EMMANUEL ADHÉMAR ESMEIN.

      Professor of Law in the University of Paris. Officer of the Legion
      of Honour. Member of the Institute of France. Author of _Cours
      élémentaire d'histoire du droit français_; &c.

    Intendant.

  J. P. Pe.
    REV. JOHN PUNNETT PETERS, PH.D., D.D.

      Canon Residentiary, Cathedral of New York. Formerly Professor of
      Hebrew in the University of Pennsylvania. Director of the
      University Expedition to Babylonia, 1888-1895. Author of _Nippur,
      or Explorations and Adventures on the Euphrates_.

    Irak-Arabi (_in part_).

  J. S. Bl.
    JOHN SUTHERLAND BLACK, M.A., LL.D.

      Assistant Editor of the 9th edition of the _Encyclopaedia
      Britannica_. Joint-editor of the _Encyclopaedia Biblica_.

    Huss, John.

  J. S. Co.
    JAMES SUTHERLAND COTTON, M.A.

      Editor of the _Imperial Gazetteer of India_. Hon. Secretary of the
      Egyptian Exploration Fund. Formerly Fellow and Lecturer of Queen's
      College, Oxford. Author of _India_; &c.

    India: _Geography and Statistics (in part); History (in part)_;
    Indore.

  J. S. F.
    JOHN SMITH FLETT, D.SC., F.G.S.

      Petrographer to the Geological Survey. Formerly Lecturer on
      Petrology in Edinburgh University. Neill Medallist of the Royal
      Society of Edinburgh. Bigsby Medallist of the Geological Society
      of London.

    Itacolumite.

  J. T. Be.
    John Thomas Bealby.

      Joint-author of Stanford's _Europe_. Formerly Editor of the
      _Scottish Geographical Magazine_. Translator of Sven Hedin's
      _Through Asia, Central Asia and Tibet_; &c.

    Irkutsk (_in part_).

  J. V.*
    JULES VIARD.

      Archivist at the National Archives, Paris. Officer of Public
      Instruction. Author of _La France sous Philippe VI. de Valois_;
      &c.

    Isabella of Bavaria.

  Jno. W.
    JOHN WESTLAKE, K.C., LL.D.

      Professor of International Law, Cambridge, 1888-1908. One of the
      Members for the United Kingdom of International Court of
      Arbitration under the Hague Convention, 1900-1906. Bencher of
      Lincoln's Inn. Author of _A Treatise on Private International Law,
      or the Conflict of Laws: Chapters on the Principles of
      International Law_, pt. i. "Peace," pt. ii. "War."

    International Law: _Private_.

  L.
    COUNT LÜTZOW, LITT.D. (OXON.), PH.D. (PRAGUE), F.R.G.S.

      Chamberlain of H.M. the Emperor of Austria, King of Bohemia. Hon.
      Member of the Royal Society of Literature. Member of the Bohemian
      Academy; &c. Author of _Bohemia, a Historical Sketch_; _The
      Historians of Bohemia_ (Ilchester Lecture, Oxford, 1904); _The
      Life and Times of John Hus_; &c.

    Hussites.

  L. C. B.
    LEWIS CAMPBELL BRUCE, M.D., F.R.C.P.

      Author of _Studies in Clinical Psychiatry_.

    Insanity: _Medical_ (_in part_).

  L. Ho.
    LAURENCE HOUSMAN.

      See the biographical article, HOUSMAN, L.

    Illustration (_in part_).

  L. J. S.
    LEONARD JAMES SPENCER, M.A.

      Assistant in Department of Mineralogy, British Museum. Formerly
      Scholar of Sidney Sussex College, Cambridge, and Harkness Scholar.
      Editor of the _Mineralogical Magazine_.

    Hypersthene;
    Ilmenite.

  L. T. D.
    SIR LEWIS TONNA DIBDIN, M.A., D.C.L., F.S.A.

      Dean of the Arches; Master of the Faculties; and First Church
      Estates Commissioner. Bencher of Lincoln's Inn. Author of
      _Monasticism in England_; &c.

    Incense: _Ritual Use._

  M. Ha.
    MARCUS HARTOG, M.A., D.SC., F.L.S.

      Professor of Zoology, University College, Cork. Author of
      "Protozoa" in _Cambridge Natural History_; and papers for various
      scientific journals.

    Infusoria.

  M. Ja.
    MORRIS JASTROW, JUN., PH.D.

      Professor of Semitic Languages, University of Pennsylvania, U.S.A.
      Author of _Religion of the Babylonians and Assyrians_; &c.

    Ishtar.

  M. O. B. C.
    MAXIMILIAN OTTO BISMARCK CASPARI, M.A.

      Reader in Ancient History at London University. Lecturer in Greek
      at Birmingham University, 1905-1908.

    Irene (752-803).

  N. M.
    NORMAN MCLEAN, M.A.

      Fellow, Lecturer and Librarian of Christ's College, Cambridge.
      University Lecturer in Aramaic. Examiner for the Oriental
      Languages Tripos and the Theological Tripos at Cambridge.

    Isaac of Antioch.

  O. J. R. H.
    OSBERT JOHN RADCLIFFE HOWARTH, M.A.

      Christ Church, Oxford. Geographical Scholar, 1901. Assistant
      Secretary of the British Association.

    Ireland: _Geography_.

  P. A.
    PAUL DANIEL ALPHANDÉRY.

      Professor of the History of Dogma, École pratique des hautes
      études, Sorbonne, Paris. Author of _Les Idées morales chez les
      hétérodoxes latines au début du XIII^e. siècle_.

    Inquisition.

  P. A. K.
    PRINCE PETER ALEXEIVITCH KROPOTKIN.

      See the biographical article, KROPOTKIN, PRINCE P. A.

    Irkutsk (_in part_).

  P. C. M.
    PETER CHALMERS MITCHELL, M.A., F.R.S., F.Z.S., D.SC., LL.D.

      Secretary to the Zoological Society of London. University
      Demonstrator in Comparative Anatomy and Assistant to Linacre
      Professor at Oxford, 1888-1891. Examiner in Zoology to the
      University of London, 1903. Author of _Outlines of Biology_; &c.

    Hybridism.

  P. Gi.
    PETER GILES, M.A., LL.D., LITT.D.

      Fellow and Classical Lecturer of Emmanuel College, Cambridge, and
      University Reader in Comparative Philology. Formerly Secretary of
      the Cambridge Philological Society. Author of _Manual of
      Comparative Philology_; &c.

    I;
    Indo-European Languages.

  P. Sm.
    PRESERVED SMITH, PH.D.

      Rufus B. Kellogg Fellow, Amherst College, Amherst, Mass.

    Innocent I., II.

  R.
    THE RIGHT HON. LORD RAYLEIGH.

      See the biographical article, RAYLEIGH, 3RD BARON.

    Interference of Light.

  R. A. S. M.
    ROBERT ALEXANDER STEWART MACALISTER, M.A., F.S.A.

      St John's College, Cambridge. Director of Excavations for the
      Palestine Exploration Fund.

    Idumaea.

  R. Ba.
    RICHARD BAGWELL, M.A., LL.D.

      Commissioner of National Education for Ireland. Author of _Ireland
      under the Tudors_; _Ireland under the Stuarts_.

    Ireland: _Modern History_.

  R. C. J.
    SIR RICHARD CLAVERHOUSE JEBB, D.C.L., LL.D.

      See the biographical article, JEBB, SIR RICHARD CLAVERHOUSE.

    Isaeus;
    Isocrates.

  R. G.
    RICHARD GARNETT. LL.D.

      See the biographical article, GARNETT, RICHARD.

    Irving, Washington.

  R. H. C.
    REV. ROBERT HENRY CHARLES, M.A., D.D., D.LITT.

      Grinfield Lecturer, and Lecturer in Biblical Studies, Oxford.
      Fellow of the British Academy. Formerly Professor of Biblical
      Greek, Trinity College, Dublin. Author of _Critical History of the
      Doctrine of a Future Life_; _Book of Jubilees_; &c.

    Isaiah, Ascension of.

  R. L.*
    RICHARD LYDEKKER, F.R.S., F.Z.S., F.G.S.

      Member of the Staff of the Geological Survey of India 1874-1882.
      Author of _Catalogues of Fossil Mammals, Reptiles and Birds in the
      British Museum_; _The Deer of all Lands_; &c.

    Hyracoidea;
    Ibex (_in part_);
    Indri;
    Insectivora.

  R. P. S.
    R. PHENÉ SPIERS, F.S.A., F.R.I.B.A.

      Formerly Master of the Architectural School, Royal Academy,
      London. Past President of Architectural Association. Associate and
      Fellow of King's College, London. Corresponding Member of the
      Institute of France. Editor of Fergusson's _History of
      Architecture_. Author of _Architecture; East and West_; &c.

    Hypaethros.

  R. S. C.
    ROBERT SEYMOUR CONWAY, M.A., D.LITT.(CANTAB.).

      Professor of Latin and Indo-European Philology in the University
      of Manchester. Formerly Professor of Latin in University College,
      Cardiff; and Fellow of Gonville and Caius College, Cambridge.
      Author of _The Italic Dialects_.

    Iguvium;
    Iovilae.

  S.
    THE RIGHT HON. THE EARL OF SELBORNE.

      See the biographical article, SELBORNE, 1ST EARL OF.

    Hymns.

  R. Tr.
    ROLAND TRUSLOVE, M.A.

      Formerly Scholar of Christ Church, Oxford. Dean, Fellow and
      Lecturer in Classics at Worcester College, Oxford.

    Indo-China, French (_in part_).

  S. A. C.
    STANLEY ARTHUR COOK, M.A.

      Lecturer in Hebrew and Syriac, and formerly Fellow, Gonville and
      Caius College, Cambridge. Editor for Palestine Exploration Fund.
      Author of _Glossary of Aramaic Inscriptions_; _The Laws of Moses
      and the Code of Hammurabi_; _Critical Notes on Old Testament
      History_; _Religion of Ancient Palestine_; &c.

    Ishmael.

  S. Bl.
    SIGFUS BLÖNDAL.

      Librarian of the University of Copenhagen.

    Iceland: _Recent Literature_.

  T. As.
    THOMAS ASHBY, M.A., D.LITT. (Oxon.).

      Director of British School of Archaeology at Rome. Formerly
      Scholar of Christ Church, Oxford. Craven Fellow, 1897. Conington
      Prizeman, 1906. Member of the Imperial German Archaeological
      Institute.

    Interamna Lirenas;
    Ischia.

  T. A. I.
    THOMAS ALLAN INGRAM, M.A., LL.D.

      Trinity College, Dublin.

    Illegitimacy;
    Insurance (_in part_).

  T. Ba.
    SIR THOMAS BARCLAY, M.P.

      Member of the Institute of International Law. Member of the
      Supreme Council of the Congo Free State. Officer of the Legion of
      Honour. Author of _Problems of International Practice and
      Diplomacy_; &c. M.P. for Blackburn, 1910.

    Immunity;
    International Law.

  T. F.
    REV. THOMAS FOWLER, M.A., D.D., LL.D. (1832-1904).

      President of Corpus Christi College, Oxford, 1881-1904. Honorary
      Fellow of Lincoln College. Professor of Logic, 1873-1888.
      Vice-Chancellor of the University of Oxford, 1899-1901. Author of
      _Elements of Deductive Logic_; _Elements of Inductive Logic_;
      _Locke_ ("English Men of Letters"); _Shaftesbury and Hutcheson_
      ("English Philosophers"); &c.

    Hutcheson, Francis (_in part_).

  T. F. C.
    THEODORE FREYLINGHUYSEN COLLIER, PH.D.

      Assistant Professor of History, Williams College, Williamstown,
      Mass., U.S.A.

    Innocent IX.-XIII.

  T. H. H.*
    COLONEL SIR THOMAS HUNGERFORD HOLDICH, K.C.M.G., K.C.I.E.,
    HON.D.SC.

      Superintendent, Frontier Surveys, India, 1892-1898. Gold
      Medallist, R.G.S., London, 1887. Author of _The Indian
      Borderland_; _The Countries of the King's Award_; _India_;
      _Tibet_; &c.

    Indus.

  T. K. C.
    REV. THOMAS KELLY CHEYNE, D.D.

      See the biographical article, CHEYNE, T. K.

    Isaiah.

  Th. T.
    THORVALDUR THORODDSEN.

      Icelandic Expert and Explorer. Honorary Professor in the
      University of Copenhagen. Author of _History of Icelandic
      Geography_; _Geological Map of Iceland_; &c.

    Iceland: _Geography and Statistics_.

  W. A. B. C.
    REV. WILLIAM AUGUSTUS BREVOORT COOLIDGE, M.A., F.R.G.S.,
    PH.D.(Bern).

      Fellow of Magdalen College, Oxford. Professor of English History,
      St David's College, Lampeter, 1880-1881. Author of _Guide du Haut
      Dauphiné_; _The Range of the Tödi_; _Guide to Grindelwald_; _Guide
      to Switzerland_; _The Alps in Nature and in History_; &c. Editor
      of _The Alpine Journal_, 1880-1881; &c.

    Hyères;
    Innsbruck;
    Interlaken;
    Iseo, Lake of;
    Isère (_River_);
    Isère (_Department_).

  W. A. P.
    WALTER ALISON PHILLIPS, M.A.

      Formerly Exhibitioner of Merton College and Senior Scholar of St
      John's College, Oxford. Author of _Modern Europe_; &c.

    Innocent III., IV.

  W. C. U.
    WILLIAM CAWTHORNE UNWIN, LL.D., F.R.S., M.INST.C.E., M.INST.M.E.,
    A.R.I.B.A.

      Emeritus Professor, Central Technical College, City and Guilds of
      London Institute. Author of _Wrought Iron Bridges and Roofs_;
      _Treatise on Hydraulics_; &c.

    Hydraulics.

  W. F. C.
    WILLIAM FEILDEN CRAIES, M.A.

      Barrister-at-Law, Inner Temple. Lecturer on Criminal Law, King's
      College, London. Editor of Archbold's _Criminal Pleading_ (23rd
      edition).

    Indictment.

  W. F. Sh.
    WILLIAM FLEETWOOD SHEPPARD, M.A.

      Senior Examiner in the Board of Education, London. Formerly Fellow
      of Trinity College, Cambridge. Senior Wrangler, 1884.

    Interpolation.

  W. G.
    WILLIAM GARNETT, M.A., D.C.L.

      Educational Adviser to the London County Council. Formerly Fellow
      and Lecturer of St John's College, Cambridge. Principal and
      Professor of Mathematics, Durham College of Science,
      Newcastle-on-Tyne. Author of _Elementary Dynamics_; &c.

    Hydrometer.

  W. Go.
    WILLIAM GOW, M.A., PH.D.

      Secretary of the British and Foreign Marine Insurance Co. Ltd.,
      Liverpool. Lecturer on Marine Insurance at University College,
      Liverpool. Author of _Marine Insurance_; &c.

    Insurance: _Marine_.

  W. H. F.
    SIR WILLIAM HENRY FLOWER, F.R.S.

      See the biographical article, FLOWER, SIR W. H.

    Ibex (_in part_).

  W. H. Po.
    W. HALDANE PORTER.

      Barrister-at-Law, Middle Temple.

    Ireland: _Statistics and Administration_.

  W. Ma.
    SIR WILLIAM MARKBY, K.C.I.E.

      See the biographical article, MARKBY, SIR WILLIAM.

    Indian Law.

  W. McD.
    WILLIAM MCDOUGALL, M.A.

      Wilde Reader in Mental Philosophy in the University of Oxford.
      Formerly Fellow of St John's College, Cambridge.

    Hypnotism.

  W. M. L.
    WALLACE MARTIN LINDSAY, M.A., LITT.D., LL.D.

      Professor of Humanity, University of St Andrews. Fellow of the
      British Academy. Formerly Fellow of Jesus College, Oxford. Author
      of _Handbook of Latin Inscriptions_; _The Latin Language_; &c.

    Inscriptions: _Latin_ (_in part_).

  W. M. Ra.
    SIR WILLIAM MITCHELL RAMSAY, LITT.D., D.C.L.

      See the biographical article, RAMSAY, SIR W. MITCHELL.

    Iconium.

  W. R. So.
    WILLIAM RITCHIE SORLEY, M.A., LITT.D., LL.D.

      Professor of Moral Philosophy in the University of Cambridge.
      Fellow of King's College, Cambridge. Fellow of the British
      Academy. Formerly Fellow of Trinity College. Author of _The Ethics
      of Naturalism_; _The Interpretation of Evolution_; &c.

    Iamblichus.

  W. T. T.-D.
    SIR WILLIAM TURNER THISELTON-DYER, F.R.S., K.C.M.G., C.I.E.,
    D.SC., LL.D., PH.D., F.L.S.

      Hon. Student of Christ Church, Oxford. Director, Royal Botanic
      Gardens, Kew, 1885-1905. Botanical Adviser to Secretary of State
      for Colonies, 1902-1906. Joint-author of _Flora of Middlesex_.
      Editor of _Flora Capenses_ and _Flora of Tropical Africa_.

    Huxley.

  W. Wn.
    WILLIAM WATSON, D.SC., F.R.S., A.R.C.S.

      Assistant Professor of Physics, Royal College of Science, London.
      Vice-President of the Physical Society. Author of _A Text Book of
      Practical Physics_; &c.

    Inclinometer.

  W. W. H.
    SIR WILLIAM WILSON HUNTER.

      See the biographical article. HUNTER, SIR WILLIAM WILSON.

    India: _History (in part); Geography and Statistics (in part)._




PRINCIPAL UNSIGNED ARTICLES

  Husband and Wife.   Image.           Ink.
  Hyacinth.           Impeachment.     Inkerman.
  Hyderabad.          Income Tax.      International, The.
  Hydrogen.           Indiana.         Intestacy.
  Hydropathy.         Indian Mutiny.   Inverness-shire.
  Hydrophobia.        Indicator.       Investiture.
  Ice.                Infant.          Iodine.
  Ice-Yachting.       Infanticide.     Iowa.
  Idaho.              Infinite.        Ipecacuanha.
  Illinois.           Influenza.       Iris.
  Illumination.       Inheritance.     Iron.
  Illyria.            Injunction.      Irrigation.


FOOTNOTE:

  [1] A complete list, showing all individual contributors, appears in
    the final volume.




  ENCYCLOPÆDIA BRITANNICA

  ELEVENTH EDITION

  VOLUME XIV




HUSBAND, properly the "head of a household," but now chiefly used in the
sense of a man legally joined by marriage to a woman, his "wife"; the
legal relations between them are treated below under HUSBAND AND WIFE.
The word appears in O. Eng. as _húsbonda_, answering to the Old
Norwegian _húsbóndi_, and means the owner or freeholder of a _hus_, or
house. The last part of the word still survives in "bondage" and
"bondman," and is derived from _bua_, to dwell, which, like Lat.
_colere_, means also to till or cultivate, and to have a household.
"Wife," in O. Eng. _wif_, appears in all Teutonic languages except
Gothic; cf. Ger. _Weib_, Dutch _wijf_, &c., and meant originally simply
a female, "woman" itself being derived from _wifman_, the pronunciation
of the plural _wimmen_ still preserving the original _i_. Many
derivations of "wife" have been given; thus it has been connected with
the root of "weave," with the Gothic _waibjan_, to fold or wrap up,
referring to the entangling clothes worn by a woman, and also with the
root of _vibrare_, to tremble. These are all merely guesses, and the
ultimate history of the word is lost. It does not appear outside
Teutonic languages. Parallel to "husband" is "housewife," the woman
managing a household. The earlier _húswif_ was pronounced _hussif_, and
this pronunciation survives in the application of the word to a small
case containing scissors, needles and pins, cottons, &c. From this form
also derives "hussy," now only used in a depreciatory sense of a light,
impertinent girl. Beyond the meaning of a husband as a married man, the
word appears in connexion with agriculture, in "husbandry" and
"husbandman." According to some authorities "husbandman" meant
originally in the north of England a holder of a "husbandland," a
manorial tenant who held two ox-gangs or virgates, and ranked next below
the yeoman (see J. C. Atkinson in _Notes and Queries_, 6th series, vol.
xii., and E. Bateson, _History of Northumberland_, ii., 1893). From the
idea of the manager of a household, "husband" was in use transferred to
the manager of an estate, and the title was held by certain officials,
especially in the great trading companies. Thus the "husband" of the
East India Company looked after the interests of the company at the
custom-house. The word in this sense is practically obsolete, but it
still appears in "ship's husband," an agent of the owners of a ship who
looks to the proper equipping of the vessel, and her repairs, procures
and adjusts freights, keeps the accounts, makes charter-parties and acts
generally as manager of the ship's employment. Where such an agent is
himself one of the owners of the vessel, the name of "managing owner" is
used. The "ship's husband" or "managing owner" must register his name
and address at the port of registry (Merchant Shipping Act 1894, § 59).
From the use of "husband" for a good and thrifty manager of a household,
the verb "to husband" means to economize, to lay up a store, to save.




HUSBAND AND WIFE, LAW RELATING TO. For the modes in which the relation
of husband and wife may be constituted and dissolved, see MARRIAGE and
DIVORCE. The present article will deal only with the effect of marriage
on the legal position of the spouses. The person chiefly affected is the
wife, who probably in all political systems becomes subject, in
consequence of marriage, to some kind of disability. The most favourable
system scarcely leaves her as free as an unmarried woman; and the most
unfavourable subjects her absolutely to the authority of her husband. In
modern times the effect of marriage on property is perhaps the most
important of its consequences, and on this point the laws of different
states show wide diversity of principles.

The history of Roman law exhibits a transition from an extreme theory to
its opposite. The position of the wife in the earliest Roman household
was regulated by the law of _Manus_. She fell under the "hand" of her
husband,--became one of his family, along with his sons and daughters,
natural or adopted, and his slaves. The dominion which, so far as the
children was concerned, was known as the _patria potestas_, was, with
reference to the wife, called the _manus_. The subject members of the
family, whether wife or children, had, broadly speaking, no rights of
their own. If this institution implied the complete subjection of the
wife to the husband, it also implied a much closer bond of union between
them than we find in the later Roman law. The wife on her husband's
death succeeded, like the children, to freedom and a share of the
inheritance. _Manus_, however, was not essential to a legal marriage;
its restraints were irksome and unpopular, and in course of time it
ceased to exist, leaving no equivalent protection of the stability of
family life. The later Roman marriage left the spouses comparatively
independent of each other. The distance between the two modes of
marriage may be estimated by the fact that, while under the former
the wife was one of the husband's immediate heirs, under the latter she
was called to the inheritance only after his kith and kin had been
exhausted, and only in preference to the treasury. It seems doubtful how
far she had, during the continuance of marriage, a legal right to
enforce aliment from her husband, although if he neglected her she had
the unsatisfactory remedy of an easy divorce. The law, in fact,
preferred to leave the parties to arrange their mutual rights and
obligations by private contracts. Hence the importance of the law of
settlements (_Dotes_). The _Dos_ and the _Donatio ante nuptias_ were
settlements by or on behalf of the husband or wife, during the
continuance of the marriage, and the law seems to have looked with some
jealousy on gifts made by one to the other in any less formal way, as
possibly tainted with undue influence. During the marriage the husband
had the administration of the property.

The _manus_ of the Roman law appears to be only one instance of an
institution common to all primitive societies. On the continent of
Europe after many centuries, during which local usages were brought
under the influence of principles derived from the Roman law, a theory
of marriage became established, the leading feature of which is the
_community of goods_ between husband and wife. Describing the principle
as it prevails in France, Story (_Conflict of Laws_, § 130) says: "This
community or nuptial partnership (in the absence of any special
contract) generally extends to all the movable property of the husband
and wife, and to the fruits, income and revenue thereof.... It extends
also to all immovable property of the husband and wife acquired during
the marriage, but not to such immovable property as either possessed at
the time of the marriage, or which came to them afterwards by title of
succession or by gift. The property thus acquired by this nuptial
partnership is liable to the debts of the parties existing at the time
of the marriage; to the debts contracted by the husband during the
community, or by the wife during the community with the consent of the
husband; and to debts contracted for the maintenance of the family....
The husband alone is entitled to administer the property of the
community, and he may alien, sell or mortgage it without the concurrence
of the wife." But he cannot dispose by will of more than his share of
the common property, nor can he part with it gratuitously _inter vivos_.
The community is dissolved by death (natural or civil), divorce,
separation of body or separation of property. On separation of body or
of property the wife is entitled to the full control of her movable
property, but cannot alien her immovable property, without her husband's
consent or legal authority. On the death of either party the property is
divided in equal moieties between the survivor and the heirs of the
deceased.

_Law of England._--The English common law as usual followed its own
course in dealing with this subject, and in no department were its rules
more entirely insular and independent. The text writers all assumed two
fundamental principles, which between them established a system of
rights totally unlike that just described. Husband and wife were said to
be one person in the eye of the law--_unica persona, quia caro una et
sanguis unus_. Hence a man could not grant or give anything to his wife,
because she was himself, and if there were any compacts between them
before marriage they were dissolved by the union of persons. Hence, too,
the old rule of law, now greatly modified, that husband and wife could
not be allowed to give evidence against each other, in any trial, civil
or criminal. The unity, however, was one-sided only; it was the wife who
was merged in the husband, not the husband in the wife. And when the
theory did not apply, the disabilities of "coverture" suspended the
active exercise of the wife's legal faculties. The old technical
phraseology described husband and wife as _baron_ and _feme_; the rights
of the husband were baronial rights. From one point of view the wife was
merged in the husband, from another she was as one of his vassals. A
curious example is the immunity of the wife in certain cases from
punishment for crime committed in the presence and on the presumed
coercion of the husband. "So great a favourite," says Blackstone, "is
the female sex of the laws of England."

The application of these principles with reference to the property of
the wife, and her capacity to contract, may now be briefly traced.

The _freehold property_ of the wife became vested in the husband and
herself during the coverture, and he had the management and the profits.
If the wife had been in actual possession at any time during the
marriage of an estate of inheritance, and if there had been a child of
the marriage capable of inheriting, then the husband became entitled on
his wife's death to hold the estate for his own life as tenant by the
_curtesy of England_ (_curialitas_).[1] Beyond this, however, the
husband's rights did not extend, and the wife's heir at last succeeded
to the inheritance. The wife could not part with her real estate without
the concurrence of the husband; and even so she must be examined apart
from her husband, to ascertain whether she freely and voluntarily
consented to the deed.

With regard to personal property, it passed absolutely at common law to
the husband. Specific things in the possession of the wife (_choses_ in
possession) became the property of the husband at once; things not in
possession, but due and recoverable from others (_choses_ in action),
might be recovered by the husband. A _chose_ in action not reduced into
actual possession, when the marriage was dissolved by death, reverted to
the wife if she was the survivor; if the husband survived he could
obtain possession by taking out letters of administration. A _chose_ in
action was to be distinguished from a specific thing which, although the
property of the wife, was for the time being in the hands of another. In
the latter case the property was in the wife, and passed at once to the
husband; in the former the wife had a mere _jus in personam_, which the
husband might enforce if he chose, but which was still capable of
reverting to the wife if the husband died without enforcing it.

The _chattels real_ of the wife (i.e., personal property, dependent on,
and partaking of, the nature of realty, such as leaseholds) passed to
the husband, subject to the wife's right of survivorship, unless barred
by the husband by some act done during his life. A disposition by will
did not bar the wife's interest; but any disposition _inter vivos_ by
the husband was valid and effective.

The courts of equity, however, greatly modified the rules of the common
law by the introduction of the wife's _separate estate_, i.e. property
settled to the wife for her separate use, independently of her husband.
The principle seems to have been originally admitted in a case of actual
separation, when a fund was given for the maintenance of the wife while
living apart from her husband. And the conditions under which separate
estate might be enjoyed had taken the Court of Chancery many generations
to develop. No particular form of words was necessary to create a
separate estate, and the intervention of trustees, though common, was
not necessary. A clear intention to deprive the husband of his common
law rights was sufficient to do so. In such a case a married woman was
entitled to deal with her property as if she was unmarried, although the
earlier decisions were in favour of requiring her binding engagements to
be in writing or under seal. But it was afterwards held that any
engagements, clearly made with reference to the separate estate, would
bind that estate, exactly as if the woman had been a _feme sole_.
Connected with the doctrine of separate use was the equitable
contrivance of _restraint on anticipation_ with which later legislation
has not interfered, whereby property might be so settled to the separate
use of a married woman that she could not, during coverture, alienate it
or anticipate the income. No such restraint is recognized in the ease of
a man or of a _feme sole_, and it depends entirely on the separate
estate; and the separate estate has its existence only during coverture,
so that a woman to whom such an estate is given may dispose of it so
long as she is unmarried, but becomes bound by the restraint as soon as
she is married. In yet another way the court of Chancery interfered to
protect the interests of married women. When a husband sought the
aid of that court to get possession of his wife's _choses_ in action, he
was required to make a provision for her and her children out of the
fund sought to be recovered. This is called the wife's _equity to a
settlement_, and is said to be based on the original maxim of Chancery
jurisprudence, that "he who seeks equity must do equity." Two other
property interests of minor importance are recognised. The wife's
_pin-money_ is a provision for the purchase of clothes and ornaments
suitable to her husband's station, but it is not an absolute gift to the
separate use of the wife; and a wife surviving her husband cannot claim
for more than one year's arrears of pin-money. _Paraphernalia_ are
jewels and other ornaments given to the wife by her husband for the
purpose of being worn by her, but not as her separate property. The
husband may dispose of them by act _inter vivos_ but not by will, unless
the will confers other benefits on the wife, in which case she must
elect between the will and the paraphernalia. She may also on the death
of the husband claim paraphernalia, provided all creditors have been
satisfied, her right being superior to that of any legatee.

The corresponding interest of the wife in the property of the husband is
much more meagre and illusory. Besides a general right to maintenance at
her husband's expense, she has at common law a right to dower (q.v.) in
her husband's lands, and to a _pars rationabilis_ (third) of his
personal estate, if he dies intestate. The former, which originally was
a solid provision for widows, has by the ingenuity of conveyancers, as
well as by positive enactment, been reduced to very slender dimensions.
It may be destroyed by a mere declaration to that effect on the part of
the husband, as well as by his conveyance of the land or by his will.

The common practice of regulating the rights of husband, wife and
children by marriage settlements obviates the hardships of the common
law--at least for the women of the wealthier classes. The legislature by
the Married Women's Property Acts of 1870, 1874, 1882 (which repealed
and consolidated the acts of 1870 and 1874), 1893 and 1907 introduced
very considerable changes. The chief provisions of the Married Women's
Property Act 1882, which enormously improved the position of women
unprotected by marriage settlement, are, shortly, that a married woman
is capable of acquiring, holding and disposing of by will or otherwise,
any real and personal property, in the same manner as if she were a
_feme sole_, without the intervention of any trustee. The property of a
woman married after the beginning of the act, whether belonging to her
at the time of marriage or acquired after marriage, is held by her as a
_feme sole_. The same is the case with property acquired after the
beginning of the act by a woman married before the act. After marriage a
woman remains liable for antenuptial debts and liabilities, and as
between her and her husband, in the absence of contract to the contrary,
her separate property is deemed primarily liable. The husband is only
liable to the extent of property acquired from or through his wife. The
act also contained provisions as to stock, investment, insurance,
evidence and other matters. The effect of the act was to render obsolete
the law as to what created a separate use or a reduction into possession
of _choses_ in action, as to equity to a settlement, as to fraud on the
husband's marital rights, and as to the inability of one of two married
persons to give a gift to the other. Also, in the case of a gift to a
husband and wife in terms which would make them joint tenants if
unmarried, they no longer take as one person but as two. The act
contained a special saving of existing and future settlements; a
settlement being still necessary where it is desired to secure only the
enjoyment of the income to the wife and to provide for children. The act
by itself would enable the wife, without regard to family claims,
instantly to part with the whole of any property which might come to
her. Restraint on anticipation was preserved by the act, subject to the
liability of such property for antenuptial debts, and to the power given
by the Conveyancing Act 1881 to bind a married woman's interest
notwithstanding a clause of restraint. The Married Women's Property Act
of 1893 repealed two clauses in the act of 1882, the exact bearing of
which had been a matter of controversy. It provided specifically that
every contract thereinafter entered into by a married woman, otherwise
than as an agent, should be deemed to be a contract entered into by her
with respect to and be binding upon her separate property, whether she
was or was not in fact possessed of or entitled to any separate property
at the time when she entered into such contract, that it should bind all
separate property which she might at any time or thereafter be possessed
of or entitled to, and that it should be enforceable by process of law
against all property which she might thereafter, while discovert, be
possessed of or entitled to. The act of 1907 enabled a married woman,
without her husband, to dispose of or join in disposing of, real or
personal property held by her solely or jointly as trustee or personal
representative, in like manner as if she were a _feme sole_. It also
provided that a settlement or agreement for settlement whether before or
after marriage, respecting the property of the woman, should not be
valid unless executed by her if she was of full age or confirmed by her
after she attained full age. The Married Women's Property Act 1908
removed a curious anomaly by enacting that a married woman having
separate property should be equally liable with single women and widows
for the maintenance of parents who are in receipt of poor relief.

The British colonies generally have adopted the principles of the
English acts of 1882 and 1893.

  _Law of Scotland._--The law of Scotland differs less from English law
  than the use of a very different terminology would lead us to suppose.
  The phrase _communio bonorum_ has been employed to express the
  interest which the spouses have in the _movable_ property of both, but
  its use has been severely censured as essentially inaccurate and
  misleading. It has been contended that there was no real community of
  goods, and no partnership or societas between the spouses. The wife's
  movable property, with certain exceptions, and subject to special
  agreements, became as absolutely the property of the husband as it did
  in English law. The notion of a _communio_ was, however, favoured by
  the peculiar rights of the wife and children on the dissolution of the
  marriage. Previous to the Intestate Movable Succession (Scotland) Act
  1855 the law stood as follows. The fund formed by the movable property
  of both spouses may be dealt with by the husband as he pleases during
  life; it is increased by his acquisitions and diminished by his debts.
  The respective shares contributed by husband and wife return on the
  dissolution of the marriage to them or their representatives if the
  marriage be dissolved within a year and a day, and without a living
  child. Otherwise the division is into two or three shares, according
  as children are existing or not at the dissolution of the marriage. On
  the death of the husband, his children take one-third (called
  _legitim_), the widow takes one-third (_jus relictae_), and the
  remaining one-third (the _dead part_) goes according to his will or to
  his next of kin. If there be no children, the _jus relictae_ and the
  dead's part are each one-half. If the wife die before the husband, her
  representatives, whether children or not, are creditors for the value
  of her share. The statute above-mentioned, however, enacts that "where
  a wife shall predecease her husband, the next of kin, executors or
  other representatives of such wife, whether testate or intestate,
  shall have no right to any share of the goods in communion; nor shall
  any legacy or bequest or testamentary disposition thereof by such
  wife, affect or attach to the said goods or any portion thereof." It
  also abolishes the rule by which the shares revert if the marriage
  does not subsist for a year and a day. Several later acts apply to
  Scotland some of the principles of the English Married Women's
  Property Acts. These are the Married Women's Property (Scotland) Act
  1877, which protects the earnings, &c., of wives, and limits the
  husband's liability for antenuptial debts of the wife, the Married
  Women's Policies of Assurance (Scotland) Act 1880, which enables a
  woman to contract for a policy of assurance for her separate use, and
  the Married Women's Property (Scotland) Act 1881, which abolished the
  _jus mariti_.

  A wife's _heritable_ property does not pass to the husband on
  marriage, but he acquires a right to the administration and profits.
  His courtesy, as in English law, is also recognized. On the other
  hand, a widow has a _terce_ or life-rent of a third part of the
  husband's heritable estate, unless she has accepted a conventional
  provision.

  _Continental Europe._--Since 1882 English legislation in the matter of
  married women's property has progressed from perhaps the most backward
  to the foremost place in Europe. By a curious contrast, the only two
  European countries where, in the absence of a settlement to the
  contrary, independence of the wife's property was recognized, were
  Russia and Italy. But there is now a marked tendency towards
  contractual emancipation. Sweden adopted a law on this subject in
  1874, Denmark in 1880, Norway in 1888. Germany followed, the Civil
  Code which came into operation in 1900 (Art. 1367) providing that the
  wife's wages or earnings shall form part of her _Vorbehaltsgut_ or
  separate property, which a previous article (1365) placed beyond
  the husband's control. As regards property accruing to the wife in
  Germany by succession, will or gift _inter vivos_, it is only separate
  property where the donor has deliberately stipulated exclusion of the
  husband's right.

  In France it seemed as if the system of community of property was
  ingrained in the institutions of the country. But a law of 1907 has
  brought France into line with other countries. This law gives a
  married woman sole control over earnings from her personal work and
  savings therefrom. She can with such money acquire personalty or
  realty, over the former of which she has absolute control. But if she
  abuses her rights by squandering her money or administering her
  property badly or imprudently the husband may apply to the court to
  have her freedom restricted.

  _American Law._--In the United States, the revolt against the common
  law theory of husband and wife was carried farther than in England,
  and legislation early tended in the direction of absolute equality
  between the sexes. Each state has, however, taken its own way and
  selected its own time for introducing modifications of the existing
  law, so that the legislation on this subject is now exceedingly
  complicated and difficult. James Schouler (_Law of Domestic
  Relations_) gives an account of the general result in the different
  states to which reference may be made. The peculiar system of
  Homestead Laws in many of the states (see HOMESTEAD and EXEMPTION
  LAWS) constitutes an inalienable provision for the wife and family of
  the householder.


FOOTNOTE:

  [1] Curtesy or courtesy has been explained by legal writers as
    "arising _by favour_ of the law of England." The word has nothing to
    do with courtesy in the sense of complaisance.




HUSHI (Rumanian _Husi_), the capital of the department of Falciu,
Rumania; on a branch of the Jassy-Galatz railway, 9 m. W. of the river
Pruth and the Russian frontier. Pop. (1900) 15,404, about one-fourth
being Jews. Hushi is an episcopal see. The cathedral was built in 1491
by Stephen the Great of Moldavia. There are no important manufactures,
but a large fair is held annually in September for the sale of
live-stock, and wine is produced in considerable quantities. Hushi is
said to have been founded in the 15th century by a colony of Hussites,
from whom its name is derived. The treaty of the Pruth between Russia
and Turkey was signed here in 1711.




HUSKISSON, WILLIAM (1770-1830), English statesman and financier, was
descended from an old Staffordshire family of moderate fortune, and was
born at Birch Moreton, Worcestershire, on the 11th of March 1770. Having
been placed in his fourteenth year under the charge of his maternal
great-uncle Dr Gem, physician to the English embassy at Paris, in 1783
he passed his early years amidst a political fermentation which led him
to take a deep interest in politics. Though he approved of the French
Revolution, his sympathies were with the more moderate party, and he
became a member of the "club of 1789," instituted to support the new
form of constitutional monarchy in opposition to the anarchical attempts
of the Jacobins. He early displayed his mastery of the principles of
finance by a _Discours_ delivered in August 1790 before this society, in
regard to the issue of assignats by the government. The _Discours_
gained him considerable reputation, but as it failed in its purpose he
withdrew from the society. In January 1793 he was appointed by Dundas to
an office created to direct the execution of the Aliens Act; and in the
discharge of his delicate duties he manifested such ability that in 1795
he was appointed under-secretary at war. In the following year he
entered parliament as member for Morpeth, but for a considerable period
he took scarcely any part in the debates. In 1800 he inherited a fortune
from Dr Gem. On the retirement of Pitt in 1801 he resigned office, and
after contesting Dover unsuccessfully he withdrew for a time into
private life. Having in 1804 been chosen to represent Liskeard, he was
on the restoration of the Pitt ministry appointed secretary of the
treasury, holding office till the dissolution of the ministry after the
death of Pitt in January 1806. After being elected for Harwich in 1807,
he accepted the same office under the duke of Portland, but he withdrew
from the ministry along with Canning in 1809. In the following year he
published a pamphlet on the currency system, which confirmed his
reputation as the ablest financier of his time; but his free-trade
principles did not accord with those of his party. In 1812 he was
returned for Chichester. When in 1814 he re-entered the public service,
it was only as chief commissioner of woods and forests, but his
influence was from this time very great in the commercial and financial
legislation of the country. He took a prominent part in the corn-law
debates of 1814 and 1815; and in 1819 he presented a memorandum to Lord
Liverpool advocating a large reduction in the unfunded debt, and
explaining a method for the resumption of cash payments, which was
embodied in the act passed the same year. In 1821 he was a member of the
committee appointed to inquire into the causes of the agricultural
distress then prevailing, and the proposed relaxation of the corn laws
embodied in the report was understood to have been chiefly due to his
strenuous advocacy. In 1823 he was appointed president of the board of
trade and treasurer of the navy, and shortly afterwards he received a
seat in the cabinet. In the same year he was returned for Liverpool as
successor to Canning, and as the only man who could reconcile the Tory
merchants to a free trade policy. Among the more important legislative
changes with which he was principally connected were a reform of the
Navigation Acts, admitting other nations to a full equality and
reciprocity of shipping duties; the repeal of the labour laws; the
introduction of a new sinking fund; the reduction of the duties on
manufactures and on the importation of foreign goods, and the repeal of
the quarantine duties. In accordance with his suggestion Canning in 1827
introduced a measure on the corn laws proposing the adoption of a
sliding scale to regulate the amount of duty. A misapprehension between
Huskisson and the duke of Wellington led to the duke proposing an
amendment, the success of which caused the abandonment of the measure by
the government. After the death of Canning in the same year Huskisson
accepted the secretaryship of the colonies under Lord Goderich, an
office which he continued to hold in the new cabinet formed by the duke
of Wellington in the following year. After succeeding with great
difficulty in inducing the cabinet to agree to a compromise on the corn
laws, Huskisson finally resigned office in May 1829 on account of a
difference with his colleagues in regard to the disfranchisement of East
Retford. On the 15th of September of the following year he was
accidentally killed by a locomotive engine while present at the opening
of the Liverpool and Manchester railway.

  See the _Life of Huskisson_, by J. Wright (London, 1831).




HUSS (or HUS), JOHN (c. 1373-1415), Bohemian reformer and martyr, was
born at Hussinecz,[1] a market village at the foot of the Böhmerwald,
and not far from the Bavarian frontier, between 1373 and 1375, the exact
date being uncertain. His parents appear to have been well-to-do Czechs
of the peasant class. Of his early life nothing is recorded except that,
notwithstanding the early loss of his father, he obtained a good
elementary education, first at Hussinecz, and afterwards at the
neighbouring town of Prachaticz. At, or only a very little beyond, the
usual age he entered the recently (1348) founded university of Prague,
where he became bachelor of arts in 1393, bachelor of theology in 1394,
and master of arts in 1396. In 1398 he was chosen by the Bohemian
"nation" of the university to an examinership for the bachelor's degree;
in the same year he began to lecture also, and there is reason to
believe that the philosophical writings of Wycliffe, with which he had
been for some years acquainted, were his text-books. In October 1401 he
was made dean of the philosophical faculty, and for the half-yearly
period from October 1402 to April 1403 he held the office of rector of
the university. In 1402 also he was made rector or curate
(_capellarius_) of the Bethlehem chapel, which had in 1391 been erected
and endowed by some zealous citizens of Prague for the purpose of
providing good popular preaching in the Bohemian tongue. This
appointment had a deep influence on the already vigorous religious life
of Huss himself; and one of the effects of the earnest and independent
study of Scripture into which it led him was a profound conviction of
the great value not only of the philosophical but also of the
theological writings of Wycliffe.

This newly-formed sympathy with the English reformer did not, in the
first instance at least, involve Huss in any conscious opposition to the
established doctrines of Catholicism, or in any direct conflict with the
authorities of the church; and for several years he continued to
act in full accord with his archbishop (Sbynjek, or Sbynko, of
Hasenburg). Thus in 1405 he, with other two masters, was commissioned to
examine into certain reputed miracles at Wilsnack, near Wittenberg,
which had caused that church to be made a resort of pilgrims from all
parts of Europe. The result of their report was that all pilgrimage
thither from the province of Bohemia was prohibited by the archbishop on
pain of excommunication, while Huss, with the full sanction of his
superior, gave to the world his first published writing, entitled _De
Omni Sanguine Christi Glorificato_, in which he declaimed in no measured
terms against forged miracles and ecclesiastical greed, urging
Christians at the same time to desist from looking for sensible signs of
Christ's presence, but rather to seek Him in His enduring word. More
than once also Huss, together with his friend Stanislaus of Znaim, was
appointed to be synod preacher, and in this capacity he delivered at the
provincial councils of Bohemia many faithful admonitions. As early as
the 28th of May 1403, it is true, there had been held a university
disputation about the new doctrines of Wycliffe, which had resulted in
the condemnation of certain propositions presumed to be his; five years
later (May 20, 1408) this decision had been refined into a declaration
that these, forty-five in number, were not to be taught in any
heretical, erroneous or offensive sense. But it was only slowly that the
growing sympathy of Huss with Wycliffe unfavourably affected his
relations with his colleagues in the priesthood. In 1408, however, the
clergy of the city and archiepiscopal diocese of Prague laid before the
archbishop a formal complaint against Huss, arising out of strong
expressions with regard to clerical abuses of which he had made use in
his public discourses; and the result was that, having been first
deprived of his appointment as synodal preacher, he was, after a vain
attempt to defend himself in writing, publicly forbidden the exercise of
any priestly function throughout the diocese. Simultaneously with these
proceedings in Bohemia, negotiations had been going on for the removal
of the long-continued papal schism, and it had become apparent that a
satisfactory solution could only be secured if, as seemed not
impossible, the supporters of the rival popes, Benedict XIII. and
Gregory XII., could be induced, in view of the approaching council of
Pisa, to pledge themselves to a strict neutrality. With this end King
Wenceslaus of Bohemia had requested the co-operation of the archbishop
and his clergy, and also the support of the university, in both
instances unsuccessfully, although in the case of the latter the
Bohemian "nation," with Huss at its head, had only been overborne by the
votes of the Bavarians, Saxons and Poles. There followed an expression
of nationalist and particularistic as opposed to ultramontane and also
to German feeling, which undoubtedly was of supreme importance for the
whole of the subsequent career of Huss. In compliance with this feeling
a royal edict (January 18, 1409) was issued, by which, in alleged
conformity with Paris usage, and with the original charter of the
university, the Bohemian "nation" received three votes, while only one
was allotted to the other three "nations" combined; whereupon all the
foreigners, to the number of several thousands, almost immediately
withdrew from Prague, an occurrence which led to the formation shortly
afterwards of the university of Leipzig.

It was a dangerous triumph for Huss; for his popularity at court and in
the general community had been secured only at the price of clerical
antipathy everywhere and of much German ill-will. Among the first
results of the changed order of things were on the one hand the election
of Huss (October 1409) to be again rector of the university, but on the
other hand the appointment by the archbishop of an inquisitor to inquire
into charges of heretical teaching and inflammatory preaching brought
against him. He had spoken disrespectfully of the church, it was said,
had even hinted that Antichrist might be found to be in Rome, had
fomented in his preaching the quarrel between Bohemians and Germans, and
had, notwithstanding all that had passed, continued to speak of Wycliffe
as both a pious man and an orthodox teacher. The direct result of this
investigation is not known, but it is impossible to disconnect from it
the promulgation by Pope Alexander V., on the 20th of December 1409, of
a bull which ordered the abjuration of all Wycliffite heresies and the
surrender of all his books, while at the same time--a measure specially
levelled at the pulpit of Bethlehem chapel--all preaching was prohibited
except in localities which had been by long usage set apart for that
use. This decree, as soon as it was published in Prague (March 9, 1410),
led to much popular agitation, and provoked an appeal by Huss to the
pope's better informed judgment; the archbishop, however, resolutely
insisted on carrying out his instructions, and in the following July
caused to be publicly burned, in the courtyard of his own palace,
upwards of 200 volumes of the writings of Wycliffe, while he pronounced
solemn sentence of excommunication against Huss and certain of his
friends, who had in the meantime again protested and appealed to the new
pope (John XXIII.). Again the populace rose on behalf of their hero,
who, in his turn, strong in the conscientious conviction that "in the
things which pertain to salvation God is to be obeyed rather than man,"
continued uninterruptedly to preach in the Bethlehem chapel, and in the
university began publicly to defend the so-called heretical treatises of
Wycliffe, while from king and queen, nobles and burghers, a petition was
sent to Rome praying that the condemnation and prohibition in the bull
of Alexander V. might be quashed. Negotiations were carried on for some
months, but in vain; in March 1411 the ban was anew pronounced upon Huss
as a disobedient son of the church, while the magistrates and
councillors of Prague who had favoured him were threatened with a
similar penalty in ease of their giving him a contumacious support.
Ultimately the whole city, which continued to harbour him, was laid
under interdict; yet he went on preaching, and masses were celebrated as
usual, so that at the date of Archbishop Sbynko's death in September
1411, it seemed as if the efforts of ecclesiastical authority had
resulted in absolute failure.

The struggle, however, entered on a new phase with the appearance at
Prague in May 1412 of the papal emissary charged with the proclamation
of the papal bulls by which a religious war was decreed against the
excommunicated King Ladislaus of Naples, and indulgence was promised to
all who should take part in it, on terms similar to those which had been
enjoyed by the earlier crusaders to the Holy Land. By his bold and
thorough-going opposition to this mode of procedure against Ladislaus,
and still more by his doctrine that indulgence could never be sold
without simony, and could not be lawfully granted by the church except
on condition of genuine contrition and repentance, Huss at last isolated
himself, not only from the archiepiscopal party under Albik of
Unitschow, but also from the theological faculty of the university, and
especially from such men as Stanislaus of Znaim and Stephen Paletz, who
until then had been his chief supporters. A popular demonstration, in
which the papal bulls had been paraded through the streets with
circumstances of peculiar ignominy and finally burnt, led to
intervention by Wenceslaus on behalf of public order; three young men,
for having openly asserted the unlawfulness of the papal indulgence
after silence had been enjoined, were sentenced to death (June 1412);
the excommunication against Huss was renewed, and the interdict again
laid on all places which should give him shelter--a measure which now
began to be more strictly regarded by the clergy, so that in the
following December Huss had no alternative but to yield to the express
wish of the king by temporarily withdrawing from Prague. A provincial
synod, held at the instance of Wenceslaus in February 1413, broke up
without having reached any practical result; and a commission appointed
shortly afterwards also failed to bring about a reconciliation between
Huss and his adversaries. The so-called heretic meanwhile spent his time
partly at Kozihradek, some 45 m. south of Prague, and partly at
Krakowitz in the immediate neighbourhood of the capital, occasionally
giving a course of open-air preaching, but finding his chief employment
in maintaining that copious correspondence of which some precious
fragments still are extant, and in the composition of the treatise, _De
Ecclesia_, which subsequently furnished most of the material for the
capital charges brought against him, and was formerly considered
the most important of his works, though it is mainly a transcript of
Wycliffe's work of the same name.

During the year 1413 the arrangements for the meeting of a general
council at Constance were agreed upon between Sigismund and Pope John
XXIII. The objects originally contemplated had been the restoration of
the unity of the church and its reform in head and members; but so great
had become the prominence of Bohemian affairs that to these also a first
place in the programme of the approaching oecumenical assembly required
to be assigned, and for their satisfactory settlement the presence of
Huss was necessary. His attendance was accordingly requested, and the
invitation was willingly accepted as giving him a long-wished-for
opportunity both of publicly vindicating himself from charges which he
felt to be grievous, and of loyally making confession for Christ. He set
out from Bohemia on the 14th of October 1414, not, however, until he had
carefully ordered all his private affairs, with a presentiment, which he
did not conceal, that in all probability he was going to his death. The
journey, which appears to have been undertaken with the usual passport,
and under the protection of several powerful Bohemian friends (John of
Chlum, Wenceslaus of Duba, Henry of Chlum) who accompanied him, was a
very prosperous one; and at almost all the halting-places he was
received with a consideration and enthusiastic sympathy which he had
hardly expected to meet with anywhere in Germany. On the 3rd of November
he arrived at Constance; shortly afterwards there was put into his hands
the famous imperial "safe conduct," the promise of which had been one of
his inducements to quit the comparative security he had enjoyed in
Bohemia. This safe conduct, which had been frequently printed, stated
that Huss should, whatever judgment might be passed on him, be allowed
to return freely to Bohemia. This by no means provided for his immunity
from punishment. If faith to him had not been broken he would have been
sent back to Bohemia to be punished by his sovereign, the king of
Bohemia. The treachery of King Sigismund is undeniable, and was indeed
admitted by the king himself. The safe conduct was probably indeed given
by him to entice Huss to Constance. On the 4th of December the pope
appointed a commission of three bishops to investigate the case against
the heretic, and to procure witnesses; to the demand of Huss that he
might be permitted to employ an agent in his defence a favourable answer
was at first given, but afterwards even this concession to the forms of
justice was denied. While the commission was engaged in the prosecution
of its enquiries, the flight of Pope John XXIII. took place on the 20th
of March, an event which furnished a pretext for the removal of Huss
from the Dominican convent to a more secure and more severe place of
confinement under the charge of the bishop of Constance at Gottlieben on
the Rhine. On the 4th of May the temper of the council on the doctrinal
questions in dispute was fully revealed in its unanimous condemnation of
Wycliffe, especially of the so-called "forty-five articles" as
erroneous, heretical, revolutionary. It was not, however, until the 5th
of June that the case of Huss came up for hearing; the meeting, which
was an exceptionally full one, took place in the refectory of the
Franciscan cloister. Autograph copies of his work _De Ecclesia_ and of
the controversial tracts which he had written against Paletz and
Stanislaus of Znaim having been acknowledged by him, the extracted
propositions on which the prosecution based their charge of heresy were
read; but as soon as the accused began to enter upon his defence, he was
assailed by violent outcries, amidst which it was impossible for him to
be heard, so that he was compelled to bring his speech to an abrupt
close, which he did with the calm remark: "In such a council as this I
had expected to find more propriety, piety and order." It was found
necessary to adjourn the sitting until the 7th of June, on which
occasion the outward decencies were better observed, partly no doubt
from the circumstance that Sigismund was present in person. The
propositions which had been extracted from the _De Ecclesia_ were again
brought up, and the relations between Wycliffe and Huss were discussed,
the object of the prosecution being to fasten upon the latter the
charge of having entirely adopted the doctrinal system of the former,
including especially a denial of the doctrine of transubstantiation. The
accused repudiated the charge of having abandoned the Catholic doctrine,
while expressing hearty admiration and respect for the memory of
Wycliffe. Being next asked to make an unqualified submission to the
council, he expressed himself as unable to do so, while stating his
willingness to amend his teaching wherever it had been shown to be
false. With this the proceedings of the day were brought to a close. On
the 8th of June the propositions extracted from the _De Ecclesia_ were
again taken up with some fulness of detail; some of these he repudiated
as incorrectly given, others he defended; but when asked to make a
general recantation he steadfastly declined, on the ground that to do so
would be a dishonest admission of previous guilt. Among the propositions
he could heartily abjure was that relating to transubstantiation; among
those he felt constrained unflinchingly to maintain was one which had
given great offence, to the effect that Christ, not Peter, is the head
of the church to whom ultimate appeal must be made. The council,
however, showed itself inaccessible to all his arguments and
explanations, and its final resolution, as announced by Pierre d'Ailly,
was threefold: first, that Huss should humbly declare that he had erred
in all the articles cited against him; secondly, that he should promise
on oath neither to hold nor teach them in the future; thirdly, that he
should publicly recant them. On his declining to make this submission he
was removed from the bar. Sigismund himself gave it as his opinion that
it had been clearly proved by many witnesses that the accused had taught
many pernicious heresies, and that even should he recant he ought never
to be allowed to preach or teach again or to return to Bohemia, but that
should he refuse recantation there was no remedy but the stake. During
the next four weeks no effort was spared to shake the determination of
Huss; but he steadfastly refused to swerve from the path which
conscience had once made clear. "I write this," says he, in a letter to
his friends at Prague, "in prison and in chains, expecting to-morrow to
receive sentence of death, full of hope in God that I shall not swerve
from the truth, nor abjure errors imputed to me by false witnesses." The
sentence he expected was pronounced on the 6th of July in the presence
of Sigismund and a full sitting of the council; once and again he
attempted to remonstrate, but in vain, and finally he betook himself to
silent prayer. After he had undergone the ceremony of degradation with
all the childish formalities usual on such occasions, his soul was
formally consigned by all those present to the devil, while he himself
with clasped hands and uplifted eyes reverently committed it to Christ.
He was then handed over to the secular arm, and immediately led to the
place of execution, the council meanwhile proceeding unconcernedly with
the rest of its business for the day. Many incidents recorded in the
histories make manifest the meekness, fortitude and even cheerfulness
with which he went to his death. After he had been tied to the stake and
the faggots had been piled, he was for the last time urged to recant,
but his only reply was: "God is my witness that I have never taught or
preached that which false witnesses have testified against me. He knows
that the great object of all my preaching and writing was to convert men
from sin. In the truth of that gospel which hitherto I have written,
taught and preached, I now joyfully die." The fire was then kindled, and
his voice as it audibly prayed in the words of the "Kyrie Eleison" was
soon stifled in the smoke. When the flames had done their office, the
ashes that were left and even the soil on which they lay were carefully
removed and thrown into the Rhine.

Not many words are needed to convey a tolerably adequate estimate of the
character and work of the "pale thin man in mean attire," who in
sickness and poverty thus completed the forty-sixth year of a busy life
at the stake. The value of Huss as a scholar was formerly underrated.
The publication of his _Super IV. Sententiarum_ has proved that he was a
man of profound learning. Yet his principal glory will always be founded
on his spiritual teaching. It might not be easy to formulate
precisely the doctrines for which he died, and certainly some of them,
as, for example, that regarding the church, were such as many
Protestants even would regard as unguarded and difficult to harmonize
with the maintenance of external church order; but his is undoubtedly
the honour of having been the chief intermediary in handing on from
Wycliffe to Luther the torch which kindled the Reformation, and of
having been one of the bravest of the martyrs who have died in the cause
of honesty and freedom, of progress and of growth towards the light.
     (J. S. Bl.)

  The works of Huss are usually classed under four heads: the dogmatical
  and polemical, the homiletical, the exegetical and the epistolary. In
  the earlier editions of his works sufficient care was not taken to
  distinguish between his own writings and those of Wycliffe and others
  who were associated with him. In connexion with his sermons it is
  worthy of note that by means of them and by his public teaching
  generally Huss exercised a considerable influence not only on the
  religious life of his time, but on the literary development of his
  native tongue. The earliest collected edition of his works, _Historia
  et monumenta Joannis Hus et Hieronymi Pragensis_, was published at
  Nuremberg in 1558 and was reprinted with a considerable quantity of
  new matter at Frankfort in 1715. A Bohemian edition of the works has
  been edited by K. J. Erben (Prague, 1865-1868), and the _Documenta J.
  Hus vitam, doctrinam, causam in Constantiensi concilio_ (1869), edited
  by F. Palacky, is very valuable. More recently _Joannis Hus. Opera
  omnia_ have been edited by W. Flojshaus (Prague, 1904 fol.). The
  _De Ecclesia_ was published by Ulrich von Hutten in 1520; other
  controversial writings by Otto Brumfels in 1524; and Luther wrote an
  interesting preface to _Epistolae Quaedam_, which were published in
  1537. These _Epistolae_ have been translated into French by E. de
  Bonnechose (1846), and the letters written during his imprisonment
  have been edited by C. von Kügelgen (Leipzig, 1902).

  The best and most easily accessible information for the English reader
  on Huss is found in J. A. W. Neander's _Allgemeine Geschichte der
  christlichen Religion und Kirche_, translated by J. Torrey
  (1850-1858); in G. von Lechler's _Wiclif und die Vorgeschichte der
  Reformation_, translated by P. Lorimer (1878); in H. H. Milman's
  _History of Latin Christianity_, vol. viii. (1867); and in M.
  Creighton's _History of the Papacy_ (1897). Among the earlier
  authorities is the _Historia Bohemica_ of Aeneas Sylvius (1475). The
  _Acta_ of the council of Constance (published by P. Labbe in his
  _Concilia_, vol. xvi., 1731; by H. von der Haardt in his _Magnum
  Constantiense concilium_, vol. vi., 1700; and by H. Finke in his _Acta
  concilii Constantiensis_, 1896); and J. Lenfant's _Histoire de la
  guerre des Hussites_ (1731) and the same writer's _Histoire du concile
  de Constance_ (1714) should be consulted. F. Palacky's _Geschichte
  Böhmens_ (1864-1867) is also very useful. Monographs on Huss are very
  numerous. Among them may be mentioned J. A. von Helfert, _Studien über
  Hus und Hieronymus_ (1853; this work is ultramontane in its
  sympathies); C. von Höfler, _Hus und der Abzug der deutschen
  Professoren und Studenten aus Prag_ (1864); W. Berger, _Johannes Hus
  und König Sigmund_ (1871); E. Denis, _Huss et la guerre des Hussites_
  (1878); P. Uhlmann, _König Sigmunds Geleit für Hus_ (1894); J.
  Loserth, _Hus und Wiclif_ (1884), translated into English by M. J.
  Evans (1884); A. Jeep, _Gerson, Wiclefus, Hussus, inter se comparati_
  (1857); and G. von Lechler, _Johannes Hus_ (1889). See also Count
  Lützow, _The Life and Times of John Hus_ (London, 1909).


FOOTNOTE:

  [1] From which the name Huss, or more properly Hus, an abbreviation
    adopted by himself about 1396, is derived. Prior to that date he was
    invariably known as Johann Hussynecz, Hussinecz, Hussenicz or de
    Hussynecz.




HUSSAR, originally the name of a soldier belonging to a corps of light
horse raised by Matthias Corvinus, king of Hungary, in 1458, to fight
against the Turks. The Magyar _huszar_, from which the word is derived,
was formerly connected with the Magyar _husz_, twenty, and was explained
by a supposed raising of the troops by the taking of each twentieth man.
According to the _New English Dictionary_ the word is an adaptation of
the Italian _corsaro_, corsair, a robber, and is found in 15th-century
documents coupled with _praedones_. The hussar was the typical Hungarian
cavalry soldier, and, in the absence of good light cavalry in the
regular armies of central and western Europe, the name and character of
the hussars gradually spread into Prussia, France, &c. Frederick the
Great sent Major H. J. von Zieten to study the work of this type of
cavalry in the Austrian service, and Zieten so far improved on the
Austrian model that he defeated his old teacher, General Baranyai, in an
encounter between the Prussian and Austrian hussars at Rothschloss in
1741. The typical uniform of the Hungarian hussar was followed with
modifications in other European armies. It consisted of a busby or a
high cylindrical cloth cap, jacket with heavy braiding, and a dolman or
pelisse, a loose coat worn hanging from the left shoulder. The hussar
regiments of the British army were converted from light dragoons at the
following dates: 7th (1805), 10th and 15th (1806), 18th (1807, and
again on revival after disbandment, 1858), 8th (1822), 11th (1840), 20th
(late 2nd Bengal European Cavalry) (1860), 13th, 14th, and 19th (late
1st Bengal European Cavalry) (1861). The 21st Lancers were hussars from
1862 to 1897.




HUSSITES, the name given to the followers of John Huss (1369-1415), the
Bohemian reformer. They were at first often called Wycliffites, as the
theological theories of Huss were largely founded on the teachings of
Wycliffe. Huss indeed laid more stress on church reform than on
theological controversy. On such matters he always writes as a disciple
of Wycliffe. The Hussite movement may be said to have sprung from three
sources, which are however closely connected. Bohemia, which had first
received Christianity from the East, was from geographical and other
causes long but very loosely connected with the Church of Rome. The
connexion became closer at the time when the schism with its violent
controversies between the rival pontiffs, waged with the coarse
invective customary to medieval theologians, had brought great discredit
on the papacy. The terrible rapacity of its representatives in Bohemia,
which increased in proportion as it became more difficult to obtain
money from western countries such as England and France, caused general
indignation; and this was still further intensified by the gross
immorality of the Roman priests. The Hussite movement was also a
democratic one, an uprising of the peasantry against the landowners at a
period when a third of the soil belonged to the clergy. Finally national
enthusiasm for the Slavic race contributed largely to its importance.
The towns, in most cases creations of the rulers of Bohemia who had
called in German immigrants, were, with the exception of the "new town"
of Prague, mainly German; and in consequence of the regulations of the
university, Germans also held almost all the more important
ecclesiastical offices--a condition of things greatly resented by the
natives of Bohemia, which at this period had reached a high degree of
intellectual development.

The Hussite movement assumed a revolutionary character as soon as the
news of the death of Huss reached Prague. The knights and nobles of
Bohemia and Moravia, who were in favour of church reform, sent to the
council at Constance (September 2nd, 1415) a protest, known as the
"_protestatio Bohemorum_" which condemned the execution of Huss in the
strongest language. The attitude of Sigismund, king of the Romans, who
sent threatening letters to Bohemia declaring that he would shortly
"drown all Wycliffites and Hussites," greatly incensed the people.
Troubles broke out in various parts of Bohemia, and many Romanist
priests were driven from their parishes. Almost from the first the
Hussites were divided into two sections, though many minor divisions
also arose among them. Shortly before his death Huss had accepted a
doctrine preached during his absence by his adherents at Prague, namely
that of "utraquism," i.e. the obligation of the faithful to receive
communion in both kinds (_sub utraque specie_). This doctrine became the
watchword of the moderate Hussites who were known as the Utraquists or
Calixtines (_calix_, the chalice), in Bohemian, _podoboji_; while the
more advanced Hussites were soon known as the Taborites, from the city
of Tabor that became their centre.

Under the influence of his brother Sigismund, king of the Romans, King
Wenceslaus endeavoured to stem the Hussite movement. A certain number of
Hussites lead by Nicolas of Hus--no relation of John Huss--left Prague.
They held meetings in various parts of Bohemia, particularly at Usti,
near the spot where the town of Tabor was founded soon afterwards. At
these meetings Sigismund was violently denounced, and the people
everywhere prepared for war. In spite of the departure of many prominent
Hussites the troubles at Prague continued. On the 30th of July 1419,
when a Hussite procession headed by the priest John of Zelivo (in Ger.
Selau) marched through the streets of Prague, stones were thrown at the
Hussites from the windows of the town-hall of the "new town." The
people, headed by John Zizka (1376-1424), threw the burgomaster and
several town-councillors, who were the instigators of this outrage, from
the windows and they were immediately killed by the crowd. On hearing
this news King Wenceslaus was seized with an apoplectic fit, and died a
few days afterwards. The death of the king resulted in renewed troubles
in Prague and in almost all parts of Bohemia. Many Romanists, mostly
Germans--for they had almost all remained faithful to the papal
cause--were expelled from the Bohemian cities. In Prague, in November
1419, severe fighting took place between the Hussites and the
mercenaries whom Queen Sophia (widow of Wenceslaus and regent after the
death of her husband) had hurriedly collected. After a considerable part
of the city had been destroyed a truce was concluded on the 13th of
November. The nobles, who though favourable to the Hussite cause yet
supported the regent, promised to act as mediators with Sigismund; while
the citizens of Prague consented to restore to the royal forces the
castle of Vysehrad, which had fallen into their hands. Zizka, who
disapproved of this compromise, left Prague and retired to Plzen
(Pilsen). Unable to maintain himself there he marched to southern
Bohemia, and after defeating the Romanists at Sudomer--the first
pitched battle of the Hussite wars--he arrived at Usti, one of the
earliest meeting-places of the Hussites. Not considering its situation
sufficiently strong, he moved to the neighbouring new settlement of the
Hussites, to which the biblical name of Tabor was given. Tabor soon
became the centre of the advanced Hussites, who differed from the
Utraquists by recognizing only two sacraments--Baptism and
Communion--and by rejecting most of the ceremonial of the Roman Church.
The ecclesiastical organization of Tabor had a somewhat puritanic
character, and the government was established on a thoroughly democratic
basis. Four captains of the people (_hejtmane_) were elected, one of
whom was Zizka; and a very strictly military discipline was instituted.

Sigismund, king of the Romans, had, by the death of his brother
Wenceslaus without issue, acquired a claim on the Bohemian crown; though
it was then, and remained till much later, doubtful whether Bohemia was
an hereditary or an elective monarchy. A firm adherent of the Church of
Rome, Sigismund was successful in obtaining aid from the pope. Martin V.
issued a bull on the 17th of March 1420 which proclaimed a crusade "for
the destruction of the Wycliffites, Hussites and all other heretics in
Bohemia." The vast army of crusaders, with which were Sigismund and many
German princes, and which consisted of adventurers attracted by the hope
of pillage from all parts of Europe, arrived before Prague on the 30th
of June and immediately began the siege of the city, which had, however,
soon to be abandoned (see [VZ]I[VZ]KA, JOHN). Negotiations took place
for a settlement of the religious differences. The united Hussites
formulated their demands in a statement known as the "articles of
Prague." This document, the most important of the Hussite period, runs
thus in the wording of the contemporary chronicler, Laurence of
Brezova:--

  I. The word of God shall be preached and made known in the kingdom of
  Bohemia freely and in an orderly manner by the priests of the Lord....

  II. The sacrament of the most Holy Eucharist shall be freely
  administered in the two kinds, that is bread and wine, to all the
  faithful in Christ who are not precluded by mortal sin--according to
  the word and disposition of Our Saviour.

  III. The secular power over riches and worldly goods which the clergy
  possesses in contradiction to Christ's precept, to the prejudice of
  its office and to the detriment of the secular arm, shall be taken and
  withdrawn from it, and the clergy itself shall be brought back to the
  evangelical rule and an apostolic life such as that which Christ and
  his apostles led....

  IV. All mortal sins, and in particular all public and other disorders,
  which are contrary to God's law shall in every rank of life be duly
  and judiciously prohibited and destroyed by those whose office it is.

These articles, which contain the essence of the Hussite doctrine, were
rejected by Sigismund, mainly through the influence of the papal
legates, who considered them prejudicial to the authority of the Roman
see. Hostilities therefore continued. Though Sigismund had retired from
Prague, the castles of Vysehrad and Hradcany remained in possession of
his troops. The citizens of Prague laid siege to the Vysehrad, and
towards the end of October (1420) the garrison was on the point of
capitulating through famine. Sigismund attempted to relieve the
fortress, but was decisively defeated by the Hussites on the 1st of
November near the village of Pankrác. The castles of Vysehrad and
Hradcany now capitulated, and shortly afterwards almost all Bohemia fell
into the hands of the Hussites. Internal troubles prevented them from
availing themselves completely of their victory. At Prague a demagogue,
the priest John of Zelivo, for a time obtained almost unlimited
authority over the lower classes of the townsmen; and at Tabor a
communistic movement (that of the so-called Adamites) was sternly
suppressed by Zizka. Shortly afterwards a new crusade against the
Hussites was undertaken. A large German army entered Bohemia, and in
August 1421 laid siege to the town of Zatec (Saaz). The crusaders hoped
to be joined in Bohemia by King Sigismund, but that prince was detained
in Hungary. After an unsuccessful attempt to storm Zatec the crusaders
retreated somewhat ingloriously, on hearing that the Hussite troops were
approaching. Sigismund only arrived in Bohemia at the end of the year
1421. He took possession of the town of Kutna Hora (Kuttenberg), but was
decisively defeated by Zizka at Nemecky Brod (Deutschbrod) on the 6th of
January 1422. Bohemia was now again for a time free from foreign
intervention, but internal discord again broke out caused partly by
theological strife, partly by the ambition of agitators. John of Zelivo
was on the 9th of March 1422 arrested by the town council of Prague and
decapitated. There were troubles at Tabor also, where a more advanced
party opposed Zizka's authority. Bohemia obtained a temporary respite
when, in 1422, Prince Sigismund Korybutovic of Poland became for a short
time ruler of the country. His authority was recognized by the Utraquist
nobles, the citizens of Prague, and the more moderate Taborites,
including Zizka. Korybutovic, however, remained but a short time in
Bohemia; after his departure civil war broke out, the Taborites opposing
in arms the more moderate Utraquists, who at this period are also called
by the chroniclers the "Praguers," as Prague was their principal
stronghold. On the 27th of April 1423, Zizka now again leading, the
Taborites defeated at Horic the Utraquist army under Cenek of
Wartemberg; shortly afterwards an armistice was concluded at Konopist.

Papal influence had meanwhile succeeded in calling forth a new crusade
against Bohemia, but it resulted in complete failure. In spite of the
endeavours of their rulers, the Slavs of Poland and Lithuania did not
wish to attack the kindred Bohemians; the Germans were prevented by
internal discord from taking joint action against the Hussites; and the
king of Denmark, who had landed in Germany with a large force intending
to take part in the crusade, soon returned to his own country. Free for
a time from foreign aggression, the Hussites invaded Moravia, where a
large part of the population favoured their creed; but, again paralysed
by dissensions, soon returned to Bohemia. The city of Königgrätz
(Králové Hradec), which had been under Utraquist rule, espoused the
doctrine of Tabor, and called Zizka to its aid. After several military
successes gained by Zizka (q.v.) in 1423 and the following year, a
treaty of peace between the Hussites was concluded on the 13th of
September 1424 at Liben, a village near Prague, now part of that city.

In 1426 the Hussites were again attacked by foreign enemies. In June of
that year their forces, led by Prokop the Great--who took the command of
the Taborites shortly after Zizka's death in October 1424--and Sigismund
Korybutovic, who had returned to Bohemia, signally defeated the Germans
at Aussig (Usti nad Labem). After this great victory, and another at
Tachau in 1427, the Hussites repeatedly invaded Germany, though they
made no attempt to occupy permanently any part of the country.

The almost uninterrupted series of victories of the Hussites now
rendered vain all hope of subduing them by force of arms. Moreover, the
conspicuously democratic character of the Hussite movement caused the
German princes, who were afraid that such views might extend to
their own countries, to desire peace. Many Hussites, particularly the
Utraquist clergy, were also in favour of peace. Negotiations for this
purpose were to take place at the oecumenical council which had been
summoned to meet at Basel on the 3rd of March 1431. The Roman see
reluctantly consented to the presence of heretics at this council, but
indignantly rejected the suggestion of the Hussites that members of the
Greek Church, and representatives of all Christian creeds, should also
be present. Before definitely giving its consent to peace negotiations,
the Roman Church determined on making a last effort to reduce the
Hussites to subjection. On the 1st of August 1431 a large army of
crusaders, under Frederick, margrave of Brandenburg, whom Cardinal
Cesarini accompanied as papal legate, crossed the Bohemian frontier; on
the 14th of August it reached the town of Domazlice (Tauss); but on
the arrival of the Hussite army under Prokop the crusaders immediately
took to flight, almost without offering resistance.

On the 15th of October the members of the council, who had already
assembled at Basel, issued a formal invitation to the Hussites to take
part in its deliberations. Prolonged negotiations ensued; but finally a
Hussite embassy, led by Prokop and including John of Rokycan, the
Taborite bishop Nicolas of Pelhrimov, the "English Hussite," Peter
Payne and many others, arrived at Basel on the 4th of January 1433. It
was found impossible to arrive at an agreement. Negotiations were not,
however, broken off; and a change in the political situation of Bohemia
finally resulted in a settlement. In 1434 war again broke out between
the Utraquists and the Taborites. On the 30th of May of that year the
Taborite army, led by Prokop the Great and Prokop the Less, who both
fell in the battle, was totally defeated and almost annihilated at
Lipan. The moderate party thus obtained the upper hand; and it
formulated its demands in a document which was finally accepted by the
Church of Rome in a slightly modified form, and which is known as "the
compacts." The compacts, mainly founded on the articles of Prague,
declare that:--

  1. The Holy Sacrament is to be given freely in both kinds to all
  Christians in Bohemia and Moravia, and to those elsewhere who adhere
  to the faith of these two countries.

  2. All mortal sins shall be punished and extirpated by those whose
  office it is so to do.

  3. The word of God is to be freely and truthfully preached by the
  priests of the Lord, and by worthy deacons.

  4. The priests in the time of the law of grace shall claim no
  ownership of worldly possessions.

On the 5th of July 1436 the compacts were formally accepted and signed
at Iglau, in Moravia, by King Sigismund, by the Hussite delegates, and
by the representatives of the Roman Church. The last-named, however,
refused to recognize as archbishop of Prague, John of Rokycan, who had
been elected to that dignity by the estates of Bohemia. The Utraquist
creed, frequently varying in its details, continued to be that of the
established church of Bohemia till all non-Roman religious services were
prohibited shortly after the battle of the White Mountain in 1620. The
Taborite party never recovered from its defeat at Lipan, and after the
town of Tabor had been captured by George of Podebrad in 1452
Utraquist religious worship was established there. The Bohemian
brethren, whose intellectual originator was Peter Chelcicky, but
whose actual founders were Brother Gregory, a nephew of Archbishop
Rokycan, and Michael, curate of Zamberk, to a certain extent continued
the Taborite traditions, and in the 15th and 16th centuries included
most of the strongest opponents of Rome in Bohemia. J. A. Komensky
(Comenius), a member of the brotherhood, claimed for the members of his
church that they were the genuine inheritors of the doctrines of Hus.
After the beginning of the German Reformation many Utraquists adopted to
a large extent the doctrines of Luther and Calvin; and in 1567 obtained
the repeal of the compacts, which no longer seemed sufficiently
far-reaching. From the end of the 16th century the inheritors of the
Hussite tradition in Bohemia were included in the more general name of
"Protestants" borne by the adherents of the Reformation.

  All histories of Bohemia devote a large amount of space to the Hussite
  movement. See Count Lützow, _Bohemia; an Historical Sketch_ (London,
  1896); Palacky, _Geschichte von Böhmen_; Bachmann, _Geschichte
  Böhmens_; L. Krummel, _Geschichte der böhmischen Reformation_ (Gotha,
  1866) and _Utraquisten und Taboriten_ (Gotha, 1871); Ernest Denis,
  _Huss et la guerre des Hussites_ (Paris, 1878); H. Toman, _Husitské
  Válecnictvi_ (Prague, 1898).     (L.)




HUSTING (O. Eng. _hústing_, from Old Norwegian _hústhing_), the "thing"
or "ting," i.e. assembly, of the household of personal followers or
retainers of a king, earl or chief, contrasted with the "folkmoot," the
assembly of the whole people. "Thing" meant an inanimate object, the
ordinary meaning at the present day, also a cause or suit, and an
assembly; a similar development of meaning is found in the Latin _res_.
The word still appears in the names of the legislative assemblies of
Norway, the _Storthing_ and of Iceland, the _Althing_. "Husting," or
more usually in the plural "hustings," was the name of a court of the
city of London. This court was formerly the county court for the city
and was held before the lord mayor, the sheriffs and aldermen, for pleas
of land, common pleas and appeals from the sheriffs. It had probate
jurisdiction and wills were registered. All this jurisdiction has long
been obsolete, but the court still sits occasionally for registering
gifts made to the city. The charter of Canute (1032) contains a
reference to "hustings" weights, which points to the early establishment
of the court. It is doubtful whether courts of this name were held in
other towns, but John Cowell (1554-1611) in his _Interpreter_ (1601)
s.v., "Hustings," says that according to Fleta there were such courts at
Winchester, York, Lincoln, Sheppey and elsewhere, but the passage from
Fleta, as the _New English Dictionary_ points out, does not necessarily
imply this (11. lv. _Habet etiam Rex curiam in civitatibus ... et in
locis ... sicut in Hustingis London, Winton, &c._). The ordinary use of
"hustings" at the present day for the platform from which a candidate
speaks at a parliamentary or other election, or more widely for a
political candidate's election campaign, is derived from the application
of the word, first to the platform in the Guildhall on which the London
court was held, and next to that from which the public nomination of
candidates for a parliamentary election was formerly made, and from
which the candidate addressed the electors. The Ballot Act of 1872 did
away with this public declaration of the nomination.




HUSUM, a town in the Prussian province of Schleswig-Holstein, in a
fertile district 2½ m. inland from the North Sea, on the canalized
Husumer Au, which forms its harbour and roadstead, 99 m. N.W. from
Hamburg on a branch line from Tönning. Pop. (1900) 8268. It has steam
communication with the North Frisian Islands (Nordstrand, Föhr and
Sylt), and is a port for the cattle trade with England. Besides a ducal
palace and park, it possesses an Evangelical church and a gymnasium.
Cattle markets are held weekly, and in them, as also in cereals, a
lively export trade is done. There are also extensive oyster fisheries,
the property of the state, the yield during the season being very
considerable. Husum is the birthplace of Johann Georg Forchhammer
(1794-1865), the mineralogist, Peter Wilhelm Forchhammer (1801-1894),
the archaeologist, and Theodore Storm (1817-1888), the poet, to the last
of whom a monument has been erected here.

Husum is first mentioned in 1252, and its first church was built in
1431. Wisby rights were granted it in 1582, and in 1603 it received
municipal privileges from the duke of Holstein. It suffered greatly from
inundations in 1634 and 1717.

  See Christiansen, _Die Geschichte Husums_ (Husum, 1903); and
  Henningsen, _Das Stiftungsbuch der Stadt Husum_ (Husum, 1904).




HUTCHESON, FRANCIS (1694-1746), English philosopher, was born on the 8th
of August 1694. His birthplace was probably the townland of Drumalig, in
the parish of Saintfield and county of Down, Ireland.[1] Though the
family had sprung from Ayrshire, in Scotland, both his father and
grandfather were ministers of dissenting congregations in the north of
Ireland. Hutcheson was educated partly by his grandfather, partly at an
academy, where according to his biographer, Dr Leechman, he was taught
"the ordinary scholastic philosophy which was in vogue in those
days." In 1710 he entered the university of Glasgow, where he spent six
years, at first in the study of philosophy, classics and general
literature, and afterwards in the study of theology. On quitting the
university, he returned to the north of Ireland, and received a licence
to preach. When, however, he was about to enter upon the pastorate of a
small dissenting congregation he changed his plans on the advice of a
friend and opened a private academy in Dublin. In Dublin his literary
attainments gained him the friendship of many prominent inhabitants.
Among these was Archbishop King (author of the _De origine mali_), who
resisted all attempts to prosecute Hutcheson in the archbishop's court
for keeping a school without the episcopal licence. Hutcheson's
relations with the clergy of the Established Church, especially with the
archbishops of Armagh and Dublin, Hugh Boulter (1672-1742) and William
King (1650-1729), seem to have been most cordial, and his biographer, in
speaking of "the inclination of his friends to serve him, the schemes
proposed to him for obtaining promotion," &c., probably refers to some
offers of preferment, on condition of his accepting episcopal
ordination. These offers, however, were unavailing.

While residing in Dublin, Hutcheson published anonymously the four
essays by which he is best known, namely, the _Inquiry concerning
Beauty, Order, Harmony and Design_, the _Inquiry concerning Moral Good
and Evil_, in 1725, the _Essay on the Nature and Conduct of the Passions
and Affections_ and _Illustrations upon the Moral Sense_, in 1728. The
alterations and additions made in the second edition of these Essays
were published in a separate form in 1726. To the period of his Dublin
residence are also to be referred the _Thoughts on Laughter_ (a
criticism of Hobbes) and the Observations on the _Fable of the Bees_,
being in all six letters contributed to _Hibernicus' Letters_, a
periodical which appeared, in Dublin (1725-1727, 2nd ed. 1734). At the
end of the same period occurred the controversy in the _London Journal_
with Gilbert Burnet (probably the second son of Dr Gilbert Burnet,
bishop of Salisbury); on the "True Foundation of Virtue or Moral
Goodness." All these letters were collected in one volume (Glasgow,
1772).

In 1729 Hutcheson succeeded his old master, Gershom Carmichael, in the
chair of moral philosophy in the university of Glasgow. It is curious
that up to this time all his essays and letters had been published
anonymously, though their authorship appears to have been well known. In
1730 he entered on the duties of his office, delivering an inaugural
lecture (afterwards published), _De naturali hominum socialitate_. It
was a great relief to him after the drudgery of school work to secure
leisure for his favourite studies; "non levi igitur laetitia commovebar
cum almam matrem Academiam me, suum olim alumnum, in libertatem
asseruisse audiveram." Yet the works on which Hutcheson's reputation
rests had already been published.

The remainder of his life he devoted to his professorial duties. His
reputation as a teacher attracted many young men, belonging to
dissenting families, from England and Ireland, and he enjoyed a
well-deserved popularity among both his pupils and his colleagues.
Though somewhat quick-tempered, he was remarkable for his warm feelings
and generous impulses. He was accused in 1738 before the Glasgow
presbytery for "following two false and dangerous doctrines: first, that
the standard of moral goodness was the promotion of the happiness of
others; and second, that we could have a knowledge of good and evil
without and prior to a knowledge of God" (Rae, _Life of Adam Smith_,
1895). The accusation seems to have had no result.

In addition to the works named, the following were published during
Hutcheson's lifetime: a pamphlet entitled _Considerations on Patronage_
(1735); _Philosophiae moralis institutio compendiaria, ethices et
jurisprudentiae naturalis elementa continens, lib. iii._ (Glasgow,
1742); _Metaphysicae synopsis ontologiam et pneumatologiam complectens_
(Glasgow, 1742). The last work was published anonymously. After his
death, his son, Francis Hutcheson (c. 1722-1773), author of a number of
popular songs (e.g. "As Colin one evening," "Jolly Bacchus," "Where
Weeping Yews"), published much the longest, though by no means the most
interesting, of his works, _A System of Moral Philosophy, in Three
Books_ (2 vols., London, 1755). To this is prefixed a life of the
author, by Dr William Leechman (1706-1785), professor of divinity in the
university of Glasgow. The only remaining work assigned to Hutcheson is
a small treatise on _Logic_ (Glasgow, 1764). This compendium, together
with the _Compendium of Metaphysics_, was republished at Strassburg in
1722.

Thus Hutcheson dealt with metaphysics, logic and ethics. His importance
is, however, due almost entirely to his ethical writings, and among
these primarily to the four essays and the letters published during his
residence in Dublin. His standpoint has a negative and a positive
aspect; he is in strong opposition to Thomas Hobbes and Bernard de
Mandeville, and in fundamental agreement with Shaftesbury (Anthony
Ashley Cooper, 3rd earl of Shaftesbury), whose name he very properly
coupled with his own on the title-page of the first two essays. There
are no two names, perhaps, in the history of English moral philosophy,
which stand in a closer connexion. The analogy drawn between beauty and
virtue, the functions assigned to the moral sense, the position that the
benevolent feelings form an original and irreducible part of our nature,
and the unhesitating adoption of the principle that the test of virtuous
action is its tendency to promote the general welfare are obvious and
fundamental points of agreement between the two authors.

  I. _Ethics._--According to Hutcheson, man has a variety of senses,
  internal as well as external, reflex as well as direct, the general
  definition of a sense being "any determination of our minds to receive
  ideas independently on our will, and to have perceptions of pleasure
  and pain" (_Essay on the Nature and Conduct of the Passions_, sect.
  1). He does not attempt to give an exhaustive enumeration of these
  "senses," but, in various parts of his works, he specifies, besides
  the five external senses commonly recognized (which, he rightly hints,
  might be added to),--(1) consciousness, by which each man has a
  perception of himself and of all that is going on in his own mind
  (_Metaph. Syn._ pars i. cap. 2); (2) the sense of beauty (sometimes
  called specifically "an internal sense"); (3) a public sense, or
  sensus communis, "a determination to be pleased with the happiness of
  others and to be uneasy at their misery"; (4) the moral sense, or
  "moral sense of beauty in actions and affections, by which we perceive
  virtue or vice, in ourselves or others"; (5) a sense of honour, or
  praise and blame, "which makes the approbation or gratitude of others
  the necessary occasion of pleasure, and their dislike, condemnation or
  resentment of injuries done by us the occasion of that uneasy
  sensation called shame"; (6) a sense of the ridiculous. It is plain,
  as the author confesses, that there may be "other perceptions,
  distinct from all these classes," and, in fact, there seems to be no
  limit to the number of "senses" in which a psychological division of
  this kind might result.

  Of these "senses" that which plays the most important part in
  Hutcheson's ethical system is the "moral sense." It is this which
  pronounces immediately on the character of actions and affections,
  approving those which are virtuous, and disapproving those which are
  vicious. "His principal design," he says in the preface to the two
  first treatises, "is to show that human nature was not left quite
  indifferent in the affair of virtue, to form to itself observations
  concerning the advantage or disadvantage of actions, and accordingly
  to regulate its conduct. The weakness of our reason, and the
  avocations arising from the infirmity and necessities of our nature,
  are so great that very few men could ever have formed those long
  deductions of reasons which show some actions to be in the whole
  advantageous to the agent, and their contraries pernicious. The Author
  of nature has much better furnished us for a virtuous conduct than our
  moralists seem to imagine, by almost as quick and powerful
  instructions as we have for the preservation of our bodies. He has
  made virtue a lovely form, to excite our pursuit of it, and has given
  us strong affections to be the springs of each virtuous action."
  Passing over the appeal to final causes involved in this and similar
  passages, as well as the assumption that the "moral sense" has had no
  growth or history, but was "implanted" in man exactly in the condition
  in which it is now to be found among the more civilized races, an
  assumption common to the systems of both Hutcheson and Butler, it may
  be remarked that this use of the term "sense" has a tendency to
  obscure the real nature of the process which goes on in an act of
  moral judgment. For, as is so clearly established by Hume, this act
  really consists of two parts: one an act of deliberation, more or less
  prolonged, resulting in an intellectual judgment; the other a reflex
  feeling, probably instantaneous, of satisfaction at actions which we
  denominate good, of dissatisfaction at those which we denominate bad.
  By the intellectual part of this process we refer the action or habit
  to a certain class; but no sooner is the intellectual process
  completed than there is excited in us a feeling similar to that
  which myriads of actions and habits of the same class, or deemed to be
  of the same class, have excited in us on former occasions. Now,
  supposing the latter part of this process to be instantaneous, uniform
  and exempt from error, the former certainly is not. All mankind may,
  apart from their selfish interests, approve that which is virtuous or
  makes for the general good, but surely they entertain the most widely
  divergent opinions, and, in fact, frequently arrive at directly
  opposite conclusions as to particular actions and habits. This obvious
  distinction is undoubtedly recognized by Hutcheson in his analysis of
  the mental process preceding moral action, nor does he invariably
  ignore it, even when treating of the moral approbation or
  disapprobation which is subsequent on action. None the less, it
  remains true that Hutcheson, both by his phraseology, and by the
  language in which he describes the process of moral approbation, has
  done much to favour that loose, popular view of morality which,
  ignoring the necessity of deliberation and reflection, encourages
  hasty resolves and unpremeditated judgments. The term "moral sense"
  (which, it may be noticed, had already been employed by Shaftesbury,
  not only, as Dr Whewell appears to intimate, in the margin, but also
  in the text of his _Inquiry_), if invariably coupled with the term
  "moral judgment," would be open to little objection; but, taken alone,
  as designating the complex process of moral approbation, it is liable
  to lead not only to serious misapprehension but to grave practical
  errors. For, if each man's decisions are solely the result of an
  immediate intuition of the moral sense, why be at any pains to test,
  correct or review them? Or why educate a faculty whose decisions are
  infallible? And how do we account for differences in the moral
  decisions of different societies, and the observable changes in a
  man's own views? The expression has, in fact, the fault of most
  metaphorical terms: it leads to an exaggeration of the truth which it
  is intended to suggest.

  But though Hutcheson usually describes the moral faculty as acting
  instinctively and immediately, he does not, like Butler, confound the
  moral faculty with the moral standard. The test or criterion of right
  action is with Hutcheson, as with Shaftesbury, its tendency to promote
  the general welfare of mankind. He thus anticipates the utilitarianism
  of Bentham--and not only in principle, but even in the use of the
  phrase "the greatest happiness for the greatest number" (_Inquiry
  concerning Moral Good and Evil_, sect. 3).

  It is curious that Hutcheson did not realize the inconsistency of this
  external criterion with his fundamental ethical principle. Intuition
  has no possible connexion with an empirical calculation of results,
  and Hutcheson in adopting such a criterion practically denies his
  fundamental assumption.

  As connected with Hutcheson's virtual adoption of the utilitarian
  standard may be noticed a kind of moral algebra, proposed for the
  purpose of "computing the morality of actions." This calculus occurs
  in the _Inquiry concerning Moral Good and Evil_, sect. 3.


    Benevolence.

  The most distinctive of Hutcheson's ethical doctrines still remaining
  to be noticed is what has been called the "benevolent theory" of
  morals. Hobbes had maintained that all our actions, however disguised
  under apparent sympathy, have their roots in self-love. Hutcheson not
  only maintains that benevolence is the sole and direct source of many
  of our actions, but, by a not unnatural recoil, that it is the only
  source of those actions of which, on reflection, we approve.
  Consistently with this position, actions which flow from self-love
  only are pronounced to be morally indifferent. But surely, by the
  common consent of civilized men, prudence, temperance, cleanliness,
  industry, self-respect and, in general, the "personal virtues," are
  regarded, and rightly regarded, as fitting objects of moral
  approbation. This consideration could hardly escape any author,
  however wedded to his own system, and Hutcheson attempts to extricate
  himself from the difficulty by laying down the position that a man may
  justly regard himself as a part of the rational system, and may thus
  "be, in part, an object of his own benevolence" (Ibid.),--a curious
  abuse of terms, which really concedes the question at issue. Moreover,
  he acknowledges that, though self-love does not merit approbation,
  neither, except in its extreme forms, does it merit condemnation,
  indeed the satisfaction of the dictates of self-love is one of the
  very conditions of the preservation of society. To press home the
  inconsistencies involved in these various statements would be a
  superfluous task.

  The vexed question of liberty and necessity appears to be carefully
  avoided in Hutcheson's professedly ethical works. But, in the
  _Synopsis metaphysicae_, he touches on it in three places, briefly
  stating both sides of the question, but evidently inclining to that
  which he designates as the opinion of the Stoics in opposition to what
  he designates as the opinion of the Peripatetics. This is
  substantially the same as the doctrine propounded by Hobbes and Locke
  (to the latter of whom Hutcheson refers in a note), namely, that our
  will is determined by motives in conjunction with our general
  character and habit of mind, and that the only true liberty is the
  liberty of acting as we will, not the liberty of willing as we will.
  Though, however, his leaning is clear, he carefully avoids
  dogmatizing, and deprecates the angry controversies to which the
  speculations on this subject had given rise.

  It is easy to trace the influence of Hutcheson's ethical theories on
  the systems of Hume and Adam Smith. The prominence given by these
  writers to the analysis of moral action and moral approbation, with
  the attempt to discriminate the respective provinces of the reason and
  the emotions in these processes, is undoubtedly due to the influence
  of Hutcheson. To a study of the writings of Shaftesbury and Hutcheson
  we might, probably, in large measure, attribute the unequivocal
  adoption of the utilitarian standard by Hume, and, if this be the
  case, the name of Hutcheson connects itself, through Hume, with the
  names of Priestley, Paley and Bentham. Butler's _Sermons_ appeared in
  1726, the year after the publication of Hutcheson's two first essays,
  and the parallelism between the "conscience" of the one writer and the
  "moral sense" of the other is, at least, worthy of remark.

  II. _Mental Philosophy._--In the sphere of mental philosophy and logic
  Hutcheson's contributions are by no means so important or original as
  in that of moral philosophy. They are interesting mainly as a link
  between Locke and the Scottish school. In the former subject the
  influence of Locke is apparent throughout. All the main outlines of
  Locke's philosophy seem, at first sight, to be accepted as a matter of
  course. Thus, in stating his theory of the moral sense, Hutcheson is
  peculiarly careful to repudiate the doctrine of innate ideas (see, for
  instance, _Inquiry concerning Moral Good and Evil_, sect. 1 ad fin.,
  and sect. 4; and compare _Synopsis Metaphysicae_, pars i. cap. 2). At
  the same time he shows more discrimination than does Locke in
  distinguishing between the two uses of this expression, and between
  the legitimate and illegitimate form of the doctrine (Syn. Metaph.
  pars i. cap. 2). All our ideas are, as by Locke, referred to external
  or internal sense, or, in other words, to sensation and reflection
  (see, for instance, _Syn. Metaph._ pars i. cap. 1; _Logicae Compend._
  pars i. cap. 1; _System of Moral Philosophy_, bk. i. ch. 1). It is,
  however, a most important modification of Locke's doctrine, and one
  which connects Hutcheson's mental philosophy with that of Reid, when
  he states that the ideas of extension, figure, motion and rest "are
  more properly ideas accompanying the sensations of sight and touch
  than the sensations of either of these senses"; that the idea of self
  accompanies every thought, and that the ideas of number, duration and
  existence accompany every other idea whatsoever (see _Essay on the
  Nature and Conduct of the Passions_, sect. i. art. 1; _Syn. Metaph._
  pars i. cap. 1, pars ii. cap. 1; Hamilton on Reid, p. 124, note).
  Other important points in which Hutcheson follows the lead of Locke
  are his depreciation of the importance of the so-called laws of
  thought, his distinction between the primary and secondary qualities
  of bodies, the position that we cannot know the inmost essences of
  things ("intimae rerum naturae sive essentiae"), though they excite
  various ideas in us, and the assumption that external things are known
  only through the medium of ideas (_Syn. Metaph._ pars i. cap. 1),
  though, at the same time, we are assured of the existence of an
  external world corresponding to these ideas. Hutcheson attempts to
  account for our assurance of the reality of an external world by
  referring it to a natural instinct (_Syn. Metaph._ pars i. cap. 1). Of
  the correspondence or similitude between our ideas of the primary
  qualities of things and the things themselves God alone can be
  assigned as the cause. This similitude has been effected by Him
  through a law of nature. "Haec prima qualitatum primariarum perceptio,
  sive mentis actio quaedam sive passio dicatur, non alia similitudinis
  aut convenientiae inter ejusmodi ideas et res ipsas causa assignari
  posse videtur, quam ipse Deus, qui certa naturae lege hoc efficit, ut
  notiones, quae rebus praesentibus excitantur, sint ipsis similes, aut
  saltem earum habitudines, si non veras quantitates, depingant" (pars
  ii. cap. 1). Locke does speak of God "annexing" certain ideas to
  certain motions of bodies; but nowhere does he propound a theory so
  definite as that here propounded by Hutcheson, which reminds us at
  least as much of the speculations of Malebranche as of those of Locke.

  Amongst the more important points in which Hutcheson diverges from
  Locke is his account of the idea of personal identity, which he
  appears to have regarded as made known to us directly by
  consciousness. The distinction between body and mind, _corpus_ or
  _materia_ and _res cogitans_, is more emphatically accentuated by
  Hutcheson than by Locke. Generally, he speaks as if we had a direct
  consciousness of mind as distinct from body (see, for instance, _Syn.
  Metaph._ pars ii. cap. 3), though, in the posthumous work on _Moral
  Philosophy_, he expressly states that we know mind as we know body "by
  qualities immediately perceived though the substance of both be
  unknown" (bk. i. ch. 1). The distinction between perception proper and
  sensation proper, which occurs by implication though it is not
  explicitly worked out (see Hamilton's _Lectures on Metaphysics_, Lect.
  24; Hamilton's edition of _Dugald Stewart's Works_, v. 420), the
  imperfection of the ordinary division of the external senses into five
  classes, the limitation of consciousness to a special mental faculty
  (severely criticized in Sir W. Hamilton's _Lectures on Metaphysics_,
  Lect. xii.) and the disposition to refer on disputed questions of
  philosophy not so much to formal arguments as to the testimony of
  consciousness and our natural instincts are also amongst the points in
  which Hutcheson supplemented or departed from the philosophy of Locke.
  The last point can hardly fail to suggest the "common-sense
  philosophy" of Reid.

  Thus, in estimating Hutcheson's position, we find that in particular
  questions he stands nearer to Locke, but in the general spirit of his
  philosophy he seems to approach more closely to his Scottish
  successors.

  The short _Compendium of Logic_, which is more original than such
  works usually are, is remarkable chiefly for the large
  proportion of psychological matter which it contains. In these parts
  of the book Hutcheson mainly follows Locke. The technicalities of the
  subject are passed lightly over, and the book is readable. It may be
  specially noticed that he distinguishes between the mental result and
  its verbal expression [idea--term; judgment--proposition], that he
  constantly employs the word "idea," and that he defines logical truth
  as "convenientia signorum cum rebus significatis" (or "propositionis
  convenientia cum rebus ipsis," _Syn. Metaph._ pars i. cap 3), thus
  implicitly repudiating a merely formal view of logic.

  III. _Aesthetics._--Hutcheson may further be regarded as one of the
  earliest modern writers on aesthetics. His speculations on this
  subject are contained in the _Inquiry concerning Beauty, Order,
  Harmony and Design_, the first of the two treatises published in 1725.
  He maintains that we are endowed with a special sense by which we
  perceive beauty, harmony and proportion. This is a _reflex_ sense,
  because it presupposes the action of the external senses of sight and
  hearing. It may be called an internal sense, both in order to
  distinguish its perceptions from the mere perceptions of sight and
  hearing, and because "in some other affairs, where our external senses
  are not much concerned, we discern a sort of beauty, very like in many
  respects to that observed in sensible objects, and accompanied with
  like pleasure" (_Inquiry, &c._, sect. 1). The latter reason leads him
  to call attention to the beauty perceived in universal truths, in the
  operations of general causes and in moral principles and actions.
  Thus, the analogy between beauty and virtue, which was so favourite a
  topic with Shaftesbury, is prominent in the writings of Hutcheson
  also. Scattered up and down the treatise there are many important and
  interesting observations which our limits prevent us from noticing.
  But to the student of mental philosophy it may be specially
  interesting to remark that Hutcheson both applies the principle of
  association to explain our ideas of beauty and also sets limits to its
  application, insisting on there being "a natural power of perception
  or sense of beauty in objects, antecedent to all custom, education or
  example" (see _Inquiry, &c._, sects. 6, 7; Hamilton's _Lectures on
  Metaphysics_, Lect. 44 ad fin.).

  Hutcheson's writings naturally gave rise to much controversy. To say
  nothing of minor opponents, such as "Philaretus" (Gilbert Burnet,
  already alluded to), Dr John Balguy (1686-1748), prebendary of
  Salisbury, the author of two tracts on "The Foundation of Moral
  Goodness," and Dr John Taylor (1694-1761) of Norwich, a minister of
  considerable reputation in his time (author of _An Examination of the
  Scheme of Morality advanced by Dr Hutcheson_), the essays appear to
  have suggested, by antagonism, at least two works which hold a
  permanent place in the literature of English ethics--Butler's
  _Dissertation on the Nature of Virtue_, and Richard Price's _Treatise
  of Moral Good and Evil_ (1757). In this latter work the author
  maintains, in opposition to Hutcheson, that actions are _in
  themselves_ right or wrong, that right and wrong are simple ideas
  incapable of analysis, and that these ideas are perceived immediately
  by the understanding. We thus see that, not only directly but also
  through the replies which it called forth, the system of Hutcheson, or
  at least the system of Hutcheson combined with that of Shaftesbury,
  contributed, in large measure, to the formation and development of
  some of the most important of the modern schools of ethics (see
  especially art. ETHICS).

  AUTHORITIES.--Notices of Hutcheson occur in most histories, both of
  general philosophy and of moral philosophy, as, for instance, in pt.
  vii. of Adam Smith's _Theory of Moral Sentiments_; Mackintosh's
  _Progress of Ethical Philosophy_; Cousin, _Cours d'histoire de la
  philosophie morale du XVIII^e siècle_; Whewell's _Lectures on the
  History of Moral Philosophy in England_; A. Bain's _Mental and Moral
  Science_; Noah Porter's Appendix to the English translation of
  Ueberweg's _History of Philosophy_; Sir Leslie Stephen's _History of
  English Thought in the Eighteenth Century_, &c. See also Martineau,
  _Types of Ethical Theory_ (London, 1902); W. R. Scott, _Francis
  Hutcheson_ (Cambridge, 1900); Albee, _History of English
  Utilitarianism_ (London, 1902); T. Fowler, _Shaftesbury and Hutcheson_
  (London, 1882); J. McCosh, _Scottish Philosophy_ (New York, 1874). Of
  Dr Leechman's _Biography_ of Hutcheson we have already spoken. J.
  Veitch gives an interesting account of his professorial work in
  Glasgow, _Mind_, ii. 209-212.     (T. F.; X.)


FOOTNOTE:

  [1] See _Belfast Magazine_ for August 1813.




HUTCHINSON, ANNE (c. 1600-1643), American religious enthusiast, leader
of the "Antinomians" in New England, was born in Lincolnshire, England,
about 1600. She was the daughter of a clergyman named Francis Marbury,
and, according to tradition, was a cousin of John Dryden. She married
William Hutchinson, and in 1634 emigrated to Boston, Massachusetts, as a
follower and admirer of the Rev. John Cotton. Her orthodoxy was
suspected and for a time she was not admitted to the church, but soon
she organized meetings among the Boston women, among whom her
exceptional ability and her services as a nurse had given her great
influence; and at these meetings she discussed and commented upon recent
sermons and gave expression to her own theological views. The meetings
became increasingly popular, and were soon attended not only by the
women but even by some of the ministers and magistrates, including
Governor Henry Vane. At these meetings she asserted that she, Cotton and
her brother-in-law, the Rev. John Wheelwright--whom she was trying to
make second "teacher" in the Boston church--were under a "covenant of
grace," that they had a special inspiration, a "peculiar indwelling of
the Holy Ghost," whereas the Rev. John Wilson, the pastor of the Boston
church, and the other ministers of the colony were under a "covenant of
works." Anne Hutchinson was, in fact, voicing a protest against the
legalism of the Massachusetts Puritans, and was also striking at the
authority of the clergy in an intensely theocratic community. In such a
community a theological controversy inevitably was carried into secular
politics, and the entire colony was divided into factions. Mrs
Hutchinson was supported by Governor Vane, Cotton, Wheelwright and the
great majority of the Boston church; opposed to her were Deputy-Governor
John Winthrop, Wilson and all of the country magistrates and churches.
At a general fast, held late in January 1637, Wheelwright preached a
sermon which was taken as a criticism of Wilson and his friends. The
strength of the parties was tested at the General Court of Election of
May 1637, when Winthrop defeated Vane for the governorship. Cotton
recanted, Vane returned to England in disgust, Wheelwright was tried and
banished and the rank and file either followed Cotton in making
submission or suffered various minor punishments. Mrs Hutchinson was
tried (November 1637) by the General Court chiefly for "traducing the
ministers," and was sentenced to banishment; later, in March 1638, she
was tried before the Boston church and was formally excommunicated. With
William Coddington (d. 1678), John Clarke and others, she established a
settlement on the island of Aquidneck (now Rhode Island) in 1638. Four
years later, after the death of her husband, she settled on Long Island
Sound near what is now New Rochelle, Westchester county, New York, and
was killed in an Indian rising in August 1643, an event regarded in
Massachusetts as a manifestation of Divine Providence. Anne Hutchinson
and her followers were called "Antinomians," probably more as a term of
reproach than with any special reference to her doctrinal theories; and
the controversy in which she was involved is known as the "Antinomian
Controversy."

  See C. F. Adams, _Antinomianism in the Colony of Massachusetts Bay_,
  vol. xiv. of the Prince Society Publications (Boston, 1894); and
  _Three Episodes of Massachusetts History_ (Boston and New York, 1896).




HUTCHINSON, JOHN (1615-1664), Puritan soldier, son of Sir Thomas
Hutchinson of Owthorpe, Nottinghamshire, and of Margaret, daughter of
Sir John Byron of Newstead, was baptized on the 18th of September 1615.
He was educated at Nottingham and Lincoln schools and at Peterhouse,
Cambridge, and in 1637 he entered Lincoln's Inn. On the outbreak of the
great Rebellion he took the side of the Parliament, and was made in 1643
governor of Nottingham Castle, which he defended against external
attacks and internal divisions, till the triumph of the parliamentary
cause. He was chosen member for Nottinghamshire in March 1646, took the
side of the Independents, opposed the offers of the king at Newport, and
signed the death-warrant. Though a member at first of the council of
state, he disapproved of the subsequent political conduct of Cromwell
and took no further part in politics during the lifetime of the
protector. He resumed his seat in the recalled Long Parliament in May
1659, and followed Monk in opposing Lambert, believing that the former
intended to maintain the commonwealth. He was returned to the Convention
Parliament for Nottingham but expelled on the 9th of June 1660, and
while not excepted from the Act of Indemnity was declared incapable of
holding public office. In October 1663, however, he was arrested upon
suspicion of being concerned in the Yorkshire plot, and after a rigorous
confinement in the Tower of London, of which he published an account
(reprinted in the Harleian _Miscellany_, vol. iii.), and in Sandown
Castle, Kent, he died on the 11th of September 1664. His career draws
its chief interest from the _Life_ by his wife, Lucy, daughter of Sir
Allen Apsley, written after the death of her husband but not
published till 1806 (since often reprinted), a work not only valuable
for the picture which it gives of the man and of the time in which he
lived, but for the simple beauty of its style, and the naïveté with
which the writer records her sentiments and opinions, and details the
incidents of her private life.

  See the edition of Lucy Hutchinson's _Memoirs of the Life of Colonel
  Hutchinson_ by C. H. Firth (1885); _Brit. Mus. Add. MSS._ 25,901 (a
  fragment of the Life), also _Add. MSS._ 19, 333, 36,247 f. 51; _Notes
  and Queries_, 7, ser. iii. 25, viii. 422; _Monk's Contemporaries_, by
  Guizot.




HUTCHINSON, JOHN (1674-1737), English theological writer, was born at
Spennithorne, Yorkshire, in 1674. He served as steward in several
families of position, latterly in that of the duke of Somerset, who
ultimately obtained for him the post of riding purveyor to the master of
the horse, a sinecure worth about £200 a year. In 1700 he became
acquainted with Dr John Woodward (1665-1728) physician to the duke and
author of a work entitled _The Natural History of the Earth_, to whom he
entrusted a large number of fossils of his own collecting, along with a
mass of manuscript notes, for arrangement and publication. A
misunderstanding as to the manner in which these should be dealt with
was the immediate occasion of the publication by Hutchinson in 1724 of
_Moses's Principia_, part i., in which Woodward's _Natural History_ was
bitterly ridiculed, his conduct with regard to the mineralogical
specimens not obscurely characterized, and a refutation of the Newtonian
doctrine of gravitation seriously attempted. It was followed by part ii.
in 1727, and by various other works, including _Moses's Sine Principio_,
1730; _The Confusion of Tongues and Trinity of the Gentiles_, 1731;
_Power Essential and Mechanical, or what power belongs to God and what
to his creatures, in which the design of Sir I. Newton and Dr Samuel
Clarke is laid open_, 1732; _Glory or Gravity_, 1733; _The Religion of
Satan, or Antichrist Delineated_, 1736. He taught that the Bible
contained the elements not only of true religion but also of all
rational philosophy. He held that the Hebrew must be read without
points, and his interpretation rested largely on fanciful symbolism.
Bishop George Home of Norwich was during some of his earlier years an
avowed Hutchinsonian; and William Jones of Nayland continued to be so to
the end of his life.

  A complete edition of his publications, edited by Robert Spearman and
  Julius Bate, appeared in 1748 (12 vols.); an _Abstract_ of these
  followed in 1753; and a _Supplement_, with _Life_ by Spearman
  prefixed, in 1765.




HUTCHINSON, SIR JONATHAN (1828-   ), English surgeon and pathologist, was
born on the 23rd of July 1828 at Selby, Yorkshire, his parents belonging
to the Society of Friends. He entered St Bartholomew's Hospital, became
a member of the Royal College of Surgeons in 1850 (F.R.C.S. 1862), and
rapidly gained reputation as a skilful operator and a scientific
inquirer. He was president of the Hunterian Society in 1869 and 1870,
professor of surgery and pathology at the College of Surgeons from 1877
to 1882, president of the Pathological Society, 1879-1880, of the
Ophthalmological Society, 1883, of the Neurological Society, 1887, of
the Medical Society, 1890, and of the Royal Medical and Chirurgical in
1894-1896. In 1889 he was president of the Royal College of Surgeons. He
was a member of two Royal Commissions, that of 1881 to inquire into the
provision for smallpox and fever cases in the London hospitals, and that
of 1889-1896 on vaccination and leprosy. He also acted as honorary
secretary to the Sydenham Society. His activity in the cause of
scientific surgery and in advancing the study of the natural sciences
was unwearying. His lectures on neuro-pathogenesis, gout, leprosy,
diseases of the tongue, &c., were full of original observation; but his
principal work was connected with the study of syphilis, on which he
became the first living authority. He was the founder of the London
Polyclinic or Postgraduate School of Medicine; and both in his native
town of Selby and at Haslemere, Surrey, he started (about 1890)
educational museums for popular instruction in natural history. He
published several volumes on his own subjects, was editor of the
quarterly _Archives of Surgery_, and was given the Hon. LL.D. degree by
both Glasgow and Cambridge. After his retirement from active
consultative work he continued to take great interest in the question of
leprosy, asserting the existence of a definite connexion between this
disease and the eating of salted fish. He received a knighthood in 1908.




HUTCHINSON, THOMAS (1711-1780), the last royal governor of the province
of Massachusetts, son of a wealthy merchant of Boston, Mass., was born
there on the 9th of September 1711. He graduated at Harvard in 1727,
then became an apprentice in his father's counting-room, and for several
years devoted himself to business. In 1737 he began his public career as
a member of the Boston Board of Selectmen, and a few weeks later he was
elected to the General Court of Massachusetts Bay, of which he was a
member until 1740 and again from 1742 to 1749, serving as speaker in
1747, 1748 and 1749. He consistently contended for a sound financial
system, and vigorously opposed the operations of the "Land Bank" and the
issue of pernicious bills of credit. In 1748 he carried through the
General Court a bill providing for the cancellation and redemption of
the outstanding paper currency. Hutchinson went to England in 1740 as
the representative of Massachusetts in a boundary dispute with New
Hampshire. He was a member of the Massachusetts Council from 1749 to
1756, was appointed judge of probate in 1752 and was chief justice of
the superior court of the province from 1761 to 1769, was
lieutenant-governor from 1758 to 1771, acting as governor in the latter
two years, and from 1771 to 1774 was governor. In 1754 he was a delegate
from Massachusetts to the Albany Convention, and, with Franklin, was a
member of the committee appointed to draw up a plan of union. Though he
recognized the legality of the Stamp Act of 1765, he considered the
measure inexpedient and impolitic and urged its repeal, but his attitude
was misunderstood; he was considered by many to have instigated the
passage of the Act, and in August 1765 a mob sacked his Boston residence
and destroyed many valuable manuscripts and documents. He was acting
governor at the time of the "Boston Massacre" in 1770, and was virtually
forced by the citizens of Boston, under the leadership of Samuel Adams,
to order the removal of the British troops from the town. Throughout the
pre-Revolutionary disturbances in Massachusetts he was the
representative of the British ministry, and though he disapproved of
some of the ministerial measures he felt impelled to enforce them and
necessarily incurred the hostility of the Whig or Patriot element. In
1774, upon the appointment of General Thomas Gage as military governor
he went to England, and acted as an adviser to George III. and the
British ministry on American affairs, uniformly counselling moderation.
He died at Brompton, now part of London, on the 3rd of June 1780.

  He wrote _A Brief Statement of the Claim of the Colonies_ (1764); a
  _Collection of Original Papers relative to the History of
  Massachusetts Bay_ (1769), reprinted as _The Hutchinson Papers_ by the
  Prince Society in 1865; and a judicious, accurate and very valuable
  _History of the Province of Massachusetts Bay_ (vol. i., 1764, vol.
  ii., 1767, and vol. iii., 1828). His _Diary and Letters, with an
  Account of his Administration_, was published at Boston in 1884-1886.

  See James K. Hosmer's _Life of Thomas Hutchinson_ (Boston, 1896), and
  a biographical chapter in John Fiske's _Essays Historical and
  Literary_ (New York, 1902). For an estimate of Hutchinson as an
  historian, see M. C. Tyler's _Literary History of the American
  Revolution_ (New York, 1897).




HUTCHINSON, a city and the county-seat of Reno county, Kansas, U.S.A.,
in the broad bottom-land on the N. side of the Arkansas river. Pop.
(1900) 9379, of whom 414 were foreign-born and 442 negroes; (1910
census) 16,364. It is served by the Atchison, Topeka & Santa Fé, the
Missouri Pacific and the Chicago, Rock Island & Pacific railways. The
principal public buildings are the Federal building and the county court
house. The city has a public library, and an industrial reformatory is
maintained here by the state. Hutchinson is situated in a stock-raising,
fruit-growing and farming region (the principal products of which are
wheat, Indian corn and fodder), with which it has a considerable
wholesale trade. An enormous deposit of rock salt underlies the city and
its vicinity, and Hutchinson's principal industry is the
manufacture (by the open-pan and grainer processes) and the shipping of
salt; the city has one of the largest salt plants in the world. Among
the other manufactures are flour, creamery products, soda-ash,
straw-board, planing-mill products and packed meats. Natural gas is
largely used as a factory fuel. The city's factory product was valued at
$2,031,048 in 1905, an increase of 31.8% since 1900. Hutchinson was
chartered as a city In 1871.




HUTTEN, PHILIPP VON (c. 1511-1546), German knight, was a relative of
Ulrich von Hutten and passed some of his early years at the court of the
emperor Charles V. Later he joined the band of adventurers which under
Georg Hohermuth, or George of Spires, sailed to Venezuela, or Venosala
as Hutten calls it, with the object of conquering and exploiting this
land in the interests of the Augsburg family of Welser. The party landed
at Coro in February 1535 and Hutten accompanied Hohermuth on his long
and toilsome expedition into the interior in search of treasure. After
the death of Hohermuth in December 1540 he became captain-general of
Venezuela. Soon after this event he vanished into the interior,
returning after five years of wandering to find that a Spaniard, Juan de
Caravazil, or Caravajil, had been appointed governor in his absence.
With his travelling companion, Bartholomew Welser the younger, he was
seized by Caravazil in April 1546 and the two were afterwards put to
death.

  Hutten left some letters, and also a narrative of the earlier part of
  his adventures, this _Zeitung aus India Junkher Philipps von Hutten_
  being published in 1785.




HUTTEN, ULRICH VON (1488-1523), was born on the 21st of April 1488, at
the castle of Steckelberg, near Fulda, in Hesse. Like Erasmus or
Pirckheimer, he was one of those men who form the bridge between
Humanists and Reformers. He lived with both, sympathized with both,
though he died before the Reformation had time fully to develop. His
life may be divided into four parts:--his youth and cloister-life
(1488-1504); his wanderings in pursuit of knowledge (1504-1515); his
strife with Ulrich of Württemberg (1515-1519); and his connexion with
the Reformation (1519-1523). Each of these periods had its own special
antagonism, which coloured Hutten's career: in the first, his horror of
dull monastic routine; in the second, the ill-treatment he met with at
Greifswald; in the third, the crime of Duke Ulrich; in the fourth, his
disgust with Rome and with Erasmus. He was the eldest son of a poor and
not undistinguished knightly family. As he was mean of stature and
sickly his father destined him for the cloister, and he was sent to the
Benedictine house at Fulda; the thirst for learning there seized on him,
and in 1505 he fled from the monastic life, and won his freedom with the
sacrifice of his worldly prospects, and at the cost of incurring his
father's undying anger. From the Fulda cloister he went first to
Cologne, next to Erfurt, and then to Frankfort-on-Oder on the opening in
1506 of the new university of that town. For a time he was in Leipzig,
and in 1508 we find him a shipwrecked beggar on the Pomeranian coast. In
1509 the university of Greifswald welcomed him, but here too those who
at first received him kindly became his foes; the sensitive
ill-regulated youth, who took the liberties of genius, wearied his
burgher patrons; they could not brook the poet's airs and vanity, and
ill-timed assertions of his higher rank. Wherefore he left Greifswald,
and as he went was robbed of clothes and books, his only baggage, by the
servants of his late friends; in the dead of winter, half starved,
frozen, penniless, he reached Rostock. Here again the Humanists received
him gladly, and under their protection he wrote against his Greifswald
patrons, thus beginning the long list of his satires and fierce attacks
on personal or public foes. Rostock could not hold him long; he wandered
on to Wittenberg and Leipzig, and thence to Vienna, where he hoped to
win the emperor Maximilian's favour by an elaborate national poem on the
war with Venice. But neither Maximilian nor the university of Vienna
would lift a hand for him, and he passed into Italy, where, at Pavia, he
sojourned throughout 1511 and part of 1512. In the latter year his
studies were interrupted by war; in the siege of Pavia by papal troops
and Swiss, he was plundered by both sides, and escaped, sick and
penniless, to Bologna; on his recovery he even took service as a private
soldier in the emperor's army.

This dark period lasted no long time; in 1514 he was again in Germany,
where, thanks to his poetic gifts and the friendship of Eitelwolf von
Stein (d. 1515), he won the favour of the elector of Mainz, Archbishop
Albert of Brandenburg. Here high dreams of a learned career rose on him;
Mainz should be made the metropolis of a grand Humanist movement, the
centre of good style and literary form. But the murder in 1515 of his
relative Hans von Hutten by Ulrich, duke of Württemberg, changed the
whole course of his life; satire, chief refuge of the weak, became
Hutten's weapon; with one hand he took his part in the famous _Epistolae
obscurorum virorum_, and with the other launched scathing letters,
eloquent Ciceronian orations, or biting satires against the duke. Though
the emperor was too lazy and indifferent to smite a great prince, he
took Hutten under his protection and bestowed on him the honour of a
laureate crown in 1517. Hutten, who had meanwhile revisited Italy, again
attached himself to the electoral court at Mainz; and he was there when
in 1518 his friend Pirckheimer wrote, urging him to abandon the court
and dedicate himself to letters. We have the poet's long reply, in an
epistle on his "way of life," an amusing mixture of earnestness and
vanity, self-satisfaction and satire; he tells his friend that his
career is just begun, that he has had twelve years of wandering, and
will now enjoy himself a while in patriotic literary work; that he has
by no means deserted the humaner studies, but carries with him a little
library of standard books. Pirckheimer in his burgher life may have ease
and even luxury; he, a knight of the empire, how can he condescend to
obscurity? He must abide where he can shine.

In 1519 he issued in one volume his attacks on Duke Ulrich, and then,
drawing sword, took part in the private war which overthrew that prince;
in this affair he became intimate with Franz von Sickingen, the champion
of the knightly order (Ritterstand). Hutten now warmly and openly
espoused the Lutheran cause, but he was at the same time mixed up in the
attempt of the "Ritterstand" to assert itself as the militia of the
empire against the independence of the German princes. Soon after this
time he discovered at Fulda a copy of the manifesto of the emperor Henry
IV. against Hildebrand, and published it with comments as an attack on
the papal claims over Germany. He hoped thereby to interest the new
emperor Charles V., and the higher orders in the empire, in behalf of
German liberties; but the appeal failed. What Luther had achieved by
speaking to cities and common folk in homely phrase, because he touched
heart and conscience, that the far finer weapons of Hutten failed to
effect, because he tried to touch the more cultivated sympathies and
dormant patriotism of princes and bishops, nobles and knights. And so he
at once gained an undying name in the republic of letters and ruined his
own career. He showed that the artificial verse-making of the Humanists
could be connected with the new outburst of genuine German poetry. The
Minnesinger was gone; the new national singer, a Luther or a Hans Sachs,
was heralded by the stirring lines of Hutten's pen. These have in them a
splendid natural swing and ring, strong and patriotic, though
unfortunately addressed to knight and landsknecht rather than to the
German people.

The poet's high dream of a knightly national regeneration had a rude
awakening. The attack on the papacy, and Luther's vast and sudden
popularity, frightened Elector Albert, who dismissed Hutten from his
court. Hoping for imperial favour, he betook himself to Charles V.; but
that young prince would have none of him. So he returned to his friends,
and they rejoiced greatly to see him still alive; for Pope Leo X. had
ordered him to be arrested and sent to Rome, and assassins dogged his
steps. He now attached himself more closely to Franz von Sickingen and
the knightly movement. This also came to a disastrous end in the capture
of the Ebernberg, and Sickingen's death; the higher nobles had
triumphed; the archbishops avenged themselves on Lutheranism as
interpreted by the knightly order. With Sickingen Hutten also finally
fell. He fled to Basel, where Erasmus refused to see him, both for fear
of his loathsome diseases, and also because the beggared knight was sure
to borrow money from him. A paper war consequently broke out between the
two Humanists, which embittered Hutten's last days, and stained the
memory of Erasmus. From Basel Ulrich dragged himself to Mülhausen; and
when the vengeance of Erasmus drove him thence, he went to Zurich. There
the large heart of Zwingli welcomed him; he helped him with money, and
found him a quiet refuge with the pastor of the little isle of Ufnau on
the Zurich lake. There the frail and worn-out poet, writing swift satire
to the end, died at the end of August or beginning of September 1523 at
the age of thirty-five. He left behind him some debts due to
compassionate friends; he did not even own a single book, and all his
goods amounted to the clothes on his back, a bundle of letters, and that
valiant pen which had fought so many a sharp battle, and had won for the
poor knight-errant a sure place in the annals of literature.

Ulrich von Hutten is one of those men of genius at whom propriety is
shocked, and whom the mean-spirited avoid. Yet through his short and
buffeted life he was befriended, with wonderful charity and patience, by
the chief leaders of the Humanist movement. For, in spite of his
irritable vanity, his immoral life and habits, his odious diseases, his
painful restlessness, Hutten had much in him that strong men could love.
He passionately loved the truth, and was ever open to all good
influences. He was a patriot, whose soul soared to ideal schemes and a
grand utopian restoration of his country. In spite of all, his was a
frank and noble nature; his faults chiefly the faults of genius
ill-controlled, and of a life cast in the eventful changes of an age of
novelty. A swarm of writings issued from his pen; at first the smooth
elegance of his Latin prose and verse seemed strangely to miss his real
character; he was the Cicero and Ovid of Germany before he became its
Lucian.

  His chief works were his _Ars versificandi_ (1511); the _Nemo_ (1518);
  a work on the _Morbus Gallicus_ (1519); the volume of Steckelberg
  complaints against Duke Ulrich (including his four _Ciceronian
  Orations_, his Letters and the _Phalarismus_) also in 1519; the
  _Vadismus_ (1520); and the controversy with Erasmus at the end of his
  life. Besides these were many admirable poems in Latin and German. It
  is not known with certainty how far Hutten was the parent of the
  celebrated _Epistolae obscurorum virorum_, that famous satire on
  monastic ignorance as represented by the theologians of Cologne with
  which the friends of Reuchlin defended him. At first the
  cloister-world, not discerning its irony, welcomed the work as a
  defence of their position; though their eyes were soon opened by the
  favour with which the learned world received it. The _Epistolae_ were
  eagerly bought up; the first part (41 letters) appeared at the end of
  1515; early in 1516 there was a second edition; later in 1516 a third,
  with an appendix of seven letters; in 1517 appeared the second part
  (62 letters), to which a fresh appendix of eight letters was subjoined
  soon after. In 1909 the Latin text of the _Epistolae_ with an English
  translation was published by F. G. Stokes. Hutten, in a letter
  addressed to Robert Crocus, denied that he was the author of the book,
  but there is no doubt as to his connexion with it. Erasmus was of
  opinion that there were three authors, of whom Crotus Rubianus was the
  originator of the idea, and Hutten a chief contributor. D. F. Strauss,
  who dedicates to the subject a chapter of his admirable work on
  Hutten, concludes that he had no share in the first part, but that his
  hand is clearly visible in the second part, which he attributes in the
  main to him. To him is due the more serious and severe tone of that
  bitter portion of the satire. See W. Brecht, _Die Verfasser der
  Epistolae obscurorum virorum_ (1904).

  For a complete catalogue of the writings of Hutten, see E. Böcking's
  _Index Bibliographicus Huttenianus_ (1858). Böcking is also the editor
  of the complete edition of Hutten's works (7 vols., 1859-1862). A
  selection of Hutten's German writings, edited by G. Balke, appeared in
  1891. Cp. S. Szamatolski, _Huttens deutsche Schriften_ (1891). The
  best biography (though it is also somewhat of a political pamphlet) is
  that of D. F. Strauss (_Ulrich von Hutten_, 1857; 4th ed., 1878;
  English translation by G. Sturge, 1874), with which may be compared
  the older monographs by A. Wagenseil (1823), A. Bürck (1846) and J.
  Zeller (Paris, 1849). See also J. Deckert, _Ulrich von Huttens Leben
  und Wirken. Eine historische Skizze_ (1901).     (G. W. K.)




HUTTER, LEONHARD (1563-1616), German Lutheran theologian, was born at
Nellingen near Ulm in January 1563. From 1581 he studied at the
universities of Strassburg, Leipzig, Heidelberg and Jena. In 1594 he
began to give theological lectures at Jena, and in 1596 accepted a call
as professor of theology at Wittenberg, where he died on the 23rd of
October 1616. Hutter was a stern champion of Lutheran orthodoxy, as set
down in the confessions and embodied in his own _Compendium locorum
theologicorum_ (1610; reprinted 1863), being so faithful to his master
as to win the title of "Luther redonatus."

  In reply to Rudolf Hospinian's _Concordia discors_ (1607), he wrote a
  work, rich in historical material but one-sided in its apologetics,
  _Concordia concors_ (1614), defending the formula of Concord, which he
  regarded as inspired. His _Irenicum vere christianum_ is directed
  against David Pareus (1548-1622), professor primarius at Heidelberg,
  who in _Irenicum sive de unione et synodo Evangelicorum_ (1614) had
  pleaded for a reconciliation of Lutheranism and Calvinism; his
  _Calvinista aulopoliticus_ (1610) was written against the "damnable
  Calvinism" which was becoming prevalent in Holstein and Brandenburg.
  Another work, based on the formula of Concord, was entitled _Loci
  communes theologici_.




HUTTON, CHARLES (1737-1823), English mathematician, was born at
Newcastle-on-Tyne on the 14th of August 1737. He was educated in a
school at Jesmond, kept by Mr Ivison, a clergyman of the church of
England. There is reason to believe, on the evidence of two pay-bills,
that for a short time in 1755 and 1756 Hutton worked in Old Long Benton
colliery; at any rate, on Ivison's promotion to a living, Hutton
succeeded to the Jesmond school, whence, in consequence of increasing
pupils, he removed to Stote's Hall. While he taught during the day at
Stote's Hall, he studied mathematics in the evening at a school in
Newcastle. In 1760 he married, and began tuition on a larger scale in
Newcastle, where he had among his pupils John Scott, afterwards Lord
Eldon, chancellor of England. In 1764 he published his first work, _The
Schoolmaster's Guide, or a Complete System of Practical Arithmetic_,
which in 1770 was followed by his _Treatise on Mensuration both in
Theory and Practice_. In 1772 appeared a tract on _The Principles of
Bridges_, suggested by the destruction of Newcastle bridge by a high
flood on the 17th of November 1771. In 1773 he was appointed professor
of mathematics at the Royal Military Academy, Woolwich, and in the
following year he was elected F.R.S. and reported on Nevil Maskelyne's
determination of the mean density and mass of the earth from
measurements taken in 1774-1776 at Mount Schiehallion in Perthshire.
This account appeared in the _Philosophical Transactions_ for 1778, was
afterwards reprinted in the second volume of his _Tracts on Mathematical
and Philosophical Subjects_, and procured for Hutton the degree of LL.D.
from the university of Edinburgh. He was elected foreign secretary to
the Royal Society in 1779, but his resignation in 1783 was brought about
by the president Sir Joseph Banks, whose behaviour to the mathematical
section of the society was somewhat high-handed (see Kippis's
_Observations on the late Contests in the Royal Society_, London, 1784).
After his _Tables of the Products and Powers of Numbers_, 1781, and his
_Mathematical Tables_, 1785, he issued, for the use of the Royal
Military Academy, in 1787 _Elements of Conic Sections_, and in 1798 his
_Course of Mathematics_. His _Mathematical and Philosophical
Dictionary_, a valuable contribution to scientific biography, was
published in 1795 (2nd ed., 1815), and the four volumes of _Recreations
in Mathematics and Natural Philosophy_, mostly a translation from the
French, in 1803. One of the most laborious of his works was the
abridgment, in conjunction with G. Shaw and R. Pearson, of the
_Philosophical Transactions_. This undertaking, the mathematical and
scientific parts of which fell to Hutton's share, was completed in 1809,
and filled eighteen volumes quarto. His name first appears in the
_Ladies' Diary_ (a poetical and mathematical almanac which was begun in
1704, and lasted till 1871) in 1764; ten years later he was appointed
editor of the almanac, a post which he retained till 1817. Previously he
had begun a small periodical, _Miscellanea Mathematica_, which extended
only to thirteen numbers; subsequently he published in five volumes _The
Diarian Miscellany_, which contained large extracts from the _Diary_. He
resigned his professorship in 1807, and died on the 27th of January
1823.

  See John Bruce, _Charles Hutton_ (Newcastle, 1823).




HUTTON, JAMES (1726-1797), Scottish geologist, was born in Edinburgh on
the 3rd of June 1726. Educated at the high school and university of his
native city, he acquired while a student a passionate love of scientific
inquiry. He was apprenticed to a lawyer, but his employer advised that a
more congenial profession should be chosen for him. The young apprentice
chose medicine as being nearest akin to his favourite pursuit of
chemistry. He studied for three years at Edinburgh, and completed his
medical education in Paris, returning by the Low Countries, and taking
his degree of doctor of medicine at Leiden in 1749. Finding, however,
that there seemed hardly any opening for him, he abandoned the medical
profession, and, having inherited a small property in Berwickshire from
his father, resolved to devote himself to agriculture. He then went to
Norfolk to learn the practical work of farming, and subsequently
travelled in Holland, Belgium and the north of France. During these
years he began to study the surface of the earth, gradually shaping in
his mind the problem to which he afterwards devoted his energies. In the
summer of 1754 he established himself on his own farm in Berwickshire,
where he resided for fourteen years, and where he introduced the most
improved forms of husbandry. As the farm was brought into excellent
order, and as its management, becoming more easy, grew less interesting,
he was induced to let it, and establish himself for the rest of his life
in Edinburgh. This took place about the year 1768. He was unmarried, and
from this period until his death in 1797 he lived with his three
sisters. Surrounded by congenial literary and scientific friends he
devoted himself to research.

At that time geology in any proper sense of the term did not exist.
Mineralogy, however, had made considerable progress. But Hutton had
conceived larger ideas than were entertained by the mineralogists of his
day. He desired to trace back the origin of the various minerals and
rocks, and thus to arrive at some clear understanding of the history of
the earth. For many years he continued to study the subject. At last, in
the spring of the year 1785, he communicated his views to the recently
established Royal Society of Edinburgh in a paper entitled _Theory of
the Earth, or an Investigation of the Laws Observable in the
Composition, Dissolution and Restoration of Land upon the Globe_. In
this remarkable work the doctrine is expounded that geology is not
cosmogony, but must confine itself to the study of the materials of the
earth; that everywhere evidence may be seen that the present rocks of
the earth's surface have been in great part formed out of the waste of
older rocks; that these materials having been laid down under the sea
were there consolidated under great pressure, and were subsequently
disrupted and upheaved by the expansive power of subterranean heat; that
during these convulsions veins and masses of molten rock were injected
into the rents of the dislocated strata; that every portion of the
upraised land, as soon as exposed to the atmosphere, is subject to
decay; and that this decay must tend to advance until the whole of the
land has been worn away and laid down on the sea-floor, whence future
upheavals will once more raise the consolidated sediments into new land.
In some of these broad and bold generalizations Hutton was anticipated
by the Italian geologists; but to him belongs the credit of having first
perceived their mutual relations, and combined them in a luminous
coherent theory based upon observation.

It was not merely the earth to which Hutton directed his attention. He
had long studied the changes of the atmosphere. The same volume in which
his _Theory of the Earth_ appeared contained also a _Theory of Rain_,
which was read to the Royal Society of Edinburgh in 1784. He contended
that the amount of moisture which the air can retain in solution
increases with augmentation of temperature, and, therefore, that on the
mixture of two masses of air of different temperatures a portion of the
moisture must be condensed and appear in visible form. He investigated
the available data regarding rainfall and climate in different regions
of the globe, and came to the conclusion that the rainfall is everywhere
regulated by the humidity of the air on the one hand, and the causes
which promote mixtures of different aerial currents in the higher
atmosphere on the other.

The vigour and versatility of his genius may be understood from the
variety of works which, during his thirty years' residence in Edinburgh,
he gave to the world. In 1792 he published a quarto volume entitled
_Dissertations on different Subjects in Natural Philosophy_, in which he
discussed the nature of matter, fluidity, cohesion, light, heat and
electricity. Some of these subjects were further illustrated by him in
papers read before the Royal Society of Edinburgh. He did not restrain
himself within the domain of physics, but boldly marched into that of
metaphysics, publishing three quarto volumes with the title _An
Investigation of the Principles of Knowledge, and of the Progress of
Reason--from Sense to Science and Philosophy_. In this work he developed
the idea that the external world, as conceived by us, is the creation of
our own minds influenced by impressions from without, that there is no
resemblance between our picture of the outer world and the reality, yet
that the impressions produced upon our minds, being constant and
consistent, become as much realities to us as if they precisely
resembled things actually existing, and, therefore, that our moral
conduct must remain the same as if our ideas perfectly corresponded to
the causes producing them. His closing years were devoted to the
extension and republication of his _Theory of the Earth_, of which two
volumes appeared in 1795. A third volume, necessary to complete the
work, was left by him in manuscript, and is referred to by his
biographer John Playfair. A portion of the MS. of this volume, which had
been given to the Geological Society of London by Leonard Horner, was
published by the Society in 1899, under the editorship of Sir A. Geikie.
The rest of the manuscript appears to be lost. Soon afterwards Hutton
set to work to collect and systematize his numerous writings on
husbandry, which he proposed to publish under the title of _Elements of
Agriculture_. He had nearly completed this labour when an incurable
disease brought his active career to a close on the 26th of March 1797.

  It is by his _Theory of the Earth_ that Hutton will be remembered with
  reverence while geology continues to be cultivated. The author's
  style, however, being somewhat heavy and obscure, the book did not
  attract during his lifetime so much attention as it deserved. Happily
  for science Hutton numbered among his friends John Playfair (q.v.),
  professor of mathematics in the university of Edinburgh, whose
  enthusiasm for the spread of Hutton's doctrine was combined with a
  rare gift of graceful and luminous exposition. Five years after
  Hutton's death he published a volume, _Illustrations of the Huttonian
  Theory of the Earth_, in which he gave an admirable summary of that
  theory, with numerous additional illustrations and arguments. This
  work is justly regarded as one of the classical contributions to
  geological literature. To its influence much of the sound progress of
  British geology must be ascribed. In the year 1805 a biographical
  account of Hutton, written by Playfair, was published in vol. v. of
  the _Transactions of the Royal Society of Edinburgh_.     (A. Ge.)




HUTTON, RICHARD HOLT (1826-1897), English writer and theologian, son of
Joseph Hutton, Unitarian minister at Leeds, was born at Leeds on the 2nd
of June 1826. His family removed to London in 1835, and he was educated
at University College School and University College, where he began a
lifelong friendship with Walter Bagehot, of whose works he afterwards was
the editor; he took the degree in 1845, being awarded the gold medal for
philosophy. Meanwhile he had also studied for short periods at Heidelberg
and Berlin, and in 1847 he entered Manchester New College with the idea
of becoming a minister like his father, and studied there under James
Martineau. He did not, however, succeed in obtaining a call to any
church, and for some little time his future was unsettled. He married in
1851 his cousin, Anne Roscoe, and became joint-editor with J. L. Sanford
of the _Inquirer_, the principal Unitarian organ. But his innovations and
his unconventional views about stereotyped Unitarian doctrines caused
alarm, and in 1853 he resigned. His health had broken down, and he
visited the West Indies, where his wife died of yellow fever. In 1855
Hutton and Bagehot became joint-editors of the _National Review_, a new
monthly, and conducted it for ten years. During this time Hutton's
theological views, influenced largely by Coleridge, and more directly by
F. W. Robertson and F. D. Maurice, gradually approached more and more to
those of the Church of England, which he ultimately joined. His interest
in theology was profound, and he brought to it a spirituality of outlook
and an aptitude for metaphysical inquiry and exposition which added a
singular attraction to his writings. In 1861 he joined Meredith Townsend
as joint-editor and part proprietor of the _Spectator_, then a well-known
liberal weekly, which, however, was not remunerative from the business
point of view. Hutton took charge of the literary side of the paper, and
by degrees his own articles became and remained up to the last one of the
best-known features of serious and thoughtful English journalism. The
_Spectator_, which gradually became a prosperous property, was his
pulpit, in which unwearyingly he gave expression to his views,
particularly on literary, religious and philosophical subjects, in
opposition to the agnostic and rationalistic opinions then current in
intellectual circles, as popularized by Huxley. A man of fearless
honesty, quick and catholic sympathies, broad culture, and many friends
in intellectual and religious circles, he became one of the most
influential journalists of the day, his fine character and conscience
earning universal respect and confidence. He was an original member of
the Metaphysical Society (1869). He was an anti-vivisectionist, and a
member of the royal commission (1875) on that subject. In 1858 he had
married Eliza Roscoe, a cousin of his first wife; she died early in 1897,
and Hutton's own death followed on the 9th of September of the same year.

  Among his other publications may be mentioned _Essays, Theological and
  Literary_ (1871; revised 1888), and _Criticisms on Contemporary
  Thought and Thinkers_ (1894); and his opinions may be studied
  compendiously in the selections from his _Spectator_ articles
  published in 1899 under the title of _Aspects of Religious and
  Scientific Thought_.




HUXLEY, THOMAS HENRY (1825-1895), English biologist, was born on the 4th
of May 1825 at Ealing, where his father, George Huxley, was senior
assistant-master in the school of Dr Nicholas. This was an establishment
of repute, and is at any rate remarkable for having produced two men
with so little in common in after life as Huxley and Cardinal Newman.
The cardinal's brother, Francis William, had been "captain" of the
school in 1821. Huxley was a seventh child (as his father had also
been), and the youngest who survived infancy. Of Huxley's ancestry no
more is ascertainable than in the case of most middle-class families. He
himself thought it sprang from the Cheshire Huxleys of Huxley Hall.
Different branches migrated south, one, now extinct, reaching London,
where its members were apparently engaged in commerce. They established
themselves for four generations at Wyre Hall, near Edmonton, and one was
knighted by Charles II. Huxley describes his paternal race as "mainly
Iberian mongrels, with a good dash of Norman and a little Saxon."[1]
From his father he thought he derived little except a quick temper and
the artistic faculty which proved of great service to him and reappeared
in an even more striking degree in his daughter, the Hon. Mrs Collier.
"Mentally and physically," he wrote, "I am a piece of my mother." Her
maiden name was Rachel Withers. "She came of Wiltshire people," he adds,
and describes her as "a typical example of the Iberian variety." He
tells us that "her most distinguishing characteristic was rapidity of
thought.... That peculiarity has been passed on to me in full strength"
(_Essays_, i. 4). One of the not least striking facts in Huxley's life
is that of education in the formal sense he received none. "I had two
years of a pandemonium of a school (between eight and ten), and after
that neither help nor sympathy in any intellectual direction till I
reached manhood" (_Life_, ii. 145). After the death of Dr Nicholas the
Ealing school broke up, and Huxley's father returned about 1835 to his
native town, Coventry, where he had obtained a small appointment. Huxley
was left to his own devices; few histories of boyhood could offer any
parallel. At twelve he was sitting up in bed to read Hutton's _Geology_.
His great desire was to be a mechanical engineer; it ended in his
devotion to "the mechanical engineering of living machines." His
curiosity in this direction was nearly fatal; a _post-mortem_ he was
taken to between thirteen and fourteen was followed by an illness which
seems to have been the starting-point of the ill-health which pursued
him all through life. At fifteen he devoured Sir William Hamilton's
_Logic_, and thus acquired the taste for metaphysics, which he
cultivated to the end. At seventeen he came under the influence of
Thomas Carlyle's writings. Fifty years later he wrote: "To make things
clear and get rid of cant and shows of all sorts. This was the lesson I
learnt from Carlyle's books when I was a boy, and it has stuck by me all
my life" (_Life_, ii. 268). Incidentally they led him to begin to learn
German; he had already acquired French. At seventeen Huxley, with his
elder brother James, commenced regular medical studies at Charing Cross
Hospital, where they had both obtained scholarships. He studied under
Wharton Jones, a physiologist who never seems to have attained the
reputation he deserved. Huxley said of him: "I do not know that I ever
felt so much respect for a teacher before or since" (_Life_, i. 20). At
twenty he passed his first M.B. examination at the University of London,
winning the gold medal for anatomy and physiology; W. H. Ransom, the
well-known Nottingham physician, obtaining the exhibition. In 1845 he
published, at the suggestion of Wharton Jones, his first scientific
paper, demonstrating the existence of a hitherto unrecognized layer in
the inner sheath of hairs, a layer that has been known since as
"Huxley's layer."

Something had to be done for a livelihood, and at the suggestion of a
fellow-student, Mr (afterwards Sir Joseph) Fayrer, he applied for an
appointment in the navy. He passed the necessary examination, and at the
same time obtained the qualification of the Royal College of Surgeons.
He was "entered on the books of Nelson's old ship, the 'Victory,' for
duty at Haslar Hospital." Its chief, Sir John Richardson, who was a
well-known Arctic explorer and naturalist, recognized Huxley's ability,
and procured for him the post of surgeon to H.M.S. "Rattlesnake," about
to start for surveying work in Torres Strait. The commander, Captain
Owen Stanley, was a son of the bishop of Norwich and brother of Dean
Stanley, and wished for an officer with some scientific knowledge.
Besides Huxley the "Rattlesnake" also carried a naturalist by
profession, John Macgillivray, who, however, beyond a dull narrative of
the expedition, accomplished nothing. The "Rattlesnake" left England on
the 3rd of December 1846, and was ordered home after the lamented death
of Captain Stanley at Sydney, to be paid off at Chatham on the 9th of
November 1850. The tropical seas teem with delicate surface-life, and to
the study of this Huxley devoted himself with unremitting devotion. At
that time no known methods existed by which it could be preserved for
study in museums at home. He gathered a magnificent harvest in the
almost unreaped field, and the conclusions he drew from it were the
beginning of the revolution in zoological science which he lived to see
accomplished.

Baron Cuvier (1769-1832), whose classification still held its ground,
had divided the animal kingdom into four great _embranchements_. Each of
these corresponded to an independent archetype, of which the "idea" had
existed in the mind of the Creator. There was no other connexion between
these classes, and the "ideas" which animated them were, as far as one
can see, arbitrary. Cuvier's groups, without their theoretical basis,
were accepted by K. E. von Baer (1792-1876). The "idea" of the group, or
archetype, admitted of endless variation within it; but this was
subordinate to essential conformity with the archetype, and hence Cuvier
deduced the important principle of the "correlation of parts," of which
he made such conspicuous use in palaeontological reconstruction.
Meanwhile the "Naturphilosophen," with J. W. Goethe (1749-1832) and L.
Oken (1779-1851), had in effect grasped the underlying principle of
correlation, and so far anticipated evolution by asserting the
possibility of deriving specialized from simpler structures. Though they
were still hampered by idealistic conceptions, they established
morphology. Cuvier's four great groups were Vertebrata, Mollusca,
Articulata and Radiata. It was amongst the members of the last
class that Huxley found most material ready to his hand in the seas of
the tropics. It included organisms of the most varied kind, with nothing
more in common than that their parts were more or less distributed round
a centre. Huxley sent home "communication after communication to the
Linnean Society," then a somewhat somnolent body, "with the same result
as that obtained by Noah when he sent the raven out of the ark"
(_Essays_, i. 13). His important paper, _On the Anatomy and the
Affinities of the Family of Medusae_, met with a better fate. It was
communicated by the bishop of Norwich to the Royal Society, and printed
by it in the _Philosophical Transactions_ in 1849. Huxley united, with
the Medusae, the Hydroid and Sertularian polyps, to form a class to
which he subsequently gave the name of Hydrozoa. This alone was no
inconsiderable feat for a young surgeon who had only had the training of
the medical school. But the ground on which it was done has led to
far-reaching theoretical developments. Huxley realized that something
more than superficial characters were necessary in determining the
affinities of animal organisms. He found that all the members of the
class consisted of two membranes enclosing a central cavity or stomach.
This is characteristic of what are now called the Coelenterata. All
animals higher than these have been termed Coelomata; they possess a
distinct body-cavity in addition to the stomach. Huxley went further
than this, and the most profound suggestion in his paper is the
comparison of the two layers with those which appear in the germ of the
higher animals. The consequences which have flowed from this prophetic
generalization of the _ectoderm_ and _endoderm_ are familiar to every
student of evolution. The conclusion was the more remarkable as at the
time he was not merely free from any evolutionary belief, but actually
rejected it. The value of Huxley's work was immediately recognized. On
returning to England in 1850 he was elected a Fellow of the Royal
Society. In the following year, at the age of twenty-six, he not merely
received the Royal medal, but was elected on the council. With
absolutely no aid from any one he had placed himself in the front rank
of English scientific men. He secured the friendship of Sir J. D. Hooker
and John Tyndall, who remained his lifelong friends. The Admiralty
retained him as a nominal assistant-surgeon, in order that he might work
up the observations he had made during the voyage of the "Rattlesnake."
He was thus enabled to produce various important memoirs, especially
those on certain Ascidians, in which he solved the problem of
_Appendicularia_--an organism whose place in the animal kingdom Johannes
Müller had found himself wholly unable to assign--and on the morphology
of the Cephalous Mollusca.

Richard Owen, then the leading comparative anatomist in Great Britain,
was a disciple of Cuvier, and adopted largely from him the deductive
explanation of anatomical fact from idealistic conceptions. He
superadded the evolutionary theories of Oken, which were equally
idealistic, but were altogether repugnant to Cuvier. Huxley would have
none of either. Imbued with the methods of von Baer and Johannes Müller,
his methods were purely inductive. He would not hazard any statement
beyond what the facts revealed. He retained, however, as has been done
by his successors, the use of archetypes, though they no longer
represented fundamental "ideas" but generalizations of the essential
points of structure common to the individuals of each class. He had not
wholly freed himself, however, from archetypal trammels. "The doctrine,"
he says, "that every natural group is organized after a definite
archetype ... seems to me as important for zoology as the doctrine of
definite proportions for chemistry." This was in 1853. He further
stated: "There is no progression from a lower to a higher type, but
merely a more or less complete evolution of one type" (_Phil. Trans._,
1853, p. 63). As Chalmers Mitchell points out, this statement is of
great historical interest. Huxley definitely uses the word "evolution,"
and admits its existence _within_ the great groups. He had not, however,
rid himself of the notion that the archetype was a property inherent in
the group. Herbert Spencer, whose acquaintance he made in 1852, was
unable to convert him to evolution in its widest sense (_Life_, i.
168). He could not bring himself to acceptance of the theory--owing, no
doubt, to his rooted aversion from à priori reasoning--without a
mechanical conception of its mode of operation. In his first interview
with Darwin, which seems to have been about the same time, he expressed
his belief "in the sharpness of the lines of demarcation between natural
groups," and was received with a humorous smile (_Life_, i. 169).

The naval medical service exists for practical purposes. It is not
surprising, therefore, that after his three years' nominal employment
Huxley was ordered on active service. Though without private means of
any kind, he resigned. The navy, however, retains the credit of having
started his scientific career as well as that of Hooker and Darwin.
Huxley was now thrown on his own resources, the immediate prospects of
which were slender enough. As a matter of fact, he had not to wait many
months. His friend, Edward Forbes, was appointed to the chair of natural
history in Edinburgh, and in July 1854 he succeeded him as lecturer at
the School of Mines and as naturalist to the Geological Survey in the
following year. The latter post he hesitated at first to accept, as he
"did not care for fossils" (_Essays_, i. 15). In 1855 he married Miss H.
A. Heathorn, whose acquaintance he had made in Sydney. They were engaged
when Huxley could offer nothing but the future promise of his ability.
The confidence of his devoted helpmate was not misplaced, and her
affection sustained him to the end, after she had seen him the recipient
of every honour which English science could bestow. His most important
research belonging to this period was the Croonian Lecture delivered
before the Royal Society in 1858 on "The Theory of the Vertebrate
Skull." In this he completely and finally demolished, by applying as
before the inductive method, the idealistic, if in some degree
evolutionary, views of its origin which Owen had derived from Goethe and
Oken. This finally disposed of the "archetype," and may be said once for
all to have liberated the English anatomical school from the deductive
method.

In 1859 _The Origin of Species_ was published. This was a momentous
event in the history of science, and not least for Huxley. Hitherto he
had turned a deaf ear to evolution. "I took my stand," he says, "upon
two grounds: firstly, that ... the evidence in favour of transmutation
was wholly insufficient; and secondly, that no suggestion respecting the
causes of the transmutation assumed, which had been made, was in any way
adequate to explain the phenomena" (_Life_, i. 168). Huxley had studied
Lamarck "attentively," but to no purpose. Sir Charles Lyell "was the
chief agent in smoothing the road for Darwin. For consistent
uniformitarianism postulates evolution as much in the organic as in the
inorganic world" (l.c.); and Huxley found in Darwin what he had failed
to find in Lamarck, an intelligible hypothesis good enough as a working
basis. Yet with the transparent candour which was characteristic of him,
he never to the end of his life concealed the fact that he thought it
wanting in rigorous proof. Darwin, however, was a naturalist; Huxley was
not. He says: "I am afraid there is very little of the genuine
naturalist in me. I never collected anything, and species-work was
always a burden to me; what I cared for was the architectural and
engineering part of the business" (_Essays_, i. 7). But the solution of
the problem of organic evolution must work upwards from the initial
stages, and it is precisely for the study of these that "species-work"
is necessary. Darwin, by observing the peculiarities in the distribution
of the plants which he had collected in the Galapagos, was started on
the path that led to his theory. Anatomical research had only so far led
to transcendental hypothesis, though in Huxley's hands it had cleared
the decks of that lumber. He quotes with approval Darwin's remark that
"no one has a right to examine the question of species who has not
minutely described many" (_Essays_, ii. 283). The rigorous proof which
Huxley demanded was the production of species sterile to one another by
selective breeding (_Life_, i. 193). But this was a misconception of the
question. Sterility is a physiological character, and the specific
differences which the theory undertook to account for are
morphological; there is no necessary nexus between the two. Huxley,
however, felt that he had at last a secure grip of evolution. He warned
Darwin: "I will stop at no point as long as clear reasoning will carry
me further" (_Life_, i. 172). Owen, who had some evolutionary
tendencies, was at first favourably disposed to Darwin's theory, and
even claimed that he had to some extent anticipated it in his own
writings. But Darwin, though he did not thrust it into the foreground,
never flinched from recognizing that man could not be excluded from his
theory. "Light will be thrown on the origin of man and his history"
(_Origin_, ed. i. 488). Owen could not face the wrath of fashionable
orthodoxy. In his Rede Lecture he endeavoured to save the position by
asserting that man was clearly marked off from all other animals by the
anatomical structure of his brain. This was actually inconsistent with
known facts, and was effectually refuted by Huxley in various papers and
lectures, summed up in 1863 in _Man's Place in Nature_. This "monkey
damnification" of mankind was too much even for the "veracity" of
Carlyle, who is said to have never forgiven it. Huxley had not the
smallest respect for authority as a basis for belief, scientific or
otherwise. He held that scientific men were morally bound "to try all
things and hold fast to that which is good" (_Life_, ii. 161). Called
upon in 1862, in the absence of the president, to deliver the
presidential address to the Geological Society, he disposed once for all
of one of the principles accepted by geologists, that similar fossils in
distinct regions indicated that the strata containing them were
contemporary. All that could be concluded, he pointed out, was that the
general order of succession was the same. In 1854 Huxley had refused the
post of palaeontologist to the Geological Survey; but the fossils for
which he then said that he "did not care" soon acquired importance in
his eyes, as supplying evidence for the support of the evolutionary
theory. The thirty-one years during which he occupied the chair of
natural history at the School of Mines were largely occupied with
palaeontological research. Numerous memoirs on fossil fishes established
many far-reaching morphological facts. The study of fossil reptiles led
to his demonstrating, in the course of lectures on birds, delivered at
the College of Surgeons in 1867, the fundamental affinity of the two
groups which he united under the title of Sauropsida. An incidental
result of the same course was his proposed rearrangement of the
zoological regions into which P. L. Sclater had divided the world in
1857. Huxley anticipated, to a large extent, the results at which
botanists have since arrived: he proposed as primary divisions,
Arctogaea--to include the land areas of the northern hemisphere--and
Notogaea for the remainder. Successive waves of life originated in and
spread from the northern area, the survivors of the more ancient types
finding successively a refuge in the south. Though Huxley had accepted
the Darwinian theory as a working hypothesis, he never succeeded in
firmly grasping it in detail. He thought "evolution might conceivably
have taken place without the development of groups possessing the
characters of species" (_Essays_, v. 41). His palaeontological
researches ultimately led him to dispense with Darwin. In 1892 he wrote:
"The doctrine of evolution is no speculation, but a generalization of
certain facts ... classed by biologists under the heads of Embryology
and of Palaeontology" (_Essays_, v. 42). Earlier in 1881 he had asserted
even more emphatically that if the hypothesis of evolution "had not
existed, the palaeontologist would have had to invent it" (_Essays_, iv.
44).

From 1870 onwards he was more and more drawn away from scientific
research by the claims of public duty. Some men yield the more readily
to such demands, as their fulfilment is not unaccompanied by public
esteem. But he felt, as he himself said of Joseph Priestley, "that he
was a man and a citizen before he was a philosopher, and that the duties
of the two former positions are at least as imperative as those of the
latter" (_Essays_, iii. 13). From 1862 to 1884 he served on no less than
ten Royal Commissions, dealing in every case with subjects of great
importance, and in many with matters of the gravest moment to the
community. He held and filled with invariable dignity and distinction
more public positions than have perhaps ever fallen to the lot of a
scientific man in England. From 1871 to 1880 he was a secretary of the
Royal Society. From 1881 to 1885 he was president. For honours he cared
little, though they were within his reach; it is said that he might have
received a peerage. He accepted, however, in 1892, a Privy
Councillorship, at once the most democratic and the most aristocratic
honour accessible to an English citizen. In 1870 he was president of the
British Association at Liverpool, and in the same year was elected a
member of the newly constituted London School Board. He resigned the
latter position in 1872, but in the brief period during which he acted,
probably more than any man, he left his mark on the foundations of
national elementary education. He made war on the scholastic methods
which wearied the mind in merely taxing the memory; the children were to
be prepared to take their place worthily in the community. Physical
training was the basis; domestic economy, at any rate for girls, was
insisted upon, and for all some development of the aesthetic sense by
means of drawing and singing. Reading, writing and arithmetic were the
indispensable tools for acquiring knowledge, and intellectual discipline
was to be gained through the rudiments of physical science. He insisted
on the teaching of the Bible partly as a great literary heritage, partly
because he was "seriously perplexed to know by what practical measures
the religious feeling, which is the essential basis of conduct, was to
be kept up, in the present utterly chaotic state of opinion in these
matters, without its use" (_Essays_, iii. 397). In 1872 the School of
Mines was moved to South Kensington, and Huxley had, for the first time
after eighteen years, those appliances for teaching beyond the lecture
room, which to the lasting injury of the interests of biological science
in Great Britain had been withheld from him by the short-sightedness of
government. Huxley had only been able to bring his influence to bear
upon his pupils by oral teaching, and had had no opportunity by personal
intercourse in the laboratory of forming a school. He was now able to
organize a system of instruction for classes of elementary teachers in
the general principles of biology, which indirectly affected the
teaching of the subject throughout the country.

The first symptoms of physical failure to meet the strain of the
scientific and public duties demanded of him made some rest imperative,
and he took a long holiday in Egypt. He still continued for some years
to occupy himself mainly with vertebrate morphology. But he seemed to
find more interest and the necessary mental stimulus to exertion in
lectures, public addresses and more or less controversial writings. His
health, which had for a time been fairly restored, completely broke down
again in 1885. In 1890 he removed from London to Eastbourne, where after
a painful illness he died on the 29th of June 1895.

  The latter years of Huxley's life were mainly occupied with
  contributions to periodical literature on subjects connected with
  philosophy and theology. The effect produced by these on popular
  opinion was profound. This was partly due to his position as a man of
  science, partly to his obvious earnestness and sincerity, but in the
  main to his strenuous and attractive method of exposition. Such
  studies were not wholly new to him, as they had more or less engaged
  his thoughts from his earliest days. That his views exhibit some
  process of development and are not wholly consistent was, therefore,
  to be expected, and for this reason it is not easy to summarize them
  as a connected body of teaching. They may be found perhaps in their
  most systematic form in the volume on _Hume_ published in 1879.

  Huxley's general attitude to the problems of theology and philosophy
  was technically that of scepticism. "I am," he wrote, "too much of a
  sceptic to deny the possibility of anything" (_Life_, ii. 127). "Doubt
  is a beneficent demon" (_Essays_, ix. 56). He was anxious,
  nevertheless, to avoid the accusation of Pyrrhonism (_Life_, ii. 280),
  but the Agnosticism which he defined to express his position in 1869
  suggests the Pyrrhonist _Aphasia_. The only approach to certainty
  which he admitted lay in the order of nature. "The conception of the
  constancy of the order of nature has become the dominant idea of
  modern thought.... Whatever may be man's speculative doctrines, it is
  quite certain that every intelligent person guides his life and risks
  his fortune upon the belief that the order of nature is constant, and
  that the chain of natural causation is never broken." He adds,
  however, that "it by no means necessarily follows that we are
  justified in expanding this generalization into the infinite past"
  (_Essays_, iv. 47, 48). This was little more than a pious
  reservation, as evolution implies the principle of continuity (l.c. p.
  55). Later he stated his belief even more absolutely: "If there is
  anything in the world which I do firmly believe in, it is the
  universal validity of the law of causation, but that universality
  cannot be proved by any amount of experience" (_Essays_, ix. 121). The
  assertion that "There is only one method by which intellectual truth
  can be reached, whether the subject-matter of investigation belongs to
  the world of physics or to the world of consciousness" (_Essays_, ix.
  126) laid him open to the charge of materialism, which he vigorously
  repelled. His defence, when he rested it on the imperfection of the
  physical analysis of matter and force (l.c. p. 131), was irrelevant;
  he was on sounder ground when he contended with Berkeley "that our
  certain knowledge does not extend beyond our states of consciousness"
  (l.c. p. 130). "Legitimate materialism, that is, the extension of the
  conceptions and of the methods of physical science to the highest as
  well as to the lowest phenomena of vitality, is neither more nor less
  than a sort of shorthand idealism" (_Essays_, i. 194). While "the
  substance of matter is a metaphysical unknown quality of the existence
  of which there is no proof ... the non-existence of a substance of
  mind is equally arguable; ... the result ... is the reduction of the
  All to co-existences and sequences of phenomena beneath and beyond
  which there is nothing cognoscible" (_Essays_, ix. 66). Hume had
  defined a miracle as a "violation of the laws of nature." Huxley
  refused to accept this. While, on the one hand, he insists that "the
  whole fabric of practical life is built upon our faith in its
  continuity" (_Hume_, p. 129), on the other "nobody can presume to say
  what the order of nature must be"; this "knocks the bottom out of all
  a priori objections either to ordinary 'miracles' or to the efficacy
  of prayer" (_Essays_, v. 133). "If by the term miracles we mean only
  extremely wonderful events, there can be no just ground for denying
  the possibility of their occurrence" (_Hume_, p. 134). Assuming the
  chemical elements to be aggregates of uniform primitive matter, he saw
  no more theoretical difficulty in water being turned into alcohol in
  the miracle at Cana, than in sugar undergoing a similar conversion
  (_Essays_, v. 81). The credibility of miracles with Huxley is a
  question of evidence. It may be remarked that a scientific explanation
  is destructive of the supernatural character of a miracle, and that
  the demand for evidence may be so framed as to preclude the
  credibility of any historical event. Throughout his life theology had
  a strong attraction, not without elements of repulsion, for Huxley.
  The circumstances of his early training, when Paley was the "most
  interesting Sunday reading allowed him when a boy" (_Life_, ii. 57),
  probably had something to do with both. In 1860 his beliefs were
  apparently theistic: "Science seems to me to teach in the highest and
  strongest manner the great truth which is embodied in the Christian
  conception of entire surrender to the will of God" (_Life_, i. 219).
  In 1885 he formulates "the perfect ideal of religion" in a passage
  which has become almost famous: "In the 8th century B.C. in the heart
  of a world of idolatrous polytheists, the Hebrew prophets put forth a
  conception of religion which appears to be as wonderful an inspiration
  of genius as the art of Pheidias or the science of Aristotle. 'And
  what doth the Lord require of thee, but to do justly, and to love
  mercy, and to walk humbly with thy God'" (_Essays_, iv. 161). Two
  years later he was writing: "That there is no evidence of the
  existence of such a being as the God of the theologians is true
  enough" (_Life_, ii. 162). He insisted, however, that "atheism is on
  purely philosophical grounds untenable" (l.c.). His theism never
  really advanced beyond the recognition of "the passionless
  impersonality of the unknown and unknowable, which science shows
  everywhere underlying the thin veil of phenomena" (_Life_, i. 239). In
  other respects his personal creed was a kind of scientific Calvinism.
  There is an interesting passage in an essay written in 1892, "An
  Apologetic Eirenicon," which has not been republished, which
  illustrates this: "It is the secret of the superiority of the best
  theological teachers to the majority of their opponents that they
  substantially recognize these realities of things, however strange the
  forms in which they clothe their conceptions. The doctrines of
  predestination, of original sin, of the innate depravity of man and
  the evil fate of the greater part of the race, of the primacy of Satan
  in this world, of the essential vileness of matter, of a malevolent
  Demiurgus subordinate to a benevolent Almighty, who has only lately
  revealed himself, faulty as they are, appear to me to be vastly nearer
  the truth than the 'liberal' popular illusions that babies are all
  born good, and that the example of a corrupt society is responsible
  for their failure to remain so; that it is given to everybody to reach
  the ethical ideal if he will only try; that all partial evil is
  universal good, and other optimistic figments, such as that which
  represents 'Providence' under the guise of a paternal philanthropist,
  and bids us believe that everything will come right (according to our
  notions) at last." But his "slender definite creed," R. H. Hutton, who
  was associated with him in the Metaphysical Society, thought--and no
  doubt rightly--in no respect "represented the cravings of his larger
  nature."

  From 1880 onwards till the very end of his life, Huxley was
  continuously occupied in a controversial campaign against orthodox
  beliefs. As Professor W. F. R. Weldon justly said of his earlier
  polemics: "They were certainly among the principal agents in winning a
  larger measure of toleration for the critical examination of
  fundamental beliefs, and for the free expression of honest reverent
  doubt." He threw Christianity overboard bodily and with little
  appreciation of its historic effect as a civilizing agency. He
  thought that "the exact nature of the teachings and the convictions of
  Jesus is extremely uncertain" (_Essays_, v. 348). "What we are usually
  pleased to call religion nowadays is, for the most part, Hellenized
  Judaism" (_Essays_, iv. 162). His final analysis of what "since the
  second century, has assumed to itself the title of Orthodox
  Christianity" is a "varying compound of some of the best and some of
  the worst elements of Paganism and Judaism, moulded in practice by the
  innate character of certain people of the Western world" (_Essays_, v.
  142). He concludes "That this Christianity is doomed to fall is, to my
  mind, beyond a doubt; but its fall will neither be sudden nor speedy"
  (l.c.). He did not omit, however, to do justice to "the bright side of
  Christianity," and was deeply impressed with the life of Catherine of
  Siena. Failing Christianity, he thought that some other "hypostasis of
  men's hopes" will arise (_Essays_, v. 254). His latest speculations on
  ethical problems are perhaps the least satisfactory of his writings.
  In 1892 he wrote: "The moral sense is a very complex affair--dependent
  in part upon associations of pleasure and pain, approbation and
  disapprobation, formed by education in early youth, but in part also
  on an innate sense of moral beauty and ugliness (how originated need
  not be discussed), which is possessed by some people in great
  strength, while some are totally devoid of it" (_Life_, ii. 305). This
  is an intuitional theory, and he compares the moral with the aesthetic
  sense, which he repeatedly declares to be intuitive; thus: "All the
  understanding in the world will neither increase nor diminish the
  force of the intuition that this is beautiful and this is ugly"
  (_Essays_, ix. 80). In the Romanes Lecture delivered in 1894, in which
  this passage occurs, he defines "law and morals" to be "restraints
  upon the struggle for existence between men in society." It follows
  that "the ethical process is in opposition to the cosmic process," to
  which the struggle for existence belongs (_Essays_, ix. 31).
  Apparently he thought that the moral sense in its origin was
  intuitional and in its development utilitarian. "Morality commenced
  with society" (_Essays_, v. 52). The "ethical process" is the "gradual
  strengthening of the social bond" (_Essays_, ix. 35). "The cosmic
  process has no sort of relation to moral ends" (l.c. p. 83); "of moral
  purpose I see no trace in nature. That is an article of exclusive
  human manufacture" (_Life_, ii. 268). The cosmic process Huxley
  identified with evil, and the ethical process with good; the two are
  in necessary conflict. "The reality at the bottom of the doctrine of
  original sin" is the "innate tendency to self-assertion" inherited by
  man from the cosmic order (_Essays_, ix. 27). "The actions we call
  sinful are part and parcel of the struggle for existence" (_Life_, ii.
  282). "The prospect of attaining untroubled happiness" is "an
  illusion" (_Essays_, ix. 44), and the cosmic process in the long run
  will get the best of the contest, and "resume its sway" when evolution
  enters on its downward course (l.c. p. 45). This approaches pure
  pessimism, and though in Huxley's view the "pessimism of Schopenhauer
  is a nightmare" (_Essays_, ix. 200), his own philosophy of life is not
  distinguishable, and is often expressed in the same language. The
  cosmic order is obviously non-moral (_Essays_, ix. 197). That it is,
  as has been said, immoral is really meaningless. Pain and suffering
  are affections which imply a complex nervous organization, and we are
  not justified in projecting them into nature external to ourselves.
  Darwin and A. R. Wallace disagreed with Huxley in seeing rather the
  joyous than the suffering side of nature. Nor can it be assumed that
  the descending scale of evolution will reproduce the ascent, or that
  man will ever be conscious of his doom.

  As has been said, Huxley never thoroughly grasped the Darwinian
  principle. He thought "transmutation may take place without
  transition" (_Life_, i. 173). In other words, that evolution is
  accomplished by leaps and not by the accumulation of small variations.
  He recognized the "struggle for existence" but not the gradual
  adjustment of the organism to its environment which is implied in
  "natural selection." In highly civilized societies he thought that the
  former was at an end (_Essays_, ix. 36) and had been replaced by the
  "struggle for enjoyment" (l.c. p. 40). But a consideration of the
  stationary population of France might have shown him that the effect
  in the one case may be as restrictive as in the other. So far from
  natural selection being in abeyance under modern social conditions,
  "it is," as Professor Karl Pearson points out, "something we run up
  against at once, almost as soon as we examine a mortality table"
  (_Biometrika_, i. 76). The inevitable conclusion, whether we like it
  or not, is that the future evolution of humanity is as much a part of
  the cosmic process as its past history, and Huxley's attempt to shut
  the door on it cannot be maintained scientifically.

  AUTHORITIES.--_Life and Letters of Thomas Henry Huxley_, by his son
  Leonard Huxley (2 vols., 1900); _Scientific Memoirs of T. H. Huxley_
  (4 vols., 1898-1901); _Collected Essays_ by T. H. Huxley (9 vols.,
  1898); _Thomas Henry Huxley, a Sketch of his Life and Work_, by P.
  Chalmers Mitchell, M.A. (Oxon., 1900); a critical study founded on
  careful research and of great value.     (W. T. T.-D.)


FOOTNOTE:

  [1] _Nature_, lxiii. 127.




HUY (Lat. _Hoium_, and Flem. _Hoey_), a town of Belgium, on the right
bank of the Meuse, at the point where it is joined by the Hoyoux. Pop.
(1904), 14,164. It is 19 m. E. of Namur and a trifle less west of Liége.
Huy certainly dates from the 7th century, and, according to some, was
founded by the emperor Antoninus in A.D. 148. Its situation is
striking, with its grey citadel crowning a grey rock, and the fine
collegiate church (with a 13th-century gateway) of Notre Dame built
against it. The citadel is now used partly as a depot of military
equipment and partly as a prison. The ruins are still shown of the abbey
of Neumoustier founded by Peter the Hermit on his return from the first
crusade. He was buried there in 1115, and a statue was erected to his
memory in the abbey grounds in 1858. Neumoustier was one of seventeen
abbeys in this town alone dependent on the bishopric of Liége. Huy is
surrounded by vineyards, and the bridge which crosses the Meuse at this
point connects the fertile Hesbaye north of the river with the rocky and
barren Condroz south of it.




HUYGENS, CHRISTIAAN (1629-1695), Dutch mathematician, mechanician,
astronomer and physicist, was born at the Hague on the 14th of April
1629. He was the second son of Sir Constantijn Huygens. From his father
he received the rudiments of his education, which was continued at
Leiden under A. Vinnius and F. van Schooten, and completed in the
juridical school of Breda. His mathematical bent, however, soon diverted
him from legal studies, and the perusal of some of his earliest theorems
enabled Descartes to predict his future greatness. In 1649 he
accompanied the mission of Henry, count of Nassau, to Denmark, and in
1651 entered the lists of science as an assailant of the unsound system
of quadratures adopted by Gregory of St Vincent. This first essay
(_Exetasis quadraturae circuli_, Leiden, 1651) was quickly succeeded by
his _Theoremata de quadratura hyperboles, ellipsis, et circuli_; while,
in a treatise entitled _De circuli magnitudine inventa_, he made, three
years later, the closest approximation so far obtained to the ratio of
the circumference to the diameter of a circle.

Another class of subjects was now to engage his attention. The
improvement of the telescope was justly regarded as a _sine qua non_ for
the advancement of astronomical knowledge. But the difficulties
interposed by spherical and chromatic aberration had arrested progress
in that direction until, in 1655, Huygens, working with his brother
Constantijn, hit upon a new method of grinding and polishing lenses. The
immediate results of the clearer definition obtained were the detection
of a satellite to Saturn (the sixth in order of distance from its
primary), and the resolution into their true form of the abnormal
appendages to that planet. Each discovery in turn was, according to the
prevailing custom, announced to the learned world under the veil of an
anagram--removed, in the case of the first, by the publication, early in
1656, of the little tract _De Saturni luna observatio nova_; but
retained, as regards the second, until 1659, when in the _Systema
Saturnium_ the varying appearances of the so-called "triple planet" were
clearly explained as the phases of a ring inclined at an angle of 28° to
the ecliptic. Huygens was also in 1656 the first effective observer of
the Orion nebula; he delineated the bright region still known by his
name, and detected the multiple character of its nuclear star. His
application of the pendulum to regulate the movement of clocks sprang
from his experience of the need for an exact measure of time in
observing the heavens. The invention dates from 1656; on the 16th of
June 1657 Huygens presented his first "pendulum-clock" to the
states-general; and the _Horologium_, containing a description of the
requisite mechanism, was published in 1658.

His reputation now became cosmopolitan. As early as 1655 the university
of Angers had distinguished him with an honorary degree of doctor of
laws. In 1663, on the occasion of his second visit to England, he was
elected a fellow of the Royal Society, and imparted to that body in
January 1669 a clear and concise statement of the laws governing the
collision of elastic bodies. Although these conclusions were arrived at
independently, and, as it would seem, several years previous to their
publication, they were in great measure anticipated by the
communications on the same subject of John Wallis and Christopher Wren,
made respectively in November and December 1668.

Huygens had before this time fixed his abode in France. In 1665 Colbert
made to him on behalf of Louis XIV. an offer too tempting to be
refused, and between the following year and 1681 his residence in the
philosophic seclusion of the Bibliothèque du Roi was only interrupted by
two short visits to his native country. His _magnum opus_ dates from
this period. The _Horologium oscillatorium_, published with a dedication
to his royal patron in 1673, contained original discoveries sufficient
to have furnished materials for half a dozen striking disquisitions. His
solution of the celebrated problem of the "centre of oscillation" formed
in itself an important event in the history of mechanics. Assuming as an
axiom that the centre of gravity of any number of interdependent bodies
cannot rise higher than the point from which it fell, he arrived, by
anticipating in the particular case the general principle of the
conservation of _vis viva_, at correct although not strictly
demonstrated conclusions. His treatment of the subject was the first
successful attempt to deal with the dynamics of a system. The
determination of the true relation between the length of a pendulum and
the time of its oscillation; the invention of the theory of evolutes;
the discovery, hence ensuing, that the cycloid is its own evolute, and
is strictly isochronous; the ingenious although practically inoperative
idea of correcting the "circular error" of the pendulum by applying
cycloidal cheeks to clocks--were all contained in this remarkable
treatise. The theorems on the composition of forces in circular motion
with which it concluded formed the true prelude to Newton's _Principia_,
and would alone suffice to establish the claim of Huygens to the highest
rank among mechanical inventors.

In 1681 he finally severed his French connexions, and returned to
Holland. The harsher measures which about that time began to be adopted
towards his co-religionists in France are usually assigned as the motive
of this step. He now devoted himself during six years to the production
of lenses of enormous focal distance, which, mounted on high poles, and
connected with the eye-piece by means of a cord, formed what were called
"aerial telescopes." Three of his object-glasses, of respectively 123,
180 and 210 ft. focal length, are in the possession of the Royal
Society. He also succeeded in constructing an almost perfectly
achromatic eye-piece, still known by his name. But his researches in
physical optics constitute his chief title-deed to immortality. Although
Robert Hooke in 1668 and Ignace Pardies in 1672 had adopted a vibratory
hypothesis of light, the conception was a mere floating possibility
until Huygens provided it with a sure foundation. His powerful
scientific imagination enabled him to realize that all the points of a
wave-front originate partial waves, the aggregate effect of which is to
reconstitute the primary disturbance at the subsequent stages of its
advance, thus accomplishing its propagation; so that each primary
undulation is the envelope of an indefinite number of secondary
undulations. This resolution of the original wave is the well-known
"Principle of Huygens," and by its means he was enabled to prove the
fundamental laws of optics, and to assign the correct construction for
the direction of the extraordinary ray in uniaxial crystals. These
investigations, together with his discovery of the "wonderful
phenomenon" of polarization, are recorded in his _Traité de la lumière_,
published at Leiden in 1690, but composed in 1678. In the appended
treatise _Sur la Cause de la pesanteur_, he rejected gravitation as a
universal quality of matter, although admitting the Newtonian theory of
the planetary revolutions. From his views on centrifugal force he
deduced the oblate figure of the earth, estimating its compression,
however, at little more than one-half its actual amount.

Huygens never married. He died at the Hague on the 8th of June 1695,
bequeathing his manuscripts to the university of Leiden, and his
considerable property to the sons of his younger brother. In character
he was as estimable as he was brilliant in intellect. Although, like
most men of strong originative power, he assimilated with difficulty the
ideas of others, his tardiness sprang rather from inability to depart
from the track of his own methods than from reluctance to acknowledge
the merits of his competitors.

  In addition to the works already mentioned, his _Cosmotheoros_--a
  speculation concerning the inhabitants of the planets--was
  printed posthumously at the Hague in 1698, and appeared almost
  simultaneously in an English translation. A volume entitled _Opera
  posthuma_ (Leiden, 1703) contained his "Dioptrica," in which the ratio
  between the respective focal lengths of object-glass and eye-glass is
  given as the measure of magnifying power, together with the shorter
  essays _De vitris figurandis_, _De corona et parheliis_, &c. An early
  tract _De ratiociniis in ludo aleae_, printed in 1657 with Schooten's
  _Exercitationes mathematicae_, is notable as one of the first formal
  treatises on the theory of probabilities; nor should his
  investigations of the properties of the cissoid, logarithmic and
  catenary curves be left unnoticed. His invention of the spiral
  watch-spring was explained in the _Journal des savants_ (Feb. 25,
  1675). An edition of his works was published by G. J.'s Gravesande, in
  four quarto volumes entitled _Opera varia_ (Leiden, 1724) and _Opera
  reliqua_ (Amsterdam, 1728). His scientific correspondence was edited
  by P. J. Uylenbroek from manuscripts preserved at Leiden, with the
  title _Christiani Hugenii aliorumque seculi XVII. virorum celebrium
  exercitationes mathematicae et philosophicae_ (the Hague, 1833).

  The publication of a monumental edition of the letters and works of
  Huygens was undertaken at the Hague by the _Société Hollandaise des
  Sciences_, with the heading _Oeuvres de Christian Huygens_ (1888),
  &c. Ten quarto volumes, comprising the whole of his correspondence,
  had already been issued in 1905. A biography of Huygens was prefixed
  to his _Opera varia_ (1724); his _Éloge_ in the character of a French
  academician was printed by J. A. N. Condorcet in 1773. Consult
  further: P. J. Uylenbroek, _Oratio de fratribus Christiano atque
  Constantino Hugenio_ (Groningen, 1838); P. Harting, _Christiaan
  Huygens in zijn Leven en Werken geschetzt_ (Groningen, 1868); J. B. J.
  Delambre, _Hist. de l'astronomie moderne_ (ii. 549); J. E. Montucla,
  _Hist. des mathématiques_ (ii. 84, 412, 549); M. Chasles, _Aperçu
  historique sur l'origine des méthodes en géometrie_, pp. 101-109; E.
  Dühring, _Kritische Geschichte der allgemeinen Principien der
  Mechanik_, Abschnitt (ii. 120, 163, iii. 227); A. Berry, _A Short
  History of Astronomy_, p. 200; R. Wolf, _Geschichte der Astronomie_,
  passim; Houzeau, _Bibliographie astronomique_ (ii. 169); F. Kaiser,
  _Astr. Nach._ (xxv. 245, 1847); _Tijdschrift voor de Wetenschappen_
  (i. 7, 1848); _Allgemeine deutsche Biographie_ (M. B. Cantor); J. C.
  Poggendorff, _Biog. lit. Handwörterbuch_.     (A. M. C.)




HUYGENS, SIR CONSTANTIJN (1596-1687), Dutch poet and diplomatist, was
born at the Hague on the 4th of September 1596. His father, Christiaan
Huygens, was secretary to the state council, and a man of great
political importance. At the baptism of the child, the city of Breda was
one of his sponsors, and the admiral Justinus van Nassau the other. He
was trained in every polite accomplishment, and before he was seven
could speak French with fluency. He was taught Latin by Johannes
Dedelus, and soon became a master of classic versification. He developed
not only extraordinary intellectual gifts but great physical beauty and
strength, and was one of the most accomplished athletes and gymnasts of
his age; his skill in playing the lute and in the arts of painting and
engraving attracted general attention before he began to develop his
genius as a writer. In 1616 he proceeded, with his elder brother, to the
university of Leiden. He stayed there only one year, and in 1618 went to
London with the English ambassador Dudley Carleton; he remained in
London for some months, and then went to Oxford, where he studied for
some time in the Bodleian Library, and to Woodstock, Windsor and
Cambridge; he was introduced at the English court, and played the lute
before James I. The most interesting feature of this visit was the
intimacy which sprang up between the young Dutch poet and Dr Donne, for
whose genius Huygens preserved through life an unbounded admiration. He
returned to Holland in company with the English contingent of the synod
of Dort, and in 1619 he proceeded to Venice in the diplomatic service of
his country; on his return he nearly lost his life by a foolhardy
exploit, namely, the scaling of the topmost spire of Strassburg
cathedral. In 1621 he published one of his most weighty and popular
poems, his _Batava Tempe_, and in the same year he proceeded again to
London, as secretary to the ambassador, Wijngaerdan, but returned in
three months. His third diplomatic visit to England lasted longer, from
the 5th of December 1621 to the 1st of March 1623. During his absence,
his volume of satires, _'t Costelick Mal_, dedicated to Jacob Cats,
appeared at the Hague. In the autumn of 1622 he was knighted by James I.
He published a large volume of miscellaneous poems in 1625 under the
title of _Otiorum libri sex_; and in the same year he was appointed
private secretary to the stadholder. In 1627 Huygens married Susanna
van Baerle, and settled at the Hague; four sons and a daughter were born
to them. In 1630 Huygens was called to a seat in the privy council, and
he continued to exercise political power with wisdom and vigour for many
years, under the title of the lord of Zuylichem. In 1634 he is supposed
to have completed his long-talked-of version of the poems of Donne,
fragments of which exist. In 1637 his wife died, and he immediately
began to celebrate the virtues and pleasures of their married life in
the remarkable didactic poem called _Dagwerck_, which was not published
till long afterwards. From 1639 to 1641 he occupied himself by building
a magnificent house and garden outside the Hague, and by celebrating
their beauties in a poem entitled _Hofwijck_, which was published in
1653. In 1647 he wrote his beautiful poem of _Oogentroost_ or "Eye
Consolation," to gratify his blind friend Lucretia van Trollo. He made
his solitary effort in the dramatic line in 1657, when he brought out
his comedy of _Trijntje Cornelis Klacht_, which deals, in rather broad
humour, with the adventures of the wife of a ship's captain at Zaandam.
In 1658 he rearranged his poems, and issued them with many additions,
under the title of _Corn Flowers_. He proposed to the government that
the present highway from the Hague to the sea at Scheveningen should be
constructed, and during his absence on a diplomatic mission to the
French court in 1666 the road was made as a compliment to the venerable
statesman, who expressed his gratitude in a descriptive poem entitled
_Zeestraet_. Huygens edited his poems for the last time in 1672, and
died in his ninety-first year, on the 28th of March 1687. He was buried,
with the pomp of a national funeral, in the church of St Jacob, on the
4th of April. His second son, Christiaan, the eminent astronomer, is
noticed separately.

  Constantijn Huygens is the most brilliant figure in Dutch literary
  history. Other statesmen surpassed him in political influence, and at
  least two other poets surpassed him in the value and originality of
  their writings. But his figure was more dignified and splendid, his
  talents were more varied, and his general accomplishments more
  remarkable than those of any other person of his age, the greatest age
  in the history of the Netherlands. Huygens is the _grand seigneur_ of
  the republic, the type of aristocratic oligarchy, the jewel and
  ornament of Dutch liberty. When we consider his imposing character and
  the positive value of his writings, we may well be surprised that he
  has not found a modern editor. It is a disgrace to Dutch scholarship
  that no complete collection of the writings of Huygens exists. His
  autobiography, _De vita propria sermonum libri duo_, did not see the
  light until 1817, and his remarkable poem, _Cluyswerck_, was not
  printed until 1841. As a poet Huygens shows a finer sense of form than
  any other early Dutch writer; the language, in his hands, becomes as
  flexible as Italian. His epistles and lighter pieces, in particular,
  display his metrical ease and facility to perfection.     (E. G.)




HUYSMANS, the name of four Flemish painters who matriculated in the
Antwerp gild in the 17th century. Cornelis the elder, apprenticed in
1633, passed for a mastership in 1636, and remained obscure. Jacob,
apprenticed to Frans Wouters in 1650, wandered to England towards the
close of the reign of Charles II., and competed with Lely as a
fashionable portrait painter. He executed a portrait of the queen,
Catherine of Braganza, now in the national portrait gallery, and Horace
Walpole assigns to him the likeness of Lady Bellasys, catalogued at
Hampton Court as a work of Lely. His portrait of Izaak Walton in the
National Gallery shows a disposition to imitate the styles of Rubens and
Van Dyke. According to most accounts he died in London in 1696. Jan
Baptist Huysmans, born at Antwerp in 1654, matriculated in 1676-1677,
and died there in 1715-1716. He was younger brother to Cornelis Huysmans
the second, who was born at Antwerp in 1648, and educated by Gaspar de
Wit and Jacob van Artois. Of Jan Baptist little or nothing has been
preserved, except that he registered numerous apprentices at Antwerp,
and painted a landscape dated 1697 now in the Brussels museum. Cornelis
the second is the only master of the name of Huysmans whose talent was
largely acknowledged. He received lessons from two artists, one of whom
was familiar with the Roman art of the Poussins, whilst the other
inherited the scenic style of the school of Rubens. He combined the two
in a rich, highly coloured, and usually effective style, which, however,
was not free from monotony. Seldom attempting anything but woodside
views with fancy backgrounds, half Italian, half Flemish, he painted
with great facility, and left numerous examples behind. At the outset of
his career he practised at Malines, where he married in 1682, and there
too he entered into some business connexion with van der Meulen, for
whom he painted some backgrounds. In 1706 he withdrew to Antwerp, where
he resided till 1717, returning then to Malines, where he died on the
1st of June 1727.

  Though most of his pictures were composed for cabinets rather than
  churches, he sometimes emulated van Artois in the production of large
  sacred pieces, and for many years his "Christ on the Road to Emmaus"
  adorned the choir of Notre Dame of Malines. In the gallery of Nantes,
  where three of his small landscapes are preserved, there hangs an
  "Investment of Luxembourg," by van der Meulen, of which he is known to
  have laid in the background. The national galleries of London and
  Edinburgh contain each one example of his skill. Blenheim, too, and
  other private galleries in England, possess one or more of his
  pictures. But most of his works are on the European continent.




HUYSMANS, JORIS KARL (1848-1907), French novelist, was born at Paris on
the 5th of February 1848. He belonged to a family of artists of Dutch
extraction; he entered the ministry of the interior, and was pensioned
after thirty years' service. His earliest venture in literature, _Le
Drageoir à épices_ (1874), contained stories and short prose poems
showing the influence of Baudelaire. _Marthe_ (1876), the life of a
courtesan, was published in Brussels, and Huysmans contributed a story,
"Sac au dos," to _Les Soirées de Médan_, the collection of stories of
the Franco-German war published by Zola. He then produced a series of
novels of everyday life, including _Les Soeurs Vatard_ (1879), _En
Ménage_ (1881), and _À vau-l'eau_ (1882), in which he outdid Zola in
minute and uncompromising realism. He was influenced, however, more
directly by Flaubert and the brothers de Goncourt than by Zola. In
_L'Art moderne_ (1883) he gave a careful study of impressionism and in
_Certains_ (1889) a series of studies of contemporary artists, _À
Rebours_ (1884), the history of the morbid tastes of a decadent
aristocrat, des Esseintes, created a literary sensation, its caricature
of literary and artistic symbolism covering much of the real beliefs of
the leaders of the aesthetic revolt. In _Là-Bas_ Huysmans's most
characteristic hero, Durtal, makes his appearance. Durtal is occupied in
writing the life of Gilles de Rais; the insight he gains into Satanism
is supplemented by modern Parisian students of the black art; but
already there are signs of a leaning to religion in the sympathetic
figures of the religious bell-ringer of Saint Sulpice and his wife. _En
Route_ (1895) relates the strange conversion of Durtal to mysticism and
Catholicism in his retreat to La Trappe. In _La Cathédrale_ (1898),
Huysmans's symbolistic interpretation of the cathedral of Chartres, he
develops his enthusiasm for the purity of Catholic ritual. The life of
_Sainte Lydwine de Schiedam_ (1901), an exposition of the value of
suffering, gives further proof of his conversion; and _L'Oblat_ (1903)
describes Durtal's retreat to the Val des Saints, where he is attached
as an oblate to a Benedictine monastery. Huysmans was nominated by
Edmond de Goncourt as a member of the Académie des Goncourt. He died as
a devout Catholic, after a long illness of cancer in the palate on the
13th of May 1907. Before his death he destroyed his unpublished MSS. His
last book was _Les Foules de Lourdes_ (1906).

  See Arthur Symons, _Studies in two Literatures_ (1897) and _The
  Symbolist Movement in Literature_ (1899); Jean Lionnet in _L'Évolution
  des idées_ (1903); Eugène Gilbert in _France et Belgique_ (1905); J.
  Sargeret in _Les Grands convertis_ (1906).




HUYSUM, JAN VAN (1682-1749), Dutch painter, was born at Amsterdam in
1682, and died in his native city on the 8th of February 1749. He was
the son of Justus van Huysum, who is said to have been expeditious in
decorating doorways, screens and vases. A picture by this artist is
preserved in the gallery of Brunswick, representing Orpheus and the
Beasts in a wooded landscape, and here we have some explanation of his
son's fondness for landscapes of a conventional and Arcadian kind; for
Jan van Huysum, though skilled as a painter of still life, believed
himself to possess the genius of a landscape painter. Half his pictures
in public galleries are landscapes, views of imaginary lakes and
harbours with impossible ruins and classic edifices, and woods of tall
and motionless trees--the whole very glossy and smooth, and entirely
lifeless. The earliest dated work of this kind is that of 1717, in the
Louvre, a grove with maidens culling flowers near a tomb, ruins of a
portico, and a distant palace on the shores of a lake bounded by
mountains.

It is doubtful whether any artist ever surpassed van Huysum in
representing fruit and flowers. It has been said that his fruit has no
savour and his flowers have no perfume--in other words, that they are
hard and artificial--but this is scarcely true. In substance fruit and
flower are delicate and finished imitations of nature in its more subtle
varieties of matter. The fruit has an incomparable blush of down, the
flowers have a perfect delicacy of tissue. Van Huysum, too, shows
supreme art in relieving flowers of various colours against each other,
and often against a light and transparent background. He is always
bright, sometimes even gaudy. Great taste and much grace and elegance
are apparent in the arrangement of bouquets and fruit in vases adorned
with bas reliefs or in baskets on marble tables. There is exquisite and
faultless finish everywhere. But what van Huysum has not is the breadth,
the bold effectiveness, and the depth of thought of de Heem, from whom
he descends through Abraham Mignon.

  Some of the finest of van Huysum's fruit and flower pieces have been
  in English private collections: those of 1723 in the earl of
  Ellesmere's gallery, others of 1730-1732 in the collections of Hope
  and Ashburton. One of the best examples is now in the National Gallery
  (1736-1737). No public museum has finer and more numerous specimens
  than the Louvre, which boasts of four landscapes and six panels with
  still life; then come Berlin and Amsterdam with four fruit and flower
  pieces; then St Petersburg, Munich, Hanover, Dresden, the Hague,
  Brunswick, Vienna, Carlsruhe and Copenhagen.




HWANG HO [HOANG HO], the second largest river in China. It is known to
foreigners as the Yellow river--a name which is a literal translation of
the Chinese. It rises among the Kuenlun mountains in central Asia, its
head-waters being in close proximity to those of the Yangtsze-Kiang. It
has a total length of about 2400 m. and drains an area of approximately
400,000 sq. m. The main stream has its source in two lakes named
Tsaring-nor and Oring-nor, lying about 35° N., 97° E., and after flowing
with a south-easterly course it bends sharply to the north-west and
north, entering China in the province of Kansuh in lat. 36°. After
passing Lanchow-fu, the capital of this province, the river takes an
immense sweep to the north and north-east, until it encounters the
rugged barrier ranges that here run north and south through the
provinces of Shansi and Chihli. By these ranges it is forced due south
for 500 m., forming the boundary between the provinces of Shansi and
Shensi, until it finds an outlet eastwards at Tung Kwan--a pass which
for centuries has been renowned as the gate of Asia, being indeed the
sole commercial passage between central China and the West. At Tung Kwan
the river is joined by its only considerable affluent in China proper,
the Wei (Wei-ho), which drains the large province of Shensi, and the
combined volume of water continues its way at first east and then
north-east across the great plain to the sea. At low water in the winter
season the discharge is only about 36,000 cub. ft. per second, whereas
during the summer flood it reaches 116,000 ft. or more. The amount of
sediment carried down is very large, though no accurate observations
have been made. In the account of Lord Macartney's embassy, which
crossed the Yellow river in 1792, it was calculated to be 17,520 million
cub. ft. a year, but this is considered very much over the mark. Two
reasons, however, combine to render it probable that the sedimentary
matter is very large in proportion to the volume of water: the first
being the great fall, and the consequently rapid current over two-thirds
of the river's course; the second that the drainage area is nearly all
covered with deposits of loess, which, being very friable, readily gives
way before the rainfall and is washed down in large quantity. The
ubiquity of this loess or yellow earth, as the Chinese call it, has in
fact given its name both to the river which carries it in solution and
to the sea (the Yellow Sea) into which it is discharged. It is
calculated by Dr Guppy (_Journal of China Branch of Royal Asiatic
Society_, vol. xvi.) that the sediment brought down by the three
northern rivers of China, viz., the Yangtsze, the Hwang-ho and the
Peiho, is 24,000 million cub. ft. per annum, and is sufficient to fill
up the whole of the Yellow Sea and the Gulf of Pechili in the space of
about 36,000 years.

  Unlike the Yangtsze, the Hwang-ho is of no practical value for
  navigation. The silt and sand form banks and bars at the mouth, the
  water is too shallow in winter and the current is too strong in
  summer, and, further, the bed of the river is continually shifting. It
  is this last feature which has earned for the river the name "China's
  sorrow." As the silt-laden waters debouch from the rocky bed of the
  upper reaches on to the plains, the current slackens, and the coarser
  detritus settles on the bottom. By degrees the bed rises, and the
  people build embankments to prevent the river from overflowing. As the
  bed rises the embankments must be raised too, until the stream is
  flowing many feet above the level of the surrounding country. As time
  goes on the situation becomes more and more dangerous; finally, a
  breach occurs, and the whole river pours over the country, carrying
  destruction and ruin with it. If the breach cannot be repaired the
  river leaves its old channel entirely and finds a new exit to the sea
  along the line of least resistance. Such in brief has been the story
  of the river since the dawn of Chinese history. At various times it
  has discharged its waters alternately on one side or the other of the
  great mass of mountains forming the promontory of Shantung, and by
  mouths as far apart from each other as 500 m. At each change it has
  worked havoc and disaster by covering the cultivated fields with 2 or
  3 ft. of sand and mud.

  A great change in the river's course occurred in 1851, when a breach
  was made in the north embankment near Kaifengfu in Honan. At this
  point the river bed was some 25 ft. above the plain; the water
  consequently forsook the old channel entirely and poured over the
  level country, finally seizing on the bed of a small river called the
  Tsing, and thereby finding an exit to the sea. Since that time the new
  channel thus carved out has remained the proper course of the river,
  the old or southerly channel being left quite dry. It required some
  fifteen or more years to repair damages from this outbreak, and to
  confine the stream by new embankments. After that there was for a time
  comparative immunity from inundations, but in 1882 fresh outbursts
  again began. The most serious of all took place in 1887, when it
  appeared probable that there would be again a permanent change in the
  river's course. By dint of great exertions, however, the government
  succeeded in closing the breach, though not till January 1889, and not
  until there had been immense destruction of life and property. The
  outbreak on this occasion occurred, as all the more serious outbreaks
  have done, in Honan, a few miles west of the city of Kaifengfu. The
  stream poured itself over the level and fertile country to the
  southwards, sweeping whole villages before it, and converting the
  plain into one vast lake. The area affected was not less than 50,000
  sq. m. and the loss of life was computed at over one million. Since
  1887 there have been a series of smaller outbreaks, mostly at points
  lower down and in the neighbourhood of Chinanfu, the capital of
  Shantung. These perpetually occurring disasters entail a heavy expense
  on the government; and from the mere pecuniary point of view it would
  well repay them to call in the best foreign engineering skill
  available, an expedient, however, which has not commended itself to
  the Chinese authorities.     (G. J.)




HWICCE, one of the kingdoms of Anglo-Saxon Britain. Its exact dimensions
are unknown; they probably coincided with those of the old diocese of
Worcester, the early bishops of which bore the title "Episcopus
Hwicciorum." It would therefore include Worcestershire, Gloucestershire
except the Forest of Dean, the southern half of Warwickshire, and the
neighbourhood of Bath. The name Hwicce survives in Wychwood in
Oxfordshire and Whichford in Warwickshire. These districts, or at all
events the southern portion of them, were according to the _Anglo-Saxon
Chronicle_, _s.a._ 577, originally conquered by the West Saxons under
Ceawlin. In later times, however, the kingdom of the Hwicce appears to
have been always subject to Mercian supremacy, and possibly it was
separated from Wessex in the time of Edwin. The first kings of whom we
read were two brothers, Eanhere and Eanfrith, probably contemporaries of
Wulfhere. They were followed by a king named Osric, a contemporary of
Æthelred, and he by a king Oshere. Oshere had three sons who reigned
after him, Æthelheard, Æthelweard and Æthelric. The two last named
appear to have been reigning in the year 706. At the beginning of Offa's
reign we again find the kingdom ruled by three brothers, named Eanberht,
Uhtred and Aldred, the two latter of whom lived until about 780. After
them the title of king seems to have been given up. Their successor
Æthelmund, who was killed in a campaign against Wessex in 802, is
described only as an earl. The district remained in possession of the
rulers of Mercia until the fall of that kingdom. Together with the rest
of English Mercia it submitted to King Alfred about 877-883 under Earl
Æthelred, who possibly himself belonged to the Hwicce. No genealogy or
list of kings has been preserved, and we do not know whether the dynasty
was connected with that of Wessex or Mercia.

  See Bede, _Historia eccles._ (edited by C. Plummer) iv. 13 (Oxford,
  1896); W. de G. Birch, _Cartularium Saxonicum_, 43, 51, 76, 85, 116,
  117, 122, 163, 187, 232, 233, 238 (Oxford, 1885-1889).
       (F. G. M. B.)




HYACINTH (Gr. hyakinthos), also called JACINTH (through Ital.
_giacinto_), one of the most popular of spring garden flowers. It was in
cultivation prior to 1597, at which date it is mentioned by Gerard. Rea
in 1665 mentions several single and double varieties as being then in
English gardens, and Justice in 1754 describes upwards of fifty
single-flowered varieties, and nearly one hundred double-flowered ones,
as a selection of the best from the catalogues of two then celebrated
Dutch growers. One of the Dutch sorts, called La Reine de Femmes, a
single white, is said to have produced from thirty-four to thirty-eight
flowers in a spike, and on its first appearance to have sold for 50
guilders a bulb; while one called Overwinnaar, or Conqueror, a double
blue, sold at first for 100 guilders, Gloria Mundi for 500 guilders, and
Koning Saloman for 600 guilders. Several sorts are at that date
mentioned as blooming well in water-glasses. Justice relates that he
himself raised several very valuable double-flowered kinds from seeds,
which many of the sorts he describes are noted for producing freely.

The original of the cultivated hyacinth, _Hyacinthus orientalis_, a
native of Greece and Asia Minor, is by comparison an insignificant
plant, bearing on a spike only a few small, narrow-lobed, washy blue
flowers, resembling in form those of our common blue-bell. So great has
been the improvement effected by the florists, and chiefly by the Dutch,
that the modern hyacinth would scarcely be recognized as the descendant
of the type above referred to, the spikes being long and dense, composed
of a large number of flowers; the spikes produced by strong bulbs not
unfrequently measure 6 to 9 in. in length and from 7 to 9 in. in
circumference, with the flowers closely set on from bottom to top. Of
late years much improvement has been effected in the size of the
individual flowers and the breadth of their recurving lobes, as well as
in securing increased brilliancy and depth of colour.

The peculiarities of the soil and climate of Holland are so very
favourable to their production that Dutch florists have made a specialty
of the growth of those and other bulbous-rooted flowers. Hundreds of
acres are devoted to the growth of hyacinths in the vicinity of Haarlem,
and bring in a revenue of several hundreds of thousands of pounds. Some
notion of the vast number imported into England annually may be formed
from the fact that, for the supply of flowering plants to Covent Garden,
one market grower alone produces from 60,000 to 70,000 in pots under
glass, their blooming period being accelerated by artificial heat, and
extending from Christmas onwards until they bloom naturally in the open
ground.

In the spring flower garden few plants make a more effective display
than the hyacinth. Dotted in clumps in the flower borders, and arranged
in masses of well-contrasted colours In beds in the flower garden, there
are no flowers which impart during their season--March and April--a
gayer tone to the parterre. The bulbs are rarely grown a second time,
either for indoor or outdoor culture, though with care they might be
utilized for the latter purpose; and hence the enormous numbers which
are procured each recurring year from Holland.

The first hyacinths were single-flowered, but towards the close of the
17th century double-flowered ones began to appear, and till a recent
period these bulbs were the most esteemed. At the present time, however,
the single-flowered sorts are in the ascendant, as they produce more
regular and symmetrical spikes of blossom, the flowers being closely set
and more or less horizontal in direction, while most of the double sorts
have the bells distant and dependent, so that the spike is loose and by
comparison ineffective. For pot culture, and for growth in
water-glasses especially, the single-flowered sorts are greatly to be
preferred. Few if any of the original kinds are now in cultivation, a
succession of new and improved varieties having been raised, the demand
for which is regulated in some respects by fashion.

  The hyacinth delights in a rich light sandy soil. The Dutch
  incorporate freely with their naturally light soil a compost
  consisting of one-third coarse sea or river sand, one-third rotten cow
  dung without litter and one-third leaf-mould. The soil thus renovated
  retains its qualities for six or seven years, but hyacinths are not
  planted upon the same place for two years successively, intermediary
  crops of narcissus, crocus or tulips being taken. A good compost for
  hyacinths is sandy loam, decayed leaf-mould, rotten cow dung and sharp
  sand in equal parts, the whole being collected and laid up in a heap
  and turned over occasionally. Well-drained beds made up of this soil,
  and refreshed with a portion of new compost annually, would grow the
  hyacinth to perfection. The best time to plant the bulbs is towards
  the end of September and during October; they should be arranged in
  rows, 6 to 8 in. asunder, there being four rows in each bed. The bulbs
  should be sunk about 4 to 6 in. deep, with a small quantity of clean
  sand placed below and around each of them. The beds should be covered
  with decayed tan-bark, coco-nut fibre or half-rotten dung litter. As
  the flower-stems appear, they are tied to rigid but slender stakes to
  preserve them from accident. If the bulbs are at all prized, the stems
  should be broken off as soon as the flowering is over, so as not to
  exhaust the bulbs; the leaves, however, must be allowed to grow on
  till matured, but as soon as they assume a yellow colour, the bulbs
  are taken up, the leaves cut off near their base, and the bulbs laid
  out in a dry, airy, shady place to ripen, after which they are cleaned
  of loose earth and skin, ready for storing. It is the practice in
  Holland, about a month after the bloom, or when the tips of the leaves
  assume a withered appearance, to take up the bulbs, and to lay them
  sideways on the ground, covering them with an inch or two of earth.
  About three weeks later they are again taken up and cleaned. In the
  store-room they should be kept dry, well-aired and apart from each
  other.

  Few plants are better adapted than the hyacinth for pot culture as
  greenhouse decorative plants; and by the aid of forcing they may be
  had in bloom as early as Christmas. They flower fairly well in 5-in.
  pots, the stronger bulbs in 6-in. pots. To bloom at Christmas, they
  should be potted early in September, in a compost resembling that
  already recommended for the open-air beds; and, to keep up a
  succession of bloom, others should be potted at intervals of a few
  weeks till the middle or end of November. The tops of the bulbs should
  be about level with the soil, and if a little sand is put immediately
  around them so much the better. The pots should be set in an open
  place on a dry hard bed of ashes, and be covered over to a depth of 6
  or 8 in. with the same material or with fibre or soil; and when the
  roots are well developed, which will take from six to eight weeks,
  they may be removed to a frame, and gradually exposed to light, and
  then placed in a forcing pit in a heat of from 60 to 70°. When the
  flowers are fairly open, they may be removed to the greenhouse or
  conservatory.

  The hyacinth may be very successfully grown in glasses for ornament in
  dwelling-houses. The glasses are filled to the neck with rain or even
  tap water, a few lumps of charcoal being dropped into them. The bulbs
  are placed in the hollow provided for them, so that their base just
  touches the water. This may be done in September or October. They are
  then set in a dark cupboard for a few weeks till roots are freely
  produced, and then gradually exposed to light. The early-flowering
  single white Roman hyacinth, a small-growing pure white variety,
  remarkable for its fragrance, is well adapted for forcing, as it can
  be had in bloom if required by November. For windows it grows well in
  the small glasses commonly used for crocuses; and for decorative
  purposes should be planted about five bulbs in a 5-in. pot, or in pans
  holding a dozen each. If grown for cut flowers it can be planted
  thickly in boxes of any convenient size. It is highly esteemed during
  the winter months by florists.

  The Spanish hyacinth (_H. amethystinus_) and _H. azureus_ are charming
  little bulbs for growing in masses in the rock garden or front of the
  flower border. The older botanists included in the genus _Hyacinthus_
  species of _Muscari_, _Scilla_ and other genera of bulbous Liliaceae,
  and the name of hyacinth is still popularly applied to several other
  bulbous plants. Thus _Muscari botryoides_ is the grape hyacinth, 6
  in., blue or white, the handsomest; _M. moschatum_, the musk hyacinth,
  10 in., has peculiar livid greenish-yellow flowers and a strong musky
  odour; _M. comosum_ var. _monstrosum_, the feather hyacinth, bears
  sterile flowers broken up into a featherlike mass; _M. racemosum_, the
  starch hyacinth, is a native with deep blue plum-scented flowers. The
  Cape hyacinth is _Galtonia candicans_, a magnificent border plant, 3-4
  ft. high, with large drooping white bell-shaped flowers; the star
  hyacinth, _Scilla amoena_; the Peruvian hyacinth or Cuban lily, _S.
  peruviana_, a native of the Mediterranean region, to which Linnaeus
  gave the species name _peruviana_ on a mistaken assumption of its
  origin; the wild hyacinth or blue-bell, known variously as _Endymion
  nonscriptum_, _Hyacinthus nonscriptus_ or _Scilla nutans_; the wild
  hyacinth of western North America, _Camassia esculenta_. They all
  flourish in good garden soil of a gritty nature.




HYACINTH, or JACINTH, in mineralogy, a variety of zircon (q.v.) of
yellowish red colour, used as a gem-stone. The _hyacinthus_ of ancient
writers must have been our sapphire, or blue corundum, while the
hyacinth of modern mineralogists may have been the stone known as
_lyncurium_ ([Greek: lynkourion]). The Hebrew word _leshem_, translated
ligure in the Authorized Version (Ex. xxviii. 19), from the [Greek:
ligyrion] of the Septuagint, appears in the Revised Version as jacinth,
but with a marginal alternative of amber. Both jacinth and amber may be
reddish yellow, but their identification is doubtful. As our jacinth
(zircon) is not known in ancient Egyptian work, Professor Flinders
Petrie has suggested that the _leshem_ may have been a yellow quartz, or
perhaps agate. Some old English writers describe the jacinth as yellow,
whilst others refer to it as a blue stone, and the _hyacinthus_ of some
authorities seems undoubtedly to have been our sapphire. In Rev. xx. 20
the Revised Version retains the word jacinth, but gives sapphire as an
alternative.

Most of the gems known in trade as hyacinth are only garnets--generally
the deep orange-brown hessonite or cinnamon-stone--and many of the
antique engraved stones reputed to be hyacinth are probably garnets. The
difference may be detected optically, since the garnet is singly and the
hyacinth doubly refracting; moreover the specific gravity affords a
simple means of diagnosis, that of garnet being only about 3.7, whilst
hyacinth may have a density as high as 4.7. Again, it was shown many
years ago by Sir A. H. Church that most hyacinths, when examined by the
spectroscope, show a series of dark absorption bands, due perhaps to the
presence of some rare element such as uranium or erbium.

Hyacinth is not a common mineral. It occurs, with other zircons, in the
gem-gravels of Ceylon, and very fine stones have been found as pebbles
at Mudgee in New South Wales. Crystals of zircon, with all the typical
characters of hyacinth, occur at Expailly, Le Puy-en-Velay, in Central
France, but they are not large enough for cutting. The stones which have
been called Compostella hyacinths are simply ferruginous quartz from
Santiago de Compostella in Spain.     (F. W. R.*)




HYACINTHUS,[1] in Greek mythology, the youngest son of the Spartan king
Amyclas, who reigned at Amyclae (so Pausanias iii. 1. 3, iii. 19. 5; and
Apollodorus i. 3. 3, iii. 10. 3). Other stories make him son of Oebalus,
of Eurotas, or of Pierus and the nymph Clio (see Hyginus, _Fabulae_,
271; Lucian, _De saltatione_, 45, and _Dial. deor._ 14). According to
the general story, which is probably late and composite, his great
beauty attracted the love of Apollo, who killed him accidentally when
teaching him to throw the _discus_ (quoit); others say that Zephyrus (or
Boreas) out of jealousy deflected the quoit so that it hit Hyacinthus on
the head and killed him. According to the representation on the tomb at
Amyclae (Pausanias, _loc. cit._) Hyacinthus was translated into heaven
with his virgin sister Polyboea. Out of his blood there grew the flower
known as the hyacinth, the petals of which were marked with the mournful
exclamation AI, AI, "alas" (cf. "that sanguine flower inscribed with
woe"). This Greek hyacinth cannot have been the flower which now bears
the name: it has been identified with a species of iris and with the
larkspur (_Delphinium Aiacis_), which appear to have the markings
described. The Greek hyacinth was also said to have sprung from the
blood of Ajax. Evidently the Greek authorities confused both the flowers
and the traditions.

The death of Hyacinthus was celebrated at Amyclae by the second most
important of Spartan festivals, the Hyacinthia, which took place in the
Spartan month Hecatombeus. What month this was is not certain. Arguing
from Xenophon (_Hell._ iv. 5) we get May; assuming that the Spartan
Hecatombeus is the Attic Hecatombaion, we get July; or again it may be
the Attic Scirophorion, June. At all events the Hyacinthia was an early
summer festival. It lasted three days, and the rites gradually passed
from mourning for Hyacinthus to rejoicings in the majesty of Apollo,
the god of light and warmth, and giver of the ripe fruits of the earth
(see a passage from Polycrates, _Laconica_, quoted by Athenaeus 139 d;
criticized by L. R. Farnell, _Cults of the Greek States_, iv. 266
foll.). This festival is clearly connected with vegetation, and marks
the passage from the youthful verdure of spring to the dry heat of
summer and the ripening of the corn.

The precise relation which Apollo bears to Hyacinthus is obscure. The
fact that at Tarentum a Hyacinthus tomb is ascribed by Polybius to
Apollo Hyacinthus (not Hyacinthius) has led some to think that the
personalities are one, and that the hero is merely an emanation from the
god; confirmation is sought in the Apolline appellation [Greek:
tetracheir], alleged by Hesychius to have been used in Laconia, and
assumed to describe a composite figure of Apollo-Hyacinthus. Against
this theory is the essential difference between the two figures.
Hyacinthus is a chthonian vegetation god whose worshippers are afflicted
and sorrowful; Apollo, though interested in vegetation, is never
regarded as inhabiting the lower world, his death is not celebrated in
any ritual, his worship is joyous and triumphant, and finally the
Amyclean Apollo is specifically the god of war and song. Moreover,
Pausanias describes the monument at Amyclae as consisting of a rude
figure of Apollo standing on an altar-shaped base which formed the tomb
of Hyacinthus. Into the latter offerings were put for the hero before
gifts were made to the god.

On the whole it is probable that Hyacinthus belongs originally to the
pre-Dorian period, and that his story was appropriated and woven into
their own Apollo myth by the conquering Dorians. Possibly he may be the
apotheosis of a pre-Dorian king of Amyclae. J. G. Frazer further
suggests that he may have been regarded as spending the winter months in
the underworld and returning to earth in the spring when the "hyacinth"
blooms. In this case his festival represents perhaps both the Dorian
conquest of Amyclae and the death of spring before the ardent heat of
the summer sun, typified as usual by the _discus_ (quoit) with which
Apollo is said to have slain him. With the growth of the hyacinth from
his blood should be compared the oriental stories of violets springing
from the blood of Attis, and roses and anemones from that of Adonis. As
a youthful vegetation god, Hyacinthus may be compared with Linus and
Scephrus, both of whom are connected with Apollo Agyieus.

  See L. R. Farnell, _Cults of the Greek States_, vol. iv. (1907), pp.
  125 foll., 264 foll.; J. G. Frazer, _Adonis, Attis, Osiris_ (1906),
  bk. ii. ch. 7; S. Wide, _Lakonische Kulte_, p. 290; E. Rhode,
  _Psyche_, 3rd ed. i. 137 foll.; Roscher, _Lexikon d. griech. u. röm.
  Myth._, s.v. "Hyakinthos" (Greve); L. Preller, _Griechische Mythol._
  4th ed. i. 248 foll.     (J. M. M.)


FOOTNOTE:

  [1] The word is probably derived from an Indo-European root, meaning
    "youthful," found in Latin, Greek, English and Sanskrit. Some have
    suggested that the first two letters are from [Greek: uein], to rain,
    (cf. Hyades).




HYADES ("the rainy ones"), in Greek mythology, the daughters of Atlas
and Aethra; their number varies between two and seven. As a reward for
having brought up Zeus at Dodona and taken care of the infant Dionysus
Hyes, whom they conveyed to Ino (sister of his mother Semele) at Thebes
when his life was threatened by Lycurgus, they were translated to heaven
and placed among the stars (Hyginus, _Poët. astron._ ii. 21). Another
form of the story combines them with the Pleiades. According to this
they were twelve (or fifteen) sisters, whose brother Hyas was killed by
a snake while hunting in Libya (Ovid, _Fasti_, v. 165; Hyginus, _Fab._
192). They lamented him so bitterly that Zeus, out of compassion,
changed them into stars--five into the Hyades, at the head of the
constellation of the Bull, the remainder into the Pleiades. Their name
is derived from the fact that the rainy season commenced when they rose
at the same time as the sun (May 7-21); the original conception of them
is that of the fertilizing principle of moisture. The Romans derived the
name from [Greek: us] (pig), and translated it by _Suculae_ (Cicero, _De
nat. deorum_, ii. 43).




HYATT, ALPHEUS (1838-1902), American naturalist, was born at Washington,
D.C., on the 5th of April 1838. From 1858 to 1862 he studied at Harvard,
where he had Louis Agassiz for his master, and in 1863 he served as a
volunteer in the Civil War, attaining the rank of captain. In 1867 he
was appointed curator of the Essex Institute at Salem, and in 1870
became professor of zoology and palaeontology at the Massachusetts
Institute of Technology (resigned 1888), and custodian of the Boston
Society of Natural History (curator in 1881). In 1886 he was appointed
assistant for palaeontology in the Cambridge museum of comparative
anatomy, and in 1889 was attached to the United States Geological Survey
as palaeontologist for the Trias and Jura. He was the chief founder of
the American Society of Naturalists, of which he acted as first
president in 1883, and he also took a leading part in establishing the
marine biological laboratories at Annisquam and Woods Hole, Mass. He
died at Cambridge on the 15th of January 1902.

  His works include _Observations on Fresh-water Polyzoa_ (1866);
  _Fossil Cephalopods of the Museum of Comparative Zoology_ (1872);
  _Revision of North American Porifera_ (1875-1877); _Genera of Fossil
  Cephalopoda_ (1883); _Larval Theory of the Origin of Cellular Tissue_
  (1884); _Genesis of the Arietidae_ (1889); and _Phylogeny of an
  acquired characteristic_ (1894). He wrote the section on Cephalopoda
  in Karl von Zittel's _Paläontologie_ (1900), and his well-known study
  on the fossil pond snails of Steinheim ("The Genesis of the Tertiary
  Species of Planorbis at Steinheim") appeared in the _Memoirs_ of the
  Boston Natural History Society in 1880. He was one of the founders and
  editors of the _American Naturalist_.




HYBLA, the name of several cities In Sicily. The best known
historically, though its exact site is uncertain, is Hybla Major, near
(or by some supposed to be identical with) Megara Hyblaea (q.v.):
another Hybla, known as Hybla Minor or Galeatis, is represented by the
modern Paternò; while the site of Hybla Heraea is to be sought near
Ragusa.




HYBRIDISM. The Latin word _hybrida_, _hibrida_ or _ibrida_ has been
assumed to be derived from the Greek [Greek: hybris], an insult or
outrage, and a hybrid or mongrel has been supposed to be an outrage on
nature, an unnatural product. As a general rule animals and plants
belonging to distinct species do not produce offspring when crossed with
each other, and the term hybrid has been employed for the result of a
fertile cross between individuals of different species, the word mongrel
for the more common result of the crossing of distinct varieties. A
closer scrutiny of the facts, however, makes the term hybridism less
isolated and more vague. The words species and genus, and still more
subspecies and variety, do not correspond with clearly marked and
sharply defined zoological categories, and no exact line can be drawn
between the various kinds of crossings from those between individuals
apparently identical to those belonging to genera universally recognized
as distinct. Hybridism therefore grades into mongrelism, mongrelism into
cross-breeding, and cross-breeding into normal pairing, and we can say
little more than that the success of the union is the more unlikely or
more unnatural the further apart the parents are in natural affinity.

The interest in hybridism was for a long time chiefly of a practical
nature, and was due to the fact that hybrids are often found to present
characters somewhat different from those of either parent. The leading
facts have been known in the case of the horse and ass from time
immemorial. The earliest recorded observation of a hybrid plant is by J.
G. Gmelin towards the end of the 17th century; the next is that of Thomas
Fairchild, who in the second decade of the 18th century, produced the
cross which is still grown in gardens under the name of "Fairchild's
Sweet William." Linnaeus made many experiments in the cross-fertilization
of plants and produced several hybrids, but Joseph Gottlieb Kölreuter
(1733-1806) laid the first real foundation of our scientific knowledge of
the subject. Later on Thomas Andrew Knight, a celebrated English
horticulturist, devoted much successful labour to the improvement of
fruit trees and vegetables by crossing. In the second quarter of the 19th
century C. F. Gärtner made and published the results of a number of
experiments that had not been equalled by any earlier worker. Next came
Charles Darwin, who first in the _Origin of Species_, and later in _Cross
and Self-Fertilization of Plants_, subjected the whole question to a
critical examination, reviewed the known facts and added many to them.

  Darwin's conclusions were summed up by G. J. Romanes in the 9th
  edition of this _Encyclopaedia_ as follows:--

  1. The laws governing the production of hybrids are identical, or
  nearly identical, in the animal and vegetable kingdoms.

  2. The sterility which so generally attends the crossing of two
  specific forms is to be distinguished as of two kinds, which, although
  often confounded by naturalists, are in reality quite distinct.
  For the sterility may obtain between the two parent species when first
  crossed, or it may first assert itself in their hybrid progeny. In the
  latter case the hybrids, although possibly produced without any
  appearance of infertility on the part of their parent species,
  nevertheless prove more or less infertile among themselves, and also
  with members of either parent species.

  3. The degree of both kinds of infertility varies in the case of
  different species, and in that of their hybrid progeny, from absolute
  sterility up to complete fertility. Thus, to take the case of plants,
  "when pollen from a plant of one family is placed on the stigma of a
  plant of a distinct family, it exerts no more influence than so much
  inorganic dust. From this absolute zero of fertility, the pollen of
  different species, applied to the stigma of some one species of the
  same genus, yields a perfect gradation in the number of seeds
  produced, up to nearly complete, or even quite complete, fertility;
  so, in hybrids themselves, there are some which never have produced,
  and probably never would produce, even with the pollen of the pure
  parents, a single fertile seed; but in some of these cases a first
  trace of fertility may be detected, by the pollen of one of the pure
  parent species causing the flower of the hybrid to wither earlier than
  it otherwise would have done; and the early withering of the flower is
  well known to be a sign of incipient fertilization. From this extreme
  degree of sterility we have self-fertilized hybrids producing a
  greater and greater number of seeds up to perfect fertility."

  4. Although there is, as a rule, a certain parallelism, there is no
  fixed relation between the degree of sterility manifested by the
  parent species when crossed and that which is manifested by their
  hybrid progeny. There are many cases in which two pure species can be
  crossed with unusual facility, while the resulting hybrids are
  remarkably sterile; and, contrariwise, there are species which can
  only be crossed with extreme difficulty, though the hybrids, when
  produced, are very fertile. Even within the limits of the same genus,
  these two opposite cases may occur.

  5. When two species are reciprocally crossed, i.e. male A with female
  B, and male B with female A, the degree of sterility often differs
  greatly in the two cases. The sterility of the resulting hybrids may
  differ likewise.

  6. The degree of sterility of first crosses and of hybrids runs, to a
  certain extent, parallel with the systematic affinity of the forms
  which are united. "For species belonging to distinct genera can
  rarely, and those belonging to distinct families can never, be
  crossed. The parallelism, however, is far from complete; for a
  multitude of closely allied species will not unite, or unite with
  extreme difficulty, whilst other species, widely different from each
  other, can be crossed with perfect facility. Nor does the difficulty
  depend on ordinary constitutional differences; for annual and
  perennial plants, deciduous and evergreen trees, plants flowering at
  different seasons, inhabiting different stations, and naturally living
  under the most opposite climates, can often be crossed with ease. The
  difficulty or facility apparently depends exclusively on the sexual
  constitution of the species which are crossed, or on their sexual
  elective affinity."

There are many new records as to the production of hybrids.
Horticulturists have been extremely active and successful in their
attempts to produce new flowers or new varieties of vegetables by
seminal or graft-hybrids, and any florist's catalogue or the account of
any special plant, such as is to be found in Foster-Melliar's _Book of
the Rose_, is in great part a history of successful hybridization. Much
special experimental work has been done by botanists, notably by de
Vries, to the results of whose experiments we shall recur. Experiments
show clearly that the obtaining of hybrids is in many cases merely a
matter of taking sufficient trouble, and the successful crossing of
genera is not infrequent.

  Focke, for instance, cites cases where hybrids were obtained between
  _Brassica_ and _Raphanus_, _Galium_ and _Asperula_, _Campanula_ and
  _Phyteuma_, _Verbascum_ and _Celsia_. Among animals, new records and
  new experiments are almost equally numerous. Boveri has crossed
  _Echinus microtuberculatus_ with _Sphaerechinus granularis_. Thomas
  Hunt Morgan even obtained hybrids between Asterias, a starfish, and
  _Arbacia_, a sea-urchin, a cross as remote as would be that between a
  fish and a mammal. Vernon got many hybrids by fertilizing the eggs of
  _Strongylocentrotus lividus_ with the sperm of _Sphaerechinus
  granularis_. Standfuss has carried on an enormous series of
  experiments with Lepidopterous insects, and has obtained a very large
  series of hybrids, of which he has kept careful record. Lepidopterists
  generally begin to suspect that many curious forms offered by dealers
  as new species are products got by crossing known species. Apellö has
  succeeded with Teleostean fish; Gebhardt and others with Amphibia.
  Elliot and Suchetet have studied carefully the question of
  hybridization occurring normally among birds, and have got together a
  very large body of evidence. Among the cases cited by Elliot the most
  striking are that of the hybrid between _Colaptes cafer_ and _C.
  auratus_, which occurs over a very wide area of North America and is
  known as _C. hybridus_, and the hybrid between _Euplocamus lineatus_
  and _E. horsfieldi_, which appears to be common in Assam. St M.
  Podmore has produced successful crosses between the wood-pigeon
  (_Columba palumbus_) and a domesticated variety of the rock pigeon
  (_C. livia_). Among mammals noteworthy results have been obtained by
  Professor Cossar Ewart, who has bred nine zebra hybrids by crossing
  mares of various sizes with a zebra stallion, and who has studied in
  addition three hybrids out of zebra mares, one sired by a donkey, the
  others by ponies. Crosses have been made between the common rabbit
  (_Lepus cuniculus_) and the guinea-pig (_Cavia cobaya_), and examples
  of the results have been exhibited in the Zoological Gardens of
  Sydney, New South Wales. The Carnivora generally are very easy to
  hybridize, and many successful experiments have been made with animals
  in captivity. Karl Hagenbeck of Hamburg has produced crosses between
  the lion (_Felis leo_) and the tiger (_F. tigris_). What was probably
  a "tri-hybrid" in which lion, leopard and jaguar were mingled was
  exhibited by a London showman in 1908. Crosses between various species
  of the smaller cats have been fertile on many occasions. The black
  bear (_Ursus americanus_) and the European brown bear (_U. arctos_)
  bred in the London Zoological Gardens in 1859, but the three cubs did
  not reach maturity. Hybrids between the brown bear and the
  grizzly-bear (_U. horribilis_) have been produced in Cologne, whilst
  at Halle since 1874 a series of successful matings of polar (_U.
  maritimus_) and brown bears have been made. Examples of these hybrid
  bears have been exhibited by the London Zoological Society. The London
  Zoological Society has also successfully mated several species of
  antelopes, for instance, the water-bucks _Kobus ellipsiprymnus_ and
  _K. unctuosus_, and Selous's antelope _Limnotragus selousi_ with _L.
  gratus_.

The causes militating against the production of hybrids have also
received considerable attention. Delage, discussing the question, states
that there is a general proportion between sexual attraction and
zoological affinity, and in many cases hybrids are not naturally
produced simply from absence of the stimulus to sexual mating, or
because of preferential mating within the species or variety. In
addition to differences of habit, temperament, time of maturity, and so
forth, gross structural differences may make mating impossible. Thus
Escherick contends that among insects the peculiar structure of the
genital appendages makes cross-impregnation impossible, and there is
reason to believe that the specific peculiarities of the modified sexual
palps in male spiders have a similar result.

  The difficulties, however, may not exist, or may be overcome by
  experiment, and frequently it is only careful management that is
  required to produce crossing. Thus it has been found that when the
  pollen of one species does not succeed in fertilizing the ovules of
  another species, yet the reciprocal cross may be successful; that is
  to say, the pollen of the second species may fertilize the ovules of
  the first. H. M. Vernon, working with sea-urchins, found that the
  obtaining of hybrids depended on the relative maturity of the sexual
  products. The difficulties in crossing apparently may extend to the
  chemiotaxic processes of the actual sexual cells. Thus when the
  spermatozoa of an urchin were placed in a drop of seawater containing
  ripe eggs of an urchin and of a starfish, the former eggs became
  surrounded by clusters of the male cells, while the latter appeared to
  exert little attraction for the alien germ-cells. Finally, when the
  actual impregnation of the egg is possible naturally, or has been
  secured by artificial means, the development of the hybrid may stop at
  an early stage. Thus hybrids between the urchin and the starfish,
  animals belonging to different classes, reached only the stage of the
  pluteus larva. A. D. Apellö, experimenting with Teleostean fish, found
  that very often impregnation and segmentation occurred, but that the
  development broke down immediately afterwards. W. Gebhardt, crossing
  _Rana esculenta_ with _R. arvalis_, found that the cleavage of the
  ovum was normal, but that abnormality began with the gastrula, and
  that development soon stopped. In a very general fashion there appears
  to be a parallel between the zoological affinity and the extent to
  which the incomplete development of the hybrid proceeds.

As to the sterility of hybrids _inter se_, or with either of the parent
forms, information is still wanted. Delage, summing up the evidence in a
general way, states that mongrels are more fertile and stronger than
their parents, while hybrids are at least equally hardy but less
fertile. While many of the hybrid products of horticulturists are
certainly infertile, others appear to be indefinitely fertile.

  Focke, it is true, states that the hybrids between _Primula auricula_
  and _P. hirsuta_ are fertile for many generations, but not
  indefinitely so; but, while this may be true for the particular case,
  there seems no reason to doubt that many plant hybrids are quite
  fertile. In the case of animals the evidence is rather against
  fertility. Standfuss, who has made experiments lasting over many
  years, and who has dealt with many genera of Lepidoptera, obtained no
  fertile hybrid females, although he found that hybrid males paired
  readily and successfully with pure-bred females of the parent races.
  Elliot, dealing with birds, concluded that no hybrids were
  fertile with one another beyond the second generation, but thought
  that they were fertile with members of the parent races. Wallace, on
  the other hand, cites from Quatrefages the case of hybrids between the
  moths _Bombyx cynthia_ and _B. arrindia_, which were stated to be
  fertile _inter se_ for eight generations. He also states that hybrids
  between the sheep and goat have a limited fertility _inter se_.
  Charles Darwin, however, had evidence that some hybrid pheasants were
  completely fertile, and he himself interbred the progeny of crosses
  between the common and Chinese geese, whilst there appears to be no
  doubt as to the complete fertility of the crosses between many species
  of ducks, J. L. Bonhote having interbred in various crosses for
  several generations the mallard (_Anas boschas_), the Indian spot-bill
  duck (_A. poecilorhyncha_), the New Zealand grey duck (_A.
  superciliosa_) and the pin-tail (_Dafila acuta_). Podmore's pigeon
  hybrids were fertile _inter se_, a specimen having been exhibited at
  the London Zoological Gardens. The hybrids between the brown and polar
  bears bred at Halle proved to be fertile, both with one of the parent
  species and with one another.

  Cornevin and Lesbre state that in 1873 an Arab mule was fertilized in
  Africa by a stallion, and gave birth to female offspring which she
  suckled. All three were brought to the Jardin d'Acclimatation in
  Paris, and there the mule had a second female colt to the same father,
  and subsequently two male colts in succession to an ass and to a
  stallion. The female progeny were fertilized, but their offspring were
  feeble and died at birth. Cossar Ewart gives an account of a recent
  Indian case in which a female mule gave birth to a male colt. He
  points out, however, that many mistakes have been made about the
  breeding of hybrids, and is not altogether inclined to accept this
  supposed case. Very little has been published with regard to the most
  important question, as to the actual condition of the sexual organs
  and cells in hybrids. There does not appear to be gross anatomical
  defect to account for the infertility of hybrids, but microscopical
  examination in a large number of cases is wanted. Cossar Ewart, to
  whom indeed much of the most interesting recent work on hybrids is
  due, states that in male zebra-hybrids the sexual cells were immature,
  the tails of the spermatozoa being much shorter than those of the
  similar cells in stallions and zebras. He adds, however, that the male
  hybrids he examined were young, and might not have been sexually
  mature. He examined microscopically the ovary of a female zebra-hybrid
  and found one large and several small Graafian follicles, in all
  respects similar to those in a normal mare or female zebra. A careful
  study of the sexual organs in animal and plant hybrids is very much to
  be desired, but it may be said that so far as our present knowledge
  goes there is not to be expected any obvious microscopical cause of
  the relative infertility of hybrids.

The relative variability of hybrids has received considerable attention
from many writers. Horticulturists, as Bateson has written, are "aware
of the great and striking variations which occur in so many orders of
plants when hybridization is effected." The phrase has been used
"breaking the constitution of a plant" to indicate the effect produced
in the offspring of a hybrid union, and the device is frequently used by
those who are seeking for novelties to introduce on the market. It may
be said generally that hybrids are variable, and that the products of
hybrids are still more variable. J. L. Bonhote found extreme variations
amongst his hybrid ducks. Y. Delage states that in reciprocal crosses
there is always a marked tendency for the offspring to resemble the male
parents; he quotes from Huxley that the mule, whose male parent is an
ass, is more like the ass, and that the hinny, whose male parent is a
horse, is more like the horse. Standfuss found among Lepidoptera that
males were produced much more often than females, and that these males
paired readily. The freshly hatched larvae closely resembled the larvae
of the female parent, but in the course of growth the resemblance to the
male increased, the extent of the final approximation to the male
depending on the relative phylogenetic age of the two parents, the
parent of the older species being prepotent. In reciprocal pairing, he
found that the male was able to transmit the characters of the parents
in a higher degree. Cossar Ewart, in relation to zebra hybrids, has
discussed the matter of resemblance to parents in very great detail, and
fuller information must be sought in his writings. He shows that the
wild parent is not necessarily prepotent, although many writers have
urged that view. He described three hybrids bred out of a zebra mare by
different horses, and found in all cases that the resemblance to the
male or horse parent was more profound. Similarly, zebra-donkey hybrids
out of zebra mares bred in France and in Australia were in characters
and disposition far more like the donkey parents. The results which he
obtained in the hybrids which he bred from a zebra stallion and
different mothers were more variable, but there was rather a balance in
favour of zebra disposition and against zebra shape and marking.

  "Of the nine zebra-horse hybrids I have bred," he says, "only two in
  their make and disposition take decidedly after the wild parent. As
  explained fully below, all the hybrids differ profoundly in the plan
  of their markings from the zebra, while in their ground colour they
  take after their respective dams or the ancestors of their dams far
  more than after the zebra--the hybrid out of the yellow and white
  Iceland pony, e.g. instead of being light in colour, as I anticipated,
  is for the most part of a dark dun colour, with but indistinct
  stripes. The hoofs, mane and tail of the hybrids are at the most
  intermediate, but this is perhaps partly owing to reversion towards
  the ancestors of these respective dams. In their disposition and
  habits they all undoubtedly agree more with the wild sire."

Ewart's experiments and his discussion of them also throw important
light on the general relation of hybrids to their parents. He found that
the coloration and pattern of his zebra hybrids resembled far more those
of the Somali or Grévy's zebra than those of their sire--a Burchell's
zebra. In a general discussion of the stripings of horses, asses and
zebras, he came to the conclusion that the Somali zebra represented the
older type, and that therefore his zebra hybrids furnished important
evidence of the effect of crossing in producing reversion to ancestral
type. The same subject has of course been discussed at length by Darwin,
in relation to the cross-breeding of varieties of pigeons; but the
modern experimentalists who are following the work of Mendel interpret
reversion differently (see MENDELISM).

_Graft-Hybridism._--It is well known that, when two varieties or allied
species are grafted together, each retains its distinctive characters.
But to this general, if not universal, rule there are on record several
alleged exceptions, in which either the scion is said to have partaken
of the qualities of the stock, the stock of the scion, or each to have
affected the other. Supposing any of these influences to have been
exerted, the resulting product would deserve to be called a
graft-hybrid. It is clearly a matter of great interest to ascertain
whether such formation of hybrids by grafting is really possible; for,
if even one instance of such formation could be unequivocally proved, it
would show that sexual and asexual reproduction are essentially
identical.

The cases of alleged graft-hybridism are exceedingly few, considering
the enormous number of grafts that are made every year by
horticulturists, and have been so made for centuries. Of these cases the
most celebrated are those of Adam's laburnum (_Cytisus Adami_) and the
bizzarria orange. Adam's laburnum is now flourishing in numerous places
throughout Europe, all the trees having been raised as cuttings from the
original graft, which was made by inserting a bud of the purple laburnum
into a stock of the yellow. M. Adam, who made the graft, has left on
record that from it there sprang the existing hybrid. There can be no
question as to the truly hybrid character of the latter--all the
peculiarities of both parent species being often blended in the same
raceme, flower or even petal; but until the experiment shall have been
successfully repeated there must always remain a strong suspicion that,
notwithstanding the assertion and doubtless the belief of M. Adam, the
hybrid arose as a cross in the ordinary way of seminal reproduction.
Similarly, the bizzarria orange, which is unquestionably a hybrid
between the bitter orange and the citron--since it presents the
remarkable spectacle of these two different fruits blended into one--is
stated by the gardener who first succeeded in producing it to have
arisen as a graft-hybrid; but here again a similar doubt, similarly due
to the need of corroboration, attaches to the statement. And the same
remark applies to the still more wonderful case of the so-called
trifacial orange, which blends three distinct kinds of fruit in one, and
which is said to have been produced by artificially splitting and
uniting the seeds taken from the three distinct species, the fruits of
which now occur blended in the triple hybrid.

The other instances of alleged graft-hybridism are too numerous to be
here noticed in detail; they refer to jessamine, ash, hazel, vine,
hyacinth, potato, beet and rose. Of these the cases of the vine, beet
and rose are the strongest as evidence of graft-hybridization, from the
fact that some of them were produced as the result of careful
experiments made by very competent experimentalists. On the whole, the
results of some of these experiments, although so few in number, must be
regarded as making out a strong case in favour of the possibility of
graft-hybridism. For it must always be remembered that, in experiments
of this kind, negative evidence, however great in amount, may be
logically dissipated by a single positive result.

_Theory of Hybridism._--Charles Darwin was interested in hybridism as an
experimental side of biology, but still more from the bearing of the
facts on the theory of the origin of species. It is obvious that
although hybridism is occasionally possible as an exception to the
general infertility of species inter se, the exception is still more
minimized when it is remembered that the hybrid progeny usually display
some degree of sterility. The main facts of hybridism appear to lend
support to the old doctrine that there are placed between all species
the barriers of mutual sterility. The argument for the fixity of species
appears still stronger when the general infertility of species crossing
is contrasted with the general fertility of the crossing of natural and
artificial varieties. Darwin himself, and afterwards G. J. Romanes,
showed, however, that the theory of natural selection did not require
the possibility of the commingling of specific types, and that there was
no reason to suppose that the mutation of species should depend upon
their mutual crossing. There existed more than enough evidence, and this
has been added to since, to show that infertility with other species is
no criterion of a species, and that there is no exact parallel between
the degree of affinity between forms and their readiness to cross. The
problem of hybridism is no more than the explanation of the generally
reduced fertility of remoter crosses as compared with the generally
increased fertility of crosses between organisms slightly different.
Darwin considered and rejected the view that the inter-sterility of
species could have been the result of natural selection.

  "At one time it appeared to me probable," he wrote (_Origin of
  Species_, 6th ed. p. 247), "as it has to others, that the sterility of
  first crosses and of hybrids might have been slowly acquired through
  the natural selection of slightly lessened degrees of fertility,
  which, like any other variation, spontaneously appeared in certain
  individuals of one variety when crossed with those of another variety.
  For it would clearly be advantageous to two varieties or incipient
  species if they could be kept from blending, on the same principle
  that, when man is selecting at the same time two varieties, it is
  necessary that he should keep them separate. In the first place, it
  may be remarked that species inhabiting distinct regions are often
  sterile when crossed; now it could clearly have been of no advantage
  to such separated species to have been rendered mutually sterile and,
  consequently, this could not have been effected through natural
  selection; but it may perhaps be argued that, if a species were
  rendered sterile with some one compatriot, sterility with other
  species would follow as a necessary contingency. In the second place,
  it is almost as much opposed to the theory of natural selection as to
  that of special creation, that in reciprocal crosses the male element
  of one form should have been rendered utterly impotent on a second
  form, whilst at the same time the male element of this second form is
  enabled freely to fertilize the first form; for this peculiar state of
  the reproductive system could hardly have been advantageous to either
  species."

Darwin came to the conclusion that the sterility of crossed species must
be due to some principle quite independent of natural selection. In his
search for such a principle he brought together much evidence as to the
instability of the reproductive system, pointing out in particular how
frequently wild animals in captivity fail to breed, whereas some
domesticated races have been so modified by confinement as to be fertile
together although they are descended from species probably mutually
infertile. He was disposed to regard the phenomena of differential
sterility as, so to speak, by-products of the process of evolution. G.
J. Romanes afterwards developed his theory of physiological selection,
in which he supposed that the appearance of differential fertility
within a species was the starting-point of new species; certain
individuals by becoming fertile only _inter se_ proceeded along lines of
modification diverging from the lines followed by other members of the
species. Physiological selection in fact would operate in the same
fashion as geographical isolation; if a portion of a species separated
on an island tends to become a new species, so also a portion separated
by infertility with the others would tend to form a new species.
According to Romanes, therefore, mutual infertility was the
starting-point, not the result, of specific modification. Romanes,
however, did not associate his interesting theory with a sufficient
number of facts, and it has left little mark on the history of the
subject. A. R. Wallace, on the other hand, has argued that sterility
between incipient species may have been increased by natural selection
in the same fashion as other favourable variations are supposed to have
been accumulated. He thought that "some slight degree of infertility was
a not infrequent accompaniment of the external differences which always
arise in a state of nature between varieties and incipient species."

Weismann concluded, from an examination of a series of plant hybrids,
that from the same cross hybrids of different character may be obtained,
but that the characters are determined at the moment of fertilization;
for he found that all the flowers on the same hybrid plant resembled one
another in the minutest details of colour and pattern. Darwin already
had pointed to the act of fertilization as the determining point, and it
is in this direction that the theory of hybridism has made the greatest
advance.

The starting-point of the modern views comes from the experiments and
conclusions on plant hybrids made by Gregor Mendel and published in
1865. It is uncertain if Darwin had paid attention to this work;
Romanes, writing in the 9th edition of this _Encyclopaedia_, cited it
without comment. First H. de Vries, then W. Bateson and a series of
observers returned to the work of Mendel (see MENDELISM), and made it
the foundation of much experimental work and still more theory. It is
still too soon to decide if the confident predictions of the Mendelians
are justified, but it seems clear that a combination of Mendel's
numerical results with Weismann's (see HEREDITY) conception of the
particulate character of the germ-plasm, or hereditary material, is at
the root of the phenomena of hybridism, and that Darwin was justified in
supposing it to lie outside the sphere of natural selection and to be a
fundamental fact of living matter.

  AUTHORITIES.--Apellö, "Über einige Resultate der Kreuzbefruchtung bei
  Knochenfischen," _Bergens mus. aarbog_ (1894); Bateson, "Hybridization
  and Cross-breeding," _Journal of the Royal Horticultural Society_
  (1900); J. L. Bonhote, "Hybrid Ducks," _Proc. Zool. Soc. of London_
  (1905), p. 147; Boveri, article "Befruchtung," in _Ergebnisse der
  Anatomie und Entwickelungsgeschichte von Merkel und Bonnet_, i.
  385-485; Cornevin et Lesbre, "Étude sur un hybride issu d'une mule
  féconde et d'un cheval," _Rev. Sci._ li. 144; Charles Darwin, _Origin
  of Species_ (1859), _The Effects of Cross and Self-Fertilization in
  the Vegetable Kingdom_ (1878); Delage, _La Structure du protoplasma et
  les théories sur l'hérédité_ (1895, with a literature); de Vries, "The
  Law of Disjunction of Hybrids," _Comptes rendus_ (1900), p. 845;
  Elliot, _Hybridism_; Escherick, "Die biologische Bedeutung der
  Genitalabhänge der Insecten," _Verh. z. B. Wien_, xlii. 225; Ewart,
  _The Penycuik Experiments_ (1899); Focke, _Die Pflanzen-Mischlinge_
  (1881); Foster-Melliar, _The Book of the Rose_ (1894); C. F. Gaertner,
  various papers in _Flora_, 1828, 1831, 1832, 1833, 1836, 1847, on
  "Bastard-Pflanzen"; Gebhardt, "Über die Bastardirung von _Rana
  esculenta_ mit _R. arvalis_," _Inaug. Dissert._ (Breslau, 1894); G.
  Mendel, "Versuche über Pflanzen-Hybriden," _Verh. Natur. Vereins in
  Brünn_ (1865), pp. 1-52; Morgan, "Experimental Studies," _Anat. Anz._
  (1893), p. 141; id. p. 803; G. J. Romanes, "Physiological Selection,"
  _Jour. Linn. Soc._ xix. 337; H. Scherren, "Notes on Hybrid Bears,"
  _Proc. Zool. Soc. of London_ (1907), p. 431; Saunders, _Proc. Roy.
  Soc._ (1897), lxii. 11; Standfuss, "Études de zoologie expérimentale,"
  _Arch. Sci. Nat._ vi. 495; Suchetet, "Les Oiseaux hybrides rencontrés
  à l'état sauvage," _Mém. Soc. Zool._ v. 253-525, and vi. 26-45;
  Vernon, "The Relation between the Hybrid and Parent Forms of Echinoid
  Larvae," _Proc. Roy. Soc._ lxv. 350; Wallace, _Darwinism_ (1889);
  Weismann, _The Germ-Plasm_ (1893).     (P. C. M)




HYDANTOIN (glycolyl urea),

                 [beta] [alpha]
                  / NH · CH2
  C3H4N2O2 or CO <          ,
                  \ NH · CO
                  [gamma]

the ureïde of glycollic acid, may be obtained by heating allantoin or
alloxan with hydriodic acid, or by heating bromacetyl urea with
alcoholic ammonia. It crystallizes in needles, melting at 216° C.

When hydrolysed with baryta water yields hydantoic (glycoluric)acid,
H2N·CO·NH·CH2·CO2H, which is readily soluble in hot water, and on
heating with hydriodic acid decomposes into ammonia, carbon dioxide and
glycocoll, CH2·NH2·CO2·H. Many substituted hydantoins are known; the
[alpha]-alkyl hydantoins are formed on fusion of aldehyde- or
ketone-cyanhydrins with urea, the [beta]-alkyl hydantoins from the
fusion of mono-alkyl glycocolls with urea, and the [gamma]-alkyl
hydantoins from the action of alkalis and alkyl iodides on the
[alpha]-compounds. [gamma]-Methyl hydantoin has been obtained as a
splitting product of caffeine (E. Fischer, _Ann._, 1882, 215, p. 253).




HYDE, the name of an English family distinguished in the 17th century.
Robert Hyde of Norbury, Cheshire, had several sons, of whom the third
was Lawrence Hyde of Gussage St Michael, Dorsetshire. Lawrence's son
Henry was father of Edward Hyde, earl of Clarendon (q.v.), whose second
son by his second wife was Lawrence, earl of Rochester (q.v.); another
son was Sir Lawrence Hyde, attorney-general to Anne of Denmark, James
I.'s consort; and a third son was Sir Nicholas Hyde (d. 1631),
chief-justice of England. Sir Nicholas entered parliament in 1601 and
soon became prominent as an opponent of the court, though he does not
appear to have distinguished himself in the law. Before long, however,
he deserted the popular party, and in 1626 he was employed by the duke
of Buckingham in his defence to impeachment by the Commons; and in the
following year he was appointed chief-justice of the king's bench, in
which office it fell to him to give judgment in the celebrated case of
Sir Thomas Darnell and others who had been committed to prison on
warrants signed by members of the privy council, which contained no
statement of the nature of the charge against the prisoners. In answer
to the writ of _habeas corpus_ the attorney-general relied on the
prerogative of the crown, supported by a precedent of Queen Elizabeth's
reign. Hyde, three other judges concurring, decided in favour of the
crown, but without going so far as to declare the right of the crown to
refuse indefinitely to show cause against the discharge of the
prisoners. In 1629 Hyde was one of the judges who condemned Eliot,
Holles and Valentine for conspiracy in parliament to resist the king's
orders; refusing to admit their plea that they could not be called upon
to answer out of parliament for acts done in parliament. Sir Nicholas
Hyde died in August 1631.

Sir Lawrence Hyde, attorney-general to Anne of Denmark, had eleven sons,
four of whom were men of some mark. Henry was an ardent royalist who
accompanied Charles II. to the continent, and returning to England was
beheaded in 1650; Alexander (1598-1667) became bishop of Salisbury in
1665; Edward (1607-1659) was a royalist divine who was nominated dean of
Windsor in 1658, but died before taking up the appointment, and who was
the author of many controversial works in Anglican theology; and Robert
(1595-1665) became recorder of Salisbury and represented that borough in
the Long Parliament, in which he professed royalist principles, voting
against the attainder of Strafford. Having been imprisoned and deprived
of his recordership by the parliament in 1645/6, Robert Hyde gave refuge
to Charles II. on his flight from Worcester in 1651, and on the
Restoration he was knighted and made a judge of the common pleas. He
died in 1665. Henry Hyde (1672-1753), only son of Lawrence, earl of
Rochester, became 4th earl of Clarendon and 2nd earl of Rochester, both
of which titles became extinct at his death. He was in no way
distinguished, but his wife Jane Hyde, countess of Clarendon and
Rochester (d. 1725), was a famous beauty celebrated by the homage of
Swift, Prior and Pope, and by the groundless scandal of Lady Mary
Wortley Montagu. Two of her daughters, Jane, countess of Essex, and
Catherine, duchess of Queensberry, were also famous beauties of the
reign of Queen Anne. Her son, Henry Hyde (1710-1753), known as Viscount
Cornbury, was a Tory and Jacobite member of parliament, and an intimate
friend of Bolingbroke, who addressed to him his _Letters on the Study
and Use of History_, and _On the Spirit of Patriotism_. In 1750 Lord
Cornbury was created Baron Hyde of Hindon, but, as he predeceased his
father, this title reverted to the latter and became extinct at his
death. Lord Cornbury was celebrated as a wit and a conversationalist.
By his will he bequeathed the papers of his great-grandfather, Lord
Clarendon, the historian, to the Bodleian Library at Oxford.

  See Lord Clarendon, _The Life of Edward, Earl of Clarendon_ (3 vols.,
  Oxford, 1827); Edward Foss, _The Judges of England_ (London,
  1848-1864); Anthony à Wood, _Athenae oxonienses_ (London, 1813-1820);
  Samuel Pepys, _Diary and Correspondence_, edited by Lord Braybrooke (4
  vols., London, 1854).




HYDE, THOMAS (1636-1703), English Orientalist, was born at Billingsley,
near Bridgnorth, in Shropshire, on the 29th of June 1636. He inherited
his taste for linguistic studies, and received his first lessons in some
of the Eastern tongues, from his father, who was rector of the parish.
In his sixteenth year Hyde entered King's College, Cambridge, where,
under Wheelock, professor of Arabic, he made rapid progress in Oriental
languages, so that, after only one year of residence, he was invited to
London to assist Brian Walton in his edition of the _Polyglott Bible_.
Besides correcting the Arabic, Persic and Syriac texts for that work,
Hyde transcribed into Persic characters the Persian translation of the
Pentateuch, which had been printed in Hebrew letters at Constantinople
in 1546. To this work, which Archbishop Ussher had thought well-nigh
impossible even for a native of Persia, Hyde appended the Latin version
which accompanies it in the _Polyglott_. In 1658 he was chosen Hebrew
reader at Queen's College, Oxford, and in 1659, in consideration of his
erudition in Oriental tongues, he was admitted to the degree of M.A. In
the same year he was appointed under-keeper of the Bodleian Library, and
in 1665 librarian-in-chief. Next year he was collated to a prebend at
Salisbury, and in 1673 to the archdeaconry of Gloucester, receiving the
degree of D.D. shortly afterwards. In 1691 the death of Edward Pococke
opened up to Hyde the Laudian professorship of Arabic; and in 1697, on
the deprivation of Roger Altham, he succeeded to the regius chair of
Hebrew and a canonry of Christ Church. Under Charles II., James II. and
William III. Hyde discharged the duties of Eastern interpreter to the
court. Worn out by his unremitting labours, he resigned his
librarianship in 1701, and died at Oxford on the 18th of February 1703.
Hyde, who was one of the first to direct attention to the vast treasures
of Oriental antiquity, was an excellent classical scholar, and there was
hardly an Eastern tongue accessible to foreigners with which he was not
familiar. He had even acquired Chinese, while his writings are the best
testimony to his mastery of Turkish, Arabic, Syriac, Persian, Hebrew and
Malay.

In his chief work, _Historia religionis veterum Persarum_ (1700), he
made the first attempt to correct from Oriental sources the errors of
the Greek and Roman historians who had described the religion of the
ancient Persians. His other writings and translations comprise _Tabulae
longitudinum et latitudinum stellarum fixarum ex observatione principis
Ulugh Beighi_ (1665), to which his notes have given additional value;
_Quatuor evangelia et acta apostolorum lingua Malaica, caracteribus
Europaeis_ (1677); _Epistola de mensuris et ponderibus serum sive
sinensium_ (1688), appended to Bernard's _De mensuris et ponderibus
antiquis; Abraham Peritsol itinera mundi_ (1691); and _De ludis
orientalibus libri II._ (1694).

  With the exception of the _Historia religionis_, which was republished
  by Hunt and Costard in 1760, the writings of Hyde, including some
  unpublished MSS., were collected and printed by Dr Gregory Sharpe in
  1767 under the title _Syntagma dissertationum quas olim ... Thomas
  Hyde separatim edidit_. There is a life of the author prefixed. Hyde
  also published a catalogue of the Bodleian Library in 1674.




HYDE, a market town and municipal borough in the Hyde parliamentary
division of Cheshire, England, 7½ m. E. of Manchester, by the Great
Central railway. Pop. (1901) 32,766. It lies in the densely populated
district in the north-east of the county, on the river Tame, which here
forms the boundary of Cheshire with Lancashire. To the east the outlying
hills of the Peak district of Derbyshire rise abruptly. The town has
cotton weaving factories, spinning mills, print-works, iron foundries
and machine works; also manufactures of hats and margarine. There are
extensive coal mines in the vicinity. Hyde is wholly of modern growth,
though it contains a few ancient houses, such as Newton Hall, in the
part of the town so called. The old family of Hyde held possession of
the manor as early as the reign of John. The borough, incorporated in
1881, is under a mayor, 6 aldermen and 18 councillors. Area, 3081 acres.




HYDE DE NEUVILLE, JEAN GUILLAUME, BARON (1776-1857), French politician,
was born at La Charité-sur-Loire (Nièvre) on the 24th of January 1776,
the son of Guillaume Hyde, who belonged to an English family which had
emigrated with the Stuarts after the rebellion of 1745. He was only
seventeen when he successfully defended a man denounced by Fouché before
the revolutionary tribunal of Nevers. From 1793 onwards he was an active
agent of the exiled princes; he took part in the Royalist rising in
Berry in 1796, and after the _coup d'état_ of the 18th Brumaire
(November 9, 1799) tried to persuade Bonaparte to recall the Bourbons.
An accusation of complicity in the infernal machine conspiracy of
1800-1801 was speedily retracted, but Hyde de Neuville retired to the
United States, only to return after the Restoration. He was sent by
Louis XVIII. to London to endeavour to persuade the British government
to transfer Napoleon to a remoter and safer place of exile than the isle
of Elba, but the negotiations were cut short by the emperor's return to
France in March 1815. In January 1816 de Neuville became French
ambassador at Washington, where he negotiated a commercial treaty. On
his return in 1821 he declined the Constantinople embassy, and in
November 1822 was elected deputy for Cosne. Shortly afterwards he was
appointed French ambassador at Lisbon, where his efforts to oust British
influence culminated, in connexion with the _coup d'état_ of Dom Miguel
(April 30, 1824), in his suggestion to the Portuguese minister to invite
the armed intervention of Great Britain. It was assumed that this would
be refused, in view of the loudly proclaimed British principle of
non-intervention, and that France would then be in a position to
undertake a duty that Great Britain had declined. The scheme broke down,
however, owing to the attitude of the reactionary party in the
government of Paris, which disapproved of the Portuguese constitution.
This destroyed his influence at Lisbon, and he returned to Paris to take
his seat in the Chamber of Deputies. In spite of his pronounced
Royalism, he now showed Liberal tendencies, opposed the policy of
Villèle's cabinet, and in 1828 became a member of the moderate
administration of Martignac as minister of marine. In this capacity he
showed active sympathy with the cause of Greek independence. During the
Polignac ministry (1829-1830) he was again in opposition, being a firm
upholder of the charter; but after the revolution of July 1830 he
entered an all but solitary protest against the exclusion of the
legitimate line of the Bourbons from the throne, and resigned his seat.
He died in Paris on the 28th of May 1857.

  His _Mémoires et souvenirs_ (3 vols., 1888), compiled from his notes
  by his nieces, the vicomtesse de Bardonnet and the baronne Laurenceau,
  are of great interest for the Revolution and the Restoration.




HYDE PARK, a small township of Norfolk county, Massachusetts, U.S.A.,
about 8 m. S.W. of the business centre of Boston. Pop. (1890) 10,193;
(1900) 13,244, of whom 3805 were foreign-born; (1910 census) 15,507. Its
area is about 4½ sq. m. It is traversed by the New York, New Haven &
Hartford railway, which has large repair shops here, and by the Neponset
river and smaller streams. The township contains the villages of Hyde
Park, Readville (in which there is the famous "Weil" trotting-track),
Fairmount, Hazelwood and Clarendon Hills. Until about 1856 Hyde Park was
a farmstead. The value of the total factory product increased from
$4,383,959 in 1900 to $6,739,307 in 1905, or 53.7%. In 1868 Hyde Park
was incorporated as a township, being formed of territory taken from
Dorchester, Dedham and Milton.




HYDERABAD, or HAIDARABAD, a city and district of British India, in the
Sind province of Bombay. The city stands on a hill about 3 m. from the
left bank of the Indus, and had a population in 1901 of 69,378. Upon the
site of the present fort is supposed to have stood the ancient town of
Nerankot, which in the 8th century submitted to Mahommed bin Kasim. In
1768 the present city was founded by Ghulam Shah Kalhora; and it
remained the capital of Sind until 1843, when, after the battle of
Meeanee, it was surrendered to the British, and the capital transferred
to Karachi. The city is built on the most northerly hills of the Ganga
range, a site of great natural strength. In the fort, which covers an
area of 36 acres, is the arsenal of the province, transferred thither
from Karachi in 1861, and the palaces of the ex-mirs of Sind. An
excellent water supply is derived from the Indus. In addition to
manufactures of silk, gold and silver embroidery, lacquered ware and
pottery, there are three factories for ginning cotton. There are three
high schools, training colleges for masters and mistresses, a medical
school, an agricultural school for village officials, and a technical
school. The city suffered from plague in 1896-1897.

The DISTRICT OF HYDERABAD has an area of 8291 sq. m., with a population
in 1901 of 989,030, showing an increase of 15% in the decade. It
consists of a vast alluvial plain, on the left bank of the Indus, 216 m.
long and 48 broad. Fertile along the course of the river, it degenerates
towards the east into sandy wastes, sparsely populated, and defying
cultivation. The monotony is relieved by the fringe of forest which
marks the course of the river, and by the avenues of trees that line the
irrigation channels branching eastward from this stream. The south of
the district has a special feature in its large natural water-courses
(called _dhoras_) and basin-like shallows (_chhaus_), which retain the
rains for a long time. A limestone range called the Ganga and the
pleasant frequency of garden lands break the monotonous landscape. The
principal crops are millets, rice, oil-seeds, cotton and wheat, which
are dependent on irrigation, mostly from government canals. There is a
special manufacture at Hala of glazed pottery and striped cotton cloth.
Three railways traverse the district: (1) one of the main lines of the
North-Western system, following the Indus valley and crossing the river
near Hyderabad; (2) a broad-gauge branch running south to Badin, which
will ultimately be extended to Bombay; and (3) a metre-gauge line from
Hyderabad city into Rajputana.




HYDERABAD, HAIDARABAD, also known as the Nizam's Dominions, the
principal native state of India in extent, population and political
importance; area, 82,698 sq. m.; pop. (1901) 11,141,142, showing a
decrease of 3.4% in the decade; estimated revenue 4½ crores of Hyderabad
rupees (£2,500,000). The state occupies a large portion of the eastern
plateau of the Deccan. It is bounded on the north and north-east by
Berar, on the south and south-east by Madras, and on the west by Bombay.
The country presents much variety of surface and feature; but it may be
broadly divided into two tracts, distinguished from one another
geologically and ethnically, which are locally known from the languages
spoken as Telingana and Marathwara. In some parts it is mountainous,
wooded and picturesque, in others flat and undulating. The open country
includes lands of all descriptions, including many rich and fertile
plains, much good land not yet brought under cultivation, and numerous
tracts too sterile ever to be cultivated. In the north-west the
geological formations are volcanic, consisting principally of trap, but
in some parts of basalt; in the middle, southern and south-western parts
the country is overlaid with gneissic formations. The territory is well
watered, rivers being numerous, and tanks or artificial pieces of water
abundant, especially in Telingana. The principal rivers are the
Godavari, with its tributaries the Dudna, Manjira and Pranhita; the
Wardha, with its tributary the Penganga; and the Kistna, with its
tributary the Tungabhadra. The climate may be considered in general
good; and as there are no arid bare deserts, hot winds are little felt.

More than half the revenue of the state is derived from the land, and
the development of the country by irrigation and railways has caused
considerable expansion in this revenue, though the rate of increase in
the decade 1891-1901 was retarded by a succession of unfavourable
seasons. The soil is generally fertile, though in some parts it consists
of _chilka_, a red and gritty mould little fitted for purposes of
agriculture. The principal crops are millets of various kinds, rice,
wheat, oil-seeds, cotton, tobacco, sugar-cane, and fruits and
garden produce in great variety. Silk, known as _tussur_, the produce of
a wild species of worm, is utilized on a large scale. Lac, suitable for
use as a resin or dye, gums and oils are found in great quantities.
Hides, raw and tanned, are articles of some importance in commerce. The
principal exports are cotton, oil-seeds, country-clothes and hides; the
imports are salt, grain, timber, European piece-goods and hardware. The
mineral wealth of the state consists of coal, copper, iron, diamonds and
gold; but the development of these resources has not hitherto been very
successful. The only coal mine now worked is the large one at Singareni,
with an annual out-turn of nearly half a million tons. This coal has
enabled the nizam's guaranteed state railway to be worked so cheaply
that it now returns a handsome profit to the state. It also gives
encouragement to much-needed schemes of railway extension, and to the
erection of cotton presses and of spinning and weaving mills. The
Hyderabad-Godavari railway (opened in 1901) traverses a rich cotton
country, and cotton presses have been erected along the line. The
currency of the state is based on the _hali sikka_, which contains
approximately the same weight of silver as the British rupee, but its
exchange value fell heavily after 1893, when free coinage ceased in the
mint. In 1904, however, a new coin (the Mahbubia rupee) was minted; the
supply was regulated, and the rate of exchange became about 115 = 100
British rupees. The state suffered from famine during 1900, the total
number of persons in receipt of relief rising to nearly 500,000 in June
of that year. The nizam met the demands for relief with great
liberality.

The nizam of Hyderabad is the principal Mahommedan ruler in India. The
family was founded by Asaf Jah, a distinguished Turkoman soldier of the
emperor Aurangzeb, who in 1713 was appointed subahdar of the Deccan,
with the title of nizam-ul-mulk (regulator of the state), but eventually
threw off the control of the Delhi court. Azaf Jah's death in 1748 was
followed by an internecine struggle for the throne among his
descendants, in which the British and the French took part. At one time
the French nominee, Salabat Jang, established himself with the help of
Bussy. But finally, in 1761, when the British had secured their
predominance throughout southern India, Nizam Ali took his place and
ruled till 1803. It was he who confirmed the grant of the Northern
Circars in 1766, and joined in the two wars against Tippoo Sultan in
1792 and 1799. The additions of territory which he acquired by these
wars was afterwards (1800) ceded to the British, as payment for the
subsidiary force which he had undertaken to maintain. By a later treaty
in 1853, the districts known as Berar were "assigned" to defray the cost
of the Hyderabad contingent. In 1857 when the Mutiny broke out, the
attitude of Hyderabad as the premier native state and the cynosure of
the Mahommedans in India became a matter of extreme importance; but
Afzul-ud-Dowla, the father of the present ruler, and his famous
minister, Sir Salar Jang, remained loyal to the British. An attack on
the residency was repulsed, and the Hyderabad contingent displayed their
loyalty in the field against the rebels. In 1902 by a treaty made by
Lord Curzon, Berar was leased in perpetuity to the British government,
and the Hyderabad contingent was merged in the Indian army. The nizam
Mir Mahbub Ali Khan Bahadur, Asaf Jah, a direct descendant of the famous
nizam-ul-mulk, was born on the 18th of August 1866. On the death of his
father in 1869 he succeeded to the throne as a minor, and was invested
with full powers in 1884. He is notable as the originator of the
Imperial Service Troops, which now form the contribution of the native
chiefs to the defence of India. On the occasion of the Panjdeh incident
in 1885 he made an offer of money and men, and subsequently on the
occasion of Queen Victoria's Jubilee in 1887 he offered 20 lakhs
(£130,000) annually for three years for the purpose of frontier defence.
It was finally decided that the native chiefs should maintain small but
well-equipped bodies of infantry and cavalry for imperial defence. For
many years past the Hyderabad finances were in a very unhealthy
condition, the expenditure consistently outran the revenue, and the
nobles, who held their tenure under an obsolete feudal system, vied
with each other in ostentatious extravagance. But in 1902, on the
revision of the Berar agreement, the nizam received 25 lakhs (£167,000)
a year for the rent of Berar, thus substituting a fixed for a
fluctuating source of income, and a British financial adviser was
appointed for the purpose of reorganizing the resources of the state.

  See S. H. Bilgrami and C. Willmott, _Historical and Descriptive Sketch
  of the Nizam's Dominions_ (Bombay, 1883-1884).




HYDERABAD or HAIDARABAD, capital of the above state, is situated on the
right bank of the river Musi, a tributary of the Kistna, with Golconda
to the west, and the residency and its bazaars and the British
cantonment of Secunderabad to the north-east. It is the fourth largest
city in India; pop. (1901) 448,466, including suburbs and cantonment.
The city itself is in shape a parallelogram, with an area of more than 2
sq. m. It was founded in 1589 by Mahommed Kuli, fifth of the Kutb Shahi
kings, of whose period several important buildings remain as monuments.
The principal of these is the Char Minar or Four Minarets (1591). The
minarets rise from arches facing the cardinal points, and stand in the
centre of the city, with four roads radiating from their base. The Ashur
Khana (1594), a ceremonial building, the hospital, the Gosha Mahal
palace and the Mecca mosque, a sombre building designed after a mosque
at Mecca, surrounding a paved quadrangle 360 ft. square, were the other
principal buildings of the Kutb Shahi period, though the mosque was only
completed in the time of Aurangzeb. The city proper is surrounded by a
stone wall with thirteen gates, completed in the time of the first
nizam, who made Hyderabad his capital. The suburbs, of which the most
important is Chadarghat, extend over an additional area of 9 sq. m.
There are several fine palaces built by various nizams, and the British
residency is an imposing building in a large park on the left bank of
the Musi, N.E. of the city. The bazaars surrounding it, and under its
jurisdiction, are extremely picturesque and are thronged with natives
from all parts of India. Four bridges crossed the Musi, the most notable
of which was the Purana Pul, of 23 arches, built in 1593. On the 27th
and 28th of September 1908, however, the Musi, swollen by torrential
rainfall (during which 15 in. fell in 36 hours), rose in flood to a
height of 12 ft. above the bridges and swept them away. The damage done
was widespread; several important buildings were involved, including the
palace of Salar Jang and the Victoria zenana hospital, while the
beautiful grounds of the residency were destroyed. A large and densely
populated part of the city was wrecked, and thousands of lives were
lost. The principal educational establishments are the Nizam college
(first grade), engineering, law, medical, normal, industrial and
Sanskrit schools, and a number of schools for Europeans and Eurasians.
Hyderabad is an important centre of general trade, and there is a cotton
mill in its vicinity. The city is supplied with water from two notable
works, the Husain Sagar and the Mir Alam, both large lakes retained by
great dams. Secunderabad, the British military cantonment, is situated
5½ m. N. of the residency; it includes Bolaram, the former headquarters
of the Hyderabad contingent.




HYDER ALI, or Haidar 'Ali (c. 1722-1782), Indian ruler and commander.
This Mahommedan soldier-adventurer, who, followed by his son Tippoo,
became the most formidable Asiatic rival the British ever encountered in
India, was the great-grandson of a _fakir_ or wandering ascetic of
Islam, who had found his way from the Punjab to Gulburga in the Deccan,
and the second son of a _naik_ or chief constable at Budikota, near
Kolar in Mysore. He was born in 1722, or according to other authorities
1717. An elder brother, who like himself was early turned out into the
world to seek his own fortune, rose to command a brigade in the Mysore
army, while Hyder, who never learned to read or write, passed the first
years of his life aimlessly in sport and sensuality, sometimes, however,
acting as the agent of his brother, and meanwhile acquiring a useful
familiarity with the tactics of the French when at the height of their
reputation under Dupleix. He is said to have induced his brother to
employ a Parsee to purchase artillery and small arms from the Bombay
government, and to enrol some thirty sailors of different European
nations as gunners, and is thus credited with having been "the first
Indian who formed a corps of sepoys armed with firelocks and bayonets,
and who had a train of artillery served by Europeans." At the siege of
Devanhalli (1749) Hyder's services attracted the attention of Nanjiraj,
the minister of the raja of Mysore, and he at once received an
independent command; within the next twelve years his energy and ability
had made him completely master of minister and raja alike, and in
everything but in name he was ruler of the kingdom. In 1763 the conquest
of Kanara gave him possession of the treasures of Bednor, which he
resolved to make the most splendid capital in India, under his own name,
thenceforth changed from Hyder Naik into Hyder Ali Khan Bahadur; and in
1765 he retrieved previous defeat at the hands of the Mahrattas by the
destruction of the Nairs or military caste of the Malabar coast, and the
conquest of Calicut. Hyder Ali now began to occupy the serious attention
of the Madras government, which in 1766 entered into an agreement with
the nizam to furnish him with troops to be used against the common foe.
But hardly had this alliance been formed when a secret arrangement was
come to between the two Indian powers, the result of which was that
Colonel Smith's small force was met with a united army of 80,000 men and
100 guns. British dash and sepoy fidelity, however, prevailed, first in
the battle of Chengam (September 3rd, 1767), and again still more
remarkably in that of Tiruvannamalai (Trinomalai). On the loss of his
recently made fleet and forts on the western coast, Hyder Ali now
offered overtures for peace; on the rejection of these, bringing all his
resources and strategy into play, he forced Colonel Smith to raise the
siege of Bangalore, and brought his army within 5 m. of Madras. The
result was the treaty of April 1769, providing for the mutual
restitution of all conquests, and for mutual aid and alliance in
defensive war; it was followed by a commercial treaty in 1770 with the
authorities of Bombay. Under these arrangements Hyder Ali, when defeated
by the Mahrattas in 1772, claimed British assistance, but in vain; this
breach of faith stung him to fury, and thenceforward he and his son did
not cease to thirst for vengeance. His time came when in 1778 the
British, on the declaration of war with France, resolved to drive the
French out of India. The capture of Mahé on the coast of Malabar in
1779, followed by the annexation of lands belonging to a dependent of
his own, gave him the needed pretext. Again master of all that the
Mahrattas had taken from him, and with empire extended to the Kistna, he
descended through the passes of the Ghats amid burning villages,
reaching Conjeeveram, only 45 m. from Madras, unopposed. Not till the
smoke was seen from St Thomas's Mount, where Sir Hector Munro commanded
some 5200 troops, was any movement made; then, however, the British
general sought to effect a junction with a smaller body under Colonel
Baillie recalled from Guntur. The incapacity of these officers,
notwithstanding the splendid courage of their men, resulted in the total
destruction of Baillie's force of 2800 (September the 10th, 1780).
Warren Hastings sent from Bengal Sir Eyre Coote, who, though repulsed at
Chidambaram, defeated Hyder thrice successively in the battles of Porto
Novo, Pollilur and Sholingarh, while Tippoo was forced to raise the
siege of Wandiwash, and Vellore was provisioned. On the arrival of Lord
Macartney as governor of Madras, the British fleet captured Negapatam,
and forced Hyder Ali to confess that he could never ruin a power which
had command of the sea. He had sent his son Tippoo to the west coast, to
seek the assistance of the French fleet, when his death took place
suddenly at Chittur in December 1782.

  See L. B. Bowring, _Haidar Ali and Tipu Sultan_, "Rulers of India"
  series (1893). For the personal character and administration of Hyder
  Ali see the _History of Hyder Naik_, written by Mir Hussein Ali Khan
  Kirmani (translated from the Persian by Colonel Miles, and published
  by the Oriental Translation Fund), and the curious work written by M.
  Le Maître de La Tour, commandant of his artillery (_Histoire
  d'Hayder-Ali Khan_, Paris, 1783). For the whole life and times see
  Wilks, _Historical Sketches of the South of India_ (1810-1817);
  Aitchison's Treaties, vol. v. (2nd ed., 1876); and Pearson, _Memoirs
  of Schwartz_ (1834).




HYDRA (or SIDRA, NIDRA, IDERO, &c.; anc. _Hydrea_), an island of Greece,
lying about 4 m. off the S.E. coast of Argolis in the Peloponnesus, and
forming along with the neighbouring island of Dokos (Dhoko) the Bay of
Hydra. Pop. about 6200. The greatest length from south-west to
north-east is about 11 m., and the area is about 21 sq. mi.; but it is
little better than a rocky and treeless ridge with hardly a patch or two
of arable soil. Hence the epigram of Antonios Kriezes to the queen of
Greece: "The island produces prickly pears in abundance, splendid sea
captains and excellent prime ministers." The highest point, Mount Ere,
so called (according to Miaoules) from the Albanian word for wind, is
1958 ft. high. The next in importance is known as the Prophet Elias,
from the large convent of that name on its summit. It was there that the
patriot Theodorus Kolokotrones was imprisoned, and a large pine tree is
still called after him. The fact that in former times the island was
richly clad with woods is indicated by the name still employed by the
Turks, _Tchamliza_, the place of pines; but it is only in some favoured
spots that a few trees are now to be found. Tradition also has it that
it was once a well-watered island (hence the designation Hydrea), but
the inhabitants are now wholly dependent on the rain supply, and they
have sometimes had to bring water from the mainland. This lack of
fountains is probably to be ascribed in part to the effect of
earthquakes, which are not infrequent; that of 1769 continued for six
whole days. Hydra, the chief town, is built near the middle of the
northern coast, on a very irregular site, consisting of three hills and
the intervening ravines. From the sea its white and handsome houses
present a picturesque appearance, and its streets though narrow are
clean and attractive. Besides the principal harbour, round which the
town is built, there are three other ports on the north coast--Mandraki,
Molo, Panagia, but none of them is sufficiently sheltered. Almost all
the population of the island is collected in the chief town, which is
the seat of a bishop, and has a local court, numerous churches and a
high school. Cotton and silk weaving, tanning and shipbuilding are
carried on, and there is a fairly active trade.

Hydra was of no importance in ancient times. The only fact in its
history is that the people of Hermione (a city on the neighbouring
mainland now known by the common name of _Kastri_) surrendered it to
Samian refugees, and that from these the people of Troezen received it
in trust. It appears to be completely ignored by the Byzantine
chroniclers. In 1580 it was chosen as a refuge by a body of Albanians
from Kokkinyas in Troezenia; and other emigrants followed in 1590, 1628,
1635, 1640, &c. At the close of the 17th century the Hydriotes took part
in the reviving commerce of the Peloponnesus; and in course of time they
extended their range. About 1716 they began to build _sakturia_ (of from
10 to 15 tons burden), and to visit the islands of the Aegean; not long
after they introduced the _latinadika_ (40-50 tons), and sailed as far
as Alexandria, Constantinople, Trieste and Venice; and by and by they
ventured to France and even America. From the grain trade of south
Russia more especially they derived great wealth. In 1813 there were
about 22,000 people in the island, and of these 10,000 were seafarers.
At the time of the outbreak of the war of Greek independence the total
population was 28,190, of whom 16,460 were natives and the rest
foreigners. One of their chief families, the Konduriotti, was worth
£2,000,000. Into the struggle the Hydriotes flung themselves with rare
enthusiasm and devotion, and the final deliverance of Greece was mainly
due to the service rendered by their fleets.

  See Pouqueville, _Voy. de la Grèce_, vol. vi.; Antonios Miaoules,
  [Greek: Hypomnêma peri tês nêsou Hydras] (Munich, 1834); Id. [Greek:
  Sunoptikê historia tôn naumachiôn dia tôn ploiôn tôn triôn nêsôn,
  Hydras, Petsôn kai Psarôn] (Nauplia, 1833); Id. [Greek: Historia tês
  nêsou Hydras] (Athens, 1874); G. D. Kriezes, [Greek: Historia tês
  nêsou Hydras] (Patras, 1860).




HYDRA (watersnake), in Greek legend, the offspring of Typhon and
Echidna, a gigantic monster with nine heads (the number is variously
given), the centre one being immortal. Its haunt was a hill beneath a
plane tree near the river Amymone, in the marshes of Lerna by Argos. The
destruction of this Lernaean hydra was one of the twelve "labours"
of Heracles, which he accomplished with the assistance of Iolaus.
Finding that as soon as one head was cut off two grew up in its place,
they burnt out the roots with firebrands, and at last severed the
immortal head from the body, and buried it under a mighty block of rock.
The arrows dipped by Heracles in the poisonous blood or gall of the
monster ever afterwards inflicted fatal wounds. The generally accepted
interpretation of the legend is that "the hydra denotes the damp, swampy
ground of Lerna with its numerous springs ([Greek: kephalai], heads);
its poison the miasmic vapours rising from the stagnant water; its death
at the hands of Heracles the introduction of the culture and consequent
purification of the soil" (Preller). A euhemeristic explanation is given
by Palaephatus (39). An ancient king named Lernus occupied a small
citadel named Hydra, which was defended by 50 bowmen. Heracles besieged
the citadel and hurled firebrands at the garrison. As often as one of
the defenders fell, two others at once stepped into his place. The
citadel was finally taken with the assistance of the army of Iolaus and
the garrison slain.

  See Hesiod, _Theog._, 313; Euripides, _Hercules furens_, 419;
  Pausanias ii. 37; Apollodorus ii. 5, 2; Diod. Sic. iv. 11; Roscher's
  _Lexikon der Mythologie_. In the article GREEK ART, fig. 20 represents
  the slaying of the Lernaean hydra by Heracles.




HYDRA, in astronomy, a constellation of the southern hemisphere,
mentioned by Eudoxus (4th century B.C.) and Aratus (3rd century B.C.),
and catalogued by Ptolemy (27 stars), Tycho Brahe (19) and Hevelius
(31). Interesting objects are: the nebula _H. IV. 27 Hydrae_, a
planetary nebula, gaseous and whose light is about equal to an 8th
magnitude star; [epsilon] _Hydrae_, a beautiful triple star, composed of
two yellow stars of the 4th and 6th magnitudes, and a blue star of the
7th magnitude; _R. Hydrae_, a long period (425 days) variable, the range
in magnitude being from 4 to 9.7; and _U. Hydrae_, an irregularly
variable, the range in magnitude being 4.5 to 6.




HYDRACRYLIC ACID (ethylene lactic acid), CH2OH·CH2·CO2H. an organic
oxyacid prepared by acting with silver oxide and water on
[beta]-iodopropionic acid, or from ethylene by the addition of
hypochlorous acid, the addition product being then treated with
potassium cyanide and hydrolysed by an acid. It may also be prepared by
oxidizing the trimethylene glycol obtained by the action of hydrobromic
acid on allylbromide. It is a syrupy liquid, which on distillation is
resolved into water and the unsaturated acrylic acid, CH2:CH·CO2H.
Chromic and nitric acids oxidize it to oxalic acid and carbon dioxide.
Hydracrylic aldehyde, CH2OH·CH2·CHO, was obtained in 1904 by J. U. Nef
(_Ann._ 335, p. 219) as a colourless oil by heating acrolein with water.
Dilute alkalis convert it into crotonaldehyde, CH3·CH:CH·CHO.




HYDRANGEA, a popular flower, the plant to which the name is most
commonly applied being _Hydrangea Hortensia_, a low deciduous shrub,
producing rather large oval strongly-veined leaves in opposite pairs
along the stem. It is terminated by a massive globular corymbose head of
flowers, which remain a long period in an ornamental condition. The
normal colour of the flowers, the majority of which have neither stamens
nor pistil, is pink; but by the influence of sundry agents in the soil,
such as alum or iron, they become changed to blue. There are numerous
varieties, one of the most noteworthy being "Thomas Hogg" with pure
white flowers. The part of the inflorescence which appears to be the
flower is an exaggerated expansion of the sepals, the other parts being
generally abortive. The perfect flowers are small, rarely produced in
the species above referred to, but well illustrated by others, in which
they occupy the inner parts of the corymb, the larger showy neuter
flowers being produced at the circumference.

There are upwards of thirty species, found chiefly in Japan, in the
mountains of India, and in North America, and many of them are familiar
in gardens. _H. Hortensia_ (a species long known in cultivation In China
and Japan) is the most useful for decoration, as the head of flowers
lasts long in a fresh state, and by the aid of forcing can be had for a
considerable period for the ornamentation of the greenhouse and
conservatory. Their natural flowering season is towards the end of the
summer, but they may be had earlier by means of forcing. _H. japonica_
is another fine conservatory plant, with foliage and habit much
resembling the last named, but this has flat corymbs of flowers, the
central ones small and perfect, and the outer ones only enlarged and
neuter. This also produces pink or blue flowers under the influence of
different soils.

The Japanese species of hydrangea are sufficiently hardy to grow in any
tolerably favourable situation, but except in the most sheltered
localities they seldom blossom to any degree of perfection in the open
air, the head of blossom depending on the uninjured development of a
well-ripened terminal bud, and this growth being frequently affected by
late spring frosts. They are much more useful for pot-culture indoors,
and should be reared from cuttings of shoots having the terminal bud
plump and prominent, put in during summer, these developing a single
head of flowers the succeeding summer. Somewhat larger plants may be had
by nipping out the terminal bud and inducing three or four shoots to
start in its place, and these, being steadily developed and well
ripened, should each yield its inflorescence in the following summer,
that is, when two years old. Large plants grown in tubs and vases are
fine subjects for large conservatories, and useful for decorating
terrace walks and similar places during summer, being housed in winter,
and started under glass in spring.

_Hydrangea paniculata_ var. _grandiflora_ is a very handsome plant; the
branched inflorescence under favourable circumstances is a yard or more
in length, and consists of large spreading masses of crowded white
neuter flowers which completely conceal the few inconspicuous fertile
ones. The plant attains a height of 8 to 10 ft. and when in flower late
in summer and in autumn is a very attractive object in the shrubbery.

The Indian and American species, especially the latter, are quite hardy,
and some of them are extremely effective.




HYDRASTINE, C21H21NO6, an alkaloid found with berberine in the root of
golden seal, _Hydrastis canadensis_, a plant indigenous to North
America. It was discovered by Durand in 1851, and its chemistry formed
the subject of numerous communications by E. Schmidt and M. Freund (see
_Ann._, 1892, 271, p. 311) who, aided by P. Fritsch (_Ann._, 1895, 286,
p. 1), established its constitution. It is related to narcotine, which
is methoxy hydrastine. The root of golden seal is used in medicine under
the name hydrastis rhizome, as a stomachic and nervine stimulant.




HYDRATE, in chemistry, a compound containing the elements of water in
combination; more specifically, a compound containing the monovalent
hydroxyl or OH group. The first and more general definition includes
substances containing water of crystallization; such salts are said to
be hydrated, and when deprived of their water to be dehydrated or
anhydrous. Compounds embraced by the second definition are more usually
termed _hydroxides_, since at one time they were regarded as
combinations of an oxide with water, for example, calcium oxide or lime
when slaked with water yielded calcium hydroxide, written formerly as
CaO·H20. The general formulae of hydroxides are: M^iOH, M^(ii)(OH)2,
M^(iii)(OH)3, M^(iv)(OH)4, &c., corresponding to the oxides M2^iO,
M^(ii)O, M2^(iii)O3, M^(iv)O2, &c., the Roman index denoting the valency
of the element. There is an important difference between non-metallic
and metallic hydroxides; the former are invariably acids (oxyacids), the
latter are more usually basic, although acidic metallic oxides yield
acidic hydroxides. Elements exhibiting strong basigenic or oxygenic
characters yield the most, stable hydroxides; in other words, stable
hydroxides are associated with elements belonging to the extreme groups
of the periodic system, and unstable hydroxides with the central
members. The most stable basic hydroxides are those of the alkali
metals, viz. lithium, sodium, potassium, rubidium and caesium, and of
the alkaline earth metals, viz. calcium, barium and strontium; the most
stable acidic hydroxides are those of the elements placed in groups VB,
VIB and VIIB of the periodic table.




HYDRAULICS (Gr. [Greek: hydôr], water, and [Greek: aulos], a pipe), the
branch of engineering science which deals with the practical
applications of the laws of hydromechanics.


I. THE DATA OF HYDRAULICS[1]

§ 1. _Properties of Fluids._--The fluids to which the laws of practical
hydraulics relate are substances the parts of which possess very great
mobility, or which offer a very small resistance to distortion
independently of inertia. Under the general heading Hydromechanics a
fluid is defined to be a substance which yields continually to the
slightest tangential stress, and hence in a fluid at rest there can be
no tangential stress. But, further, in fluids such as water, air, steam,
&c., to which the present division of the article relates, the
tangential stresses that are called into action between contiguous
portions during distortion or change of figure are always small compared
with the weight, inertia, pressure, &c., which produce the visible
motions it is the object of hydraulics to estimate. On the other hand,
while a fluid passes easily from one form to another, it opposes
considerable resistance to change of volume.

It is easily deduced from the absence or smallness of the tangential
stress that contiguous portions of fluid act on each other with a
pressure which is exactly or very nearly normal to the interface which
separates them. The stress must be a pressure, not a tension, or the
parts would separate. Further, at any point in a fluid the pressure in
all directions must be the same; or, in other words, the pressure on any
small element of surface is independent of the orientation of the
surface.

§ 2. Fluids are divided into liquids, or incompressible fluids, and
gases, or compressible fluids. Very great changes of pressure change the
volume of liquids only by a small amount, and if the pressure on them is
reduced to zero they do not sensibly dilate. In gases or compressible
fluids the volume alters sensibly for small changes of pressure, and if
the pressure is indefinitely diminished they dilate without limit.

In ordinary hydraulics, liquids are treated as absolutely
incompressible. In dealing with gases the changes of volume which
accompany changes of pressure must be taken into account.

§ 3. Viscous fluids are those in which change of form under a continued
stress proceeds gradually and increases indefinitely. A very viscous
fluid opposes great resistance to change of form in a short time, and
yet may be deformed considerably by a small stress acting for a long
period. A block of pitch is more easily splintered than indented by a
hammer, but under the action of the mere weight of its parts acting for
a long enough time it flattens out and flows like a liquid.

[Illustration: FIG. 1.]

All actual fluids are viscous. They oppose a resistance to the relative
motion of their parts. This resistance diminishes with the velocity of
the relative motion, and becomes zero in a fluid the parts of which are
relatively at rest. When the relative motion of different parts of a
fluid is small, the viscosity may be neglected without introducing
important errors. On the other hand, where there is considerable
relative motion, the viscosity may be expected to have an influence too
great to be neglected.

  _Measurement of Viscosity. Coefficient of Viscosity._--Suppose the
  plane ab, fig. 1 of area [omega], to move with the velocity V
  relatively to the surface cd and parallel to it. Let the space between
  be filled with liquid. The layers of liquid in contact with ab and cd
  adhere to them. The intermediate layers all offering an equal
  resistance to shearing or distortion, the rectangle of fluid abcd will
  take the form of the parallelogram a´b´cd. Further, the resistance to
  the motion of ab may be expressed in the form

    R = [kappa][omega]V,   (1)

  where [kappa] is a coefficient the nature of which remains to be
  determined.

  If we suppose the liquid between ab and cd divided into layers as
  shown in fig. 2, it will be clear that the stress R acts, at each
  dividing face, forwards in the direction of motion if we consider the
  upper layer, backwards if we consider the lower layer. Now suppose the
  original thickness of the layer T increased to nT; if the bounding
  plane in its new position has the velocity nV, the shearing at each
  dividing face will be exactly the same as before, and the resistance
  must therefore be the same. Hence,

    R = [kappa]´[omega](nV).   (2)

  But equations (1) and (2) may both be expressed in one equation if
  [kappa] and [kappa]´ are replaced by a constant varying inversely as
  the thickness of the layer. Putting [kappa] = [mu]/T, [kappa]´ =
  [mu]/nT,

    R = [mu][omega]V/T;

  or, for an indefinitely thin layer,

    R = [mu][omega]dV/dt,   (3)

  an expression first proposed by L. M. H. Navier. The coefficient [mu]
  is termed the coefficient of viscosity.

  According to J. Clerk Maxwell, the value of [mu] for air at [theta]°
  Fahr. in pounds, when the velocities are expressed in feet per second,
  is

    [mu] = 0.0000000256 (461° + [theta]);

  that is, the coefficient of viscosity is proportional to the absolute
  temperature and independent of the pressure.

  The value of [mu] for water at 77° Fahr. is, according to H. von
  Helmholtz and G. Piotrowski,

    [mu] = 0.0000188,

  the units being the same as before. For water [mu] decreases rapidly
  with increase of temperature.

[Illustration: FIG. 2.]

§ 4. When a fluid flows in a very regular manner, as for instance when
It flows in a capillary tube, the velocities vary gradually at any
moment from one point of the fluid to a neighbouring point. The layer
adjacent to the sides of the tube adheres to it and is at rest. The
layers more interior than this slide on each other. But the resistance
developed by these regular movements is very small. If in large pipes
and open channels there were a similar regularity of movement, the
neighbouring filaments would acquire, especially near the sides, very
great relative velocities. V. J. Boussinesq has shown that the central
filament in a semicircular canal of 1 metre radius, and inclined at a
slope of only 0.0001, would have a velocity of 187 metres per second,[2]
the layer next the boundary remaining at rest. But before such a
difference of velocity can arise, the motion of the fluid becomes much
more complicated. Volumes of fluid are detached continually from the
boundaries, and, revolving, form eddies traversing the fluid in all
directions, and sliding with finite relative velocities against those
surrounding them. These slidings develop resistances incomparably
greater than the viscous resistance due to movements varying
continuously from point to point. The movements which produce the
phenomena commonly ascribed to fluid friction must be regarded as
rapidly or even suddenly varying from one point to another. The internal
resistances to the motion of the fluid do not depend merely on the
general velocities of translation at different points of the fluid (or
what Boussinesq terms the mean local velocities), but rather on the
intensity at each point of the eddying agitation. The problems of
hydraulics are therefore much more complicated than problems in which a
regular motion of the fluid is assumed, hindered by the viscosity of the
fluid.


RELATION OF PRESSURE, DENSITY, AND TEMPERATURE OF LIQUIDS

  § 5. _Units of Volume._--In practical calculations the cubic foot and
  gallon are largely used, and in metric countries the litre and cubic
  metre (= 1000 litres). The imperial gallon is now exclusively used in
  England, but the United States have retained the old English wine
  gallon.

    1 cub. ft.    = 6.236 imp. gallons = 7.481 U.S. gallons.
    1 imp. gallon = 0.1605 cub. ft.    = 1.200 U.S. gallons.
    1 U.S. gallon = 0.1337 cub. ft.    = 0.8333 imp. gallon.
    1 litre       = 0.2201 imp. gallon = 0.2641 U.S. gallon.

  _Density of Water._--Water at 53° F. and ordinary pressure contains
  62.4 lb. per cub. ft., or 10 lb. per imperial gallon at 62° F. The
  litre contains one kilogram of water at 4° C. or 1000 kilograms per
  cubic metre. River and spring water is not sensibly denser than pure
  water. But average sea water weighs 64 lb. per cub. ft. at 53° F. The
  weight of water per cubic unit will be denoted by G. Ice free from air
  weighs 57.28 lb. per cub. ft. (Leduc).

  § 6. _Compressibility of Liquids._--The most accurate experiments show
  that liquids are sensibly compressed by very great pressures, and that
  up to a pressure of 65 atmospheres, or about 1000 lb. per sq. in., the
  compression is proportional to the pressure. The chief results of
  experiment are given in the following table. Let V1 be the volume of a
  liquid in cubic feet under a pressure p1 lb. per sq. ft., and V2 its
  volume under a pressure p2. Then the cubical compression is (V2 -
  V1)/V1, and the ratio of the increase of pressure p2 - p1 to the
  cubical compression is sensibly constant. That is, k = (p2 - p1)V1/(V2
  - V1) is constant. This constant is termed the elasticity of volume.
  With the notation of the differential calculus,

             / /  dV \        dp
    k = dp  / ( - --  ) = - V --.
           /   \   V /        dV

    _Elasticity of Volume of Liquids._

    +-----------+------------+-----------+------------+------------+
    |           |   Canton.  |  Oersted. |  Colladon  |  Regnault. |
    |           |            |           | and Sturm. |            |
    +-----------+------------+-----------+------------+------------+
    | Water     | 45,990,000 | 45,900,000| 42,660,000 | 44,000,000 |
    | Sea water | 52,900,000 |     ..    |            |     ..     |
    | Mercury   |705,300,000 |     ..    |626,100,000 |604,500,000 |
    | Oil       | 44,090,000 |     ..    |            |     ..     |
    | Alcohol   | 32,060,000 |     ..    | 23,100,000 |     ..     |
    +-----------+------------+-----------+------------+------------+

  According to the experiments of Grassi, the compressibility of water
  diminishes as the temperature increases, while that of ether, alcohol
  and chloroform is increased.

  § 7. _Change of Volume and Density of Water with Change of
  Temperature._--Although the change of volume of water with change of
  temperature is so small that it may generally be neglected in ordinary
  hydraulic calculations, yet it should be noted that there is a change
  of volume which should be allowed for in very exact calculations. The
  values of [rho] in the following short table, which gives data enough
  for hydraulic purposes, are taken from Professor Everett's _System of
  Units_.

    _Density of Water at Different Temperatures._

    +-------------+----------+----------+
    |             |          |    G     |
    | Temperature.|  [rho]   |Weight of |
    +-----+-------+Density of|1 cub. ft.|
    |Cent.| Fahr. |  Water.  |  in lb.  |
    +-----+-------+----------+----------+
    |  0  |  32.0 |  .999884 |  62.417  |
    |  1  |  33.8 |  .999941 |  62.420  |
    |  2  |  35.6 |  .999982 |  62.423  |
    |  3  |  37.4 | 1.000004 |  62.424  |
    |  4  |  39.2 | 1.000013 |  62.425  |
    |  5  |  41.0 | 1.000003 |  62.424  |
    |  6  |  42.8 |  .999983 |  62.423  |
    |  7  |  44.6 |  .999946 |  62.421  |
    |  8  |  46.4 |  .999899 |  62.418  |
    |  9  |  48.2 |  .999837 |  62.414  |
    | 10  |  50.0 |  .999760 |  62.409  |
    | 11  |  51.8 |  .999668 |  62.403  |
    | 12  |  53.6 |  .999562 |  62.397  |
    | 13  |  55.4 |  .999443 |  62.389  |
    | 14  |  57.2 |  .999312 |  62.381  |
    | 15  |  59.0 |  .999173 |  62.373  |
    | 16  |  60.8 |  .999015 |  62.363  |
    | 17  |  62.6 |  .998854 |  62.353  |
    | 18  |  64.4 |  .998667 |  62.341  |
    | 19  |  66.2 |  .998473 |  62.329  |
    | 20  |  68.0 |  .998272 |  62.316  |
    | 22  |  71.6 |  .997839 |  62.289  |
    | 24  |  75.2 |  .997380 |  62.261  |
    | 26  |  78.8 |  .996879 |  62.229  |
    | 28  |  82.4 |  .996344 |  62.196  |
    | 30  |  86   |  .995778 |  62.161  |
    | 35  |  95   |  .99469  |  62.093  |
    | 40  | 104   |  .99236  |  61.947  |
    | 45  | 113   |  .99038  |  61.823  |
    | 50  | 122   |  .98821  |  61.688  |
    | 55  | 131   |  .98583  |  61.540  |
    | 60  | 140   |  .98339  |  61.387  |
    | 65  | 149   |  .98075  |  61.222  |
    | 70  | 158   |  .97795  |  61.048  |
    | 75  | 167   |  .97499  |  60.863  |
    | 80  | 176   |  .97195  |  60.674  |
    | 85  | 185   |  .96880  |  60.477  |
    | 90  | 194   |  .96557  |  60.275  |
    |100  | 212   |  .95866  |  59.844  |
    +-----+-------+----------+----------+

  The weight per cubic foot has been calculated from the values of
  [rho], on the assumption that 1 cub. ft. of water at 39.2° Fahr. is
  62.425 lb. For ordinary calculations in hydraulics, the density of
  water (which will in future be designated by the symbol G) will be
  taken at 62.4 lb. per cub. ft., which is its density at 53° Fahr. It
  may be noted also that ice at 32° Fahr. contains 57.3 lb. per cub. ft.
  The values of [rho] are the densities in grammes per cubic centimetre.

  § 8. _Pressure Column. Free Surface Level._--Suppose a small vertical
  pipe introduced into a liquid at any point P (fig. 3). Then the liquid
  will rise in the pipe to a level OO, such that the pressure due to the
  column in the pipe exactly balances the pressure on its mouth. If the
  fluid is in motion the mouth of the pipe must be supposed accurately
  parallel to the direction of motion, or the impact of the liquid at
  the mouth of the pipe will have an influence on the height of the
  column. If this condition is complied with, the height h of the
  column is a measure of the pressure at the point P. Let [omega] be the
  area of section of the pipe, h the height of the pressure column, p
  the intensity of pressure at P; then

    p[omega] = Gh[omega] lb.,

    p/G = h;

  that is, h is the height due to the pressure at p. The level OO will
  be termed the free surface level corresponding to the pressure at P.


  RELATION OF PRESSURE, TEMPERATURE, AND DENSITY OF GASES

  § 9. _Relation of Pressure, Volume, Temperature and Density in
  Compressible Fluids._--Certain problems on the flow of air and steam
  are so similar to those relating to the flow of water that they are
  conveniently treated together. It is necessary, therefore, to state as
  briefly as possible the properties of compressible fluids so far as
  knowledge of them is requisite in the solution of these problems. Air
  may be taken as a type of these fluids, and the numerical data here
  given will relate to air.

  [Illustration: FIG. 3.]

  _Relation of Pressure and Volume at Constant Temperature._--At
  constant temperature the product of the pressure p and volume V of a
  given quantity of air is a constant (Boyle's law).

  Let p0 be mean atmospheric pressure (2116.8 lb. per sq. ft.), V0 the
  volume of 1 lb. of air at 32° Fahr. under the pressure p0. Then

    p0V0 = 26214.   (1)

  If G0 is the weight per cubic foot of air in the same conditions,

    G0 = 1/V0 = 2116.8/26214 = .08075.   (2)

  For any other pressure p, at which the volume of 1 lb. is V and the
  weight per cubic foot is G, the temperature being 32° Fahr.,

    pV = p/G = 26214; or G = p/26214.   (3)

  _Change of Pressure or Volume by Change of Temperature._--Let p0, V0,
  G0, as before be the pressure, the volume of a pound in cubic feet,
  and the weight of a cubic foot in pounds, at 32° Fahr. Let p, V, G be
  the same quantities at a temperature t (measured strictly by the air
  thermometer, the degrees of which differ a little from those of a
  mercurial thermometer). Then, by experiment,

    pV = p0V0(460.6 + t)/(460.6 + 32) = p0V0[tau]/[tau]0,   (4)

  where [tau], [tau]0 are the temperatures t and 32° reckoned from the
  absolute zero, which is -460.6° Fahr.;

    p/G = p0[tau]/G0[tau]0;   (4a)

    G = p[tau]0G0/p0[tau].   (5)

  If p0 = 2116.8, G0 = .08075, [tau]0 = 460.6 + 32 = 492.6, then

    p/G = 53.2[tau].   (5a)

  Or quite generally p/G = R[tau] for all gases, if R is a constant
  varying inversely as the density of the gas at 32° F. For steam R =
  85.5.


II. KINEMATICS OF FLUIDS

§ 10. Moving fluids as commonly observed are conveniently classified
thus:

(1) _Streams_ are moving masses of indefinite length, completely or
incompletely bounded laterally by solid boundaries. When the solid
boundaries are complete, the flow is said to take place in a pipe. When
the solid boundary is incomplete and leaves the upper surface of the
fluid free, it is termed a stream bed or channel or canal.

(2) A stream bounded laterally by differently moving fluid of the same
kind is termed a _current_.

(3) A _jet_ is a stream bounded by fluid of a different kind.

(4) An _eddy_, _vortex_ or _whirlpool_ is a mass of fluid the particles
of which are moving circularly or spirally.

(5) In a stream we may often regard the particles as flowing along
definite paths in space. A chain of particles following each other along
such a constant path may be termed a fluid filament or elementary
stream.

  § 11. _Steady and Unsteady, Uniform and Varying, Motion._--There are
  two quite distinct ways of treating hydrodynamical questions. We may
  either fix attention on a given mass of fluid and consider its changes
  of position and energy under the action of the stresses to which it is
  subjected, or we may have regard to a given fixed portion of space,
  and consider the volume and energy of the fluid entering and leaving
  that space.

  If, in following a given path ab (fig. 4), a mass of water a has a
  constant velocity, the motion is said to be uniform. The kinetic
  energy of the mass a remains unchanged. If the velocity varies from
  point to point of the path, the motion is called varying motion. If at
  a given point a in space, the particles of water always arrive with
  the same velocity and in the same direction, during any given time,
  then the motion is termed steady motion. On the contrary, if at the
  point a the velocity or direction varies from moment to moment the
  motion is termed unsteady. A river which excavates its own bed is in
  unsteady motion so long as the slope and form of the bed is changing.
  It, however, tends always towards a condition in which the bed ceases
  to change, and it is then said to have reached a condition of
  permanent regime. No river probably is in absolutely permanent regime,
  except perhaps in rocky channels. In other cases the bed is scoured
  more or less during the rise of a flood, and silted again during the
  subsidence of the flood. But while many streams of a torrential
  character change the condition of their bed often and to a large
  extent, in others the changes are comparatively small and not easily
  observed.

  [Illustration: FIG. 4.]

  As a stream approaches a condition of steady motion, its regime
  becomes permanent. Hence steady motion and permanent regime are
  sometimes used as meaning the same thing. The one, however, is a
  definite term applicable to the motion of the water, the other a less
  definite term applicable in strictness only to the condition of the
  stream bed.

  § 12. _Theoretical Notions on the Motion of Water._--The actual motion
  of the particles of water is in most cases very complex. To simplify
  hydrodynamic problems, simpler modes of motion are assumed, and the
  results of theory so obtained are compared experimentally with the
  actual motions.

  _Motion in Plane Layers._--The simplest kind of motion in a stream is
  one in which the particles initially situated in any plane cross
  section of the stream continue to be found in plane cross sections
  during the subsequent motion. Thus, if the particles in a thin plane
  layer ab (fig. 5) are found again in a thin plane layer a´b´ after any
  interval of time, the motion is said to be motion in plane layers. In
  such motion the internal work in deforming the layer may usually be
  disregarded, and the resistance to the motion is confined to the
  circumference.

  [Illustration: FIG. 5.]

  _Laminar Motion._--In the case of streams having solid boundaries, it
  is observed that the central parts move faster than the lateral parts.
  To take account of these differences of velocity, the stream may be
  conceived to be divided into thin laminae, having cross sections
  somewhat similar to the solid boundary of the stream, and sliding on
  each other. The different laminae can then be treated as having
  differing velocities according to any law either observed or deduced
  from their mutual friction. A much closer approximation to the real
  motion of ordinary streams is thus obtained.

  _Stream Line Motion._--In the preceding hypothesis, all the particles
  in each lamina have the same velocity at any given cross section of
  the stream. If this assumption is abandoned, the cross section of the
  stream must be supposed divided into indefinitely small areas, each
  representing the section of a fluid filament. Then these filaments may
  have any law of variation of velocity assigned to them. If the motion
  is steady motion these fluid filaments (or as they are then termed
  _stream lines_) will have fixed positions in space.

  _Periodic Unsteady Motion._--In ordinary streams with rough
  boundaries, it is observed that at any given point the velocity varies
  from moment to moment in magnitude and direction, but that the average
  velocity for a sensible period (say for 5 or 10 minutes) varies very
  little either in magnitude or velocity. It has hence been conceived
  that the variations of direction and magnitude of the velocity are
  periodic, and that, if for each point of the stream the mean velocity
  and direction of motion were substituted for the actual more or less
  varying motions, the motion of the stream might be treated as steady
  stream line or steady laminar motion.

  [Illustration: FIG. 6.]

  § 13. _Volume of Flow._--Let A (fig. 6) be any ideal plane surface, of
  area [omega], in a stream, normal to the direction of motion, and let
  V be the velocity of the fluid. Then the volume flowing through the
  surface A in unit time is

    Q = [omega]V.   (1)

  Thus, if the motion is rectilinear, all the particles at any instant
  in the surface A will be found after one second in a similar surface
  A´, at a distance V, and as each particle is followed by a continuous
  thread of other particles, the volume of flow is the right prism AA´
  having a base [omega] and length V.

  If the direction of motion makes an angle [theta] with the normal to
  the surface, the volume of flow is represented by an oblique prism AA´
  (fig. 7), and in that case

    Q = [omega]V cos [theta].

  [Illustration: FIG. 7.]

  If the velocity varies at different points of the surface, let the
  surface be divided into very small portions, for each of which the
  velocity may be regarded as constant. If d[omega] is the area and v,
  or v cos [theta], the normal velocity for this element of the surface,
  the volume of flow is
         _                _
        /                /
    Q = | v d[omega], or | v cos [theta] d[omega],
       _/               _/

  as the case may be.

  § 14. _Principle of Continuity._--If we consider any completely
  bounded fixed space in a moving liquid initially and finally filled
  continuously with liquid, the inflow must be equal to the outflow.
  Expressing the inflow with a positive and the outflow with a negative
  sign, and estimating the volume of flow Q for all the boundaries,

    [Sigma]Q = 0.

  In general the space will remain filled with fluid if the pressure at
  every point remains positive. There will be a break of continuity, if
  at any point the pressure becomes negative, indicating that the stress
  at that point is tensile. In the case of ordinary water this statement
  requires modification. Water contains a variable amount of air in
  solution, often about one-twentieth of its volume. This air is
  disengaged and breaks the continuity of the liquid, if the pressure
  falls below a point corresponding to its tension. It is for this
  reason that pumps will not draw water to the full height due to
  atmospheric pressure.

  _Application of the Principle of Continuity to the case of a
  Stream._--If A1, A2 are the areas of two normal cross sections of a
  stream, and V1, V2 are the velocities of the stream at those sections,
  then from the principle of continuity,

    V1A1 = V2A2;

    V1/V2 = A2/A1   (2)

  that is, the normal velocities are inversely as the areas of the cross
  sections. This is true of the mean velocities, if at each section the
  velocity of the stream varies. In a river of varying slope the
  velocity varies with the slope. It is easy therefore to see that in
  parts of large cross section the slope is smaller than in parts of
  small cross section.

  If we conceive a space in a liquid bounded by normal sections at A1,
  A2 and between A1, A2 by stream lines (fig. 8), then, as there is no
  flow across the stream lines,

    V1/V2 = A2/A1,

  as in a stream with rigid boundaries.

  [Illustration: FIG. 8.]

  In the case of compressible fluids the variation of volume due to the
  difference of pressure at the two sections must be taken into account.
  If the motion is steady the weight of fluid between two cross sections
  of a stream must remain constant. Hence the weight flowing in must be
  the same as the weight flowing out. Let p1, p2 be the pressures, v1,
  v2 the velocities, G1, G2 the weight per cubic foot of fluid, at cross
  sections of a stream of areas A1, A2. The volumes of inflow and
  outflow are

    A1v1 and A2v2,

  and, if the weights of these are the same,

    G1A1v1 = G2A2v2;

  and hence, from (5a) § 9, if the temperature is constant,

    p1A1v1 = p2A2v2.   (3)

  § 15. _Stream Lines._--The characteristic of a perfect fluid, that is,
  a fluid free from viscosity, is that the pressure between any two
  parts into which it is divided by a plane must be normal to the plane.
  One consequence of this is that the particles can have no rotation
  impressed upon them, and the motion of such a fluid is irrotational. A
  stream line is the line, straight or curved, traced by a particle in a
  current of fluid in irrotational movement. In a steady current each
  stream line preserves its figure and position unchanged, and marks the
  track of a stream of particles forming a fluid filament or elementary
  stream. A current in steady irrotational movement may be conceived to
  be divided by insensibly thin partitions following the course of the
  stream lines into a number of elementary streams. If the positions of
  these partitions are so adjusted that the volumes of flow in all the
  elementary streams are equal, they represent to the mind the velocity
  as well as the direction of motion of the particles in different parts
  of the current, for the velocities are inversely proportional to the
  cross sections of the elementary streams. No actual fluid is devoid of
  viscosity, and the effect of viscosity is to render the motion of a
  fluid sinuous, or rotational or eddying under most ordinary
  conditions. At very low velocities in a tube of moderate size the
  motion of water may be nearly pure stream line motion. But at some
  velocity, smaller as the diameter of the tube is greater, the motion
  suddenly becomes tumultuous. The laws of simple stream line motion
  have hitherto been investigated theoretically, and from mathematical
  difficulties have only been determined for certain simple cases.
  Professor H. S. Hele Shaw has found means of exhibiting stream line
  motion in a number of very interesting cases experimentally. Generally
  in these experiments a thin sheet of fluid is caused to flow between
  two parallel plates of glass. In the earlier experiments streams of
  very small air bubbles introduced into the water current rendered
  visible the motions of the water. By the use of a lantern the image of
  a portion of the current can be shown on a screen or photographed. In
  later experiments streams of coloured liquid at regular distances were
  introduced into the sheet and these much more clearly marked out the
  forms of the stream lines. With a fluid sheet 0.02 in. thick, the
  stream lines were found to be stable at almost any required velocity.
  For certain simple cases Professor Hele Shaw has shown that the
  experimental stream lines of a viscous fluid are so far as can be
  measured identical with the calculated stream lines of a perfect
  fluid. Sir G. G. Stokes pointed out that in this case, either from the
  thinness of the stream between its glass walls, or the slowness of the
  motion, or the high viscosity of the liquid, or from a combination of
  all these, the flow is regular, and the effects of inertia disappear,
  the viscosity dominating everything. Glycerine gives the stream lines
  very satisfactorily.

  [Illustration: FIG. 9.]

  [Illustration: FIG. 10.]

  [Illustration: FIG. 11.]

  [Illustration: FIG. 12.]

  [Illustration: FIG. 13.]

  Fig. 9 shows the stream lines of a sheet of fluid passing a fairly
  shipshape body such as a screwshaft strut. The arrow shows the
  direction of motion of the fluid. Fig. 10 shows the stream lines for a
  very thin glycerine sheet passing a non-shipshape body, the stream
  lines being practically perfect. Fig. 11 shows one of the earlier
  air-bubble experiments with a thicker sheet of water. In this case the
  stream lines break up behind the obstruction, forming an eddying wake.
  Fig. 12 shows the stream lines of a fluid passing a sudden contraction
  or sudden enlargement of a pipe. Lastly, fig. 13 shows the stream
  lines of a current passing an oblique plane. H. S. Hele Shaw,
  "Experiments on the Nature of the Surface Resistance in Pipes and on
  Ships," _Trans. Inst. Naval Arch._ (1897). "Investigation of Stream
  Line Motion under certain Experimental Conditions," _Trans. Inst.
  Naval Arch._ (1898); "Stream Line Motion of a Viscous Fluid," _Report
  of British Association_ (1898).


  III. PHENOMENA OF THE DISCHARGE OF LIQUIDS FROM ORIFICES AS
  ASCERTAINABLE BY EXPERIMENTS

  § 16. When a liquid issues vertically from a small orifice, it forms a
  jet which rises nearly to the level of the free surface of the liquid
  in the vessel from which it flows. The difference of level h_r (fig.
  14) is so small that it may be at once suspected to be due either to
  air resistance on the surface of the jet or to the viscosity of the
  liquid or to friction against the sides of the orifice. Neglecting for
  the moment this small quantity, we may infer, from the elevation of
  the jet, that each molecule on leaving the orifice possessed the
  velocity required to lift it against gravity to the height h. From
  ordinary dynamics, the relation between the velocity and height of
  projection is given by the equation

    v = [root](2gh).   (1)

  As this velocity is nearly reached in the flow from well-formed
  orifices, it is sometimes called the theoretical velocity of
  discharge. This relation was first obtained by Torricelli.

  [Illustration: FIG. 14.]

  If the orifice is of a suitable conoidal form, the water issues in
  filaments normal to the plane of the orifice. Let [omega] be the area
  of the orifice, then the discharge per second must be, from eq. (1),

    Q = [omega]v = [omega][root](2gh) nearly.   (2)

  This is sometimes quite improperly called the theoretical discharge
  for any kind of orifice. Except for a well-formed conoidal orifice the
  result is not approximate even, so that if it is supposed to be based
  on a theory the theory is a false one.

  _Use of the term Head in Hydraulics._--The term _head_ is an old
  millwright's term, and meant primarily the height through which a mass
  of water descended in actuating a hydraulic machine. Since the water
  in fig. 14 descends through a height h to the orifice, we may say
  there are h ft. of head above the orifice. Still more generally any
  mass of liquid h ft. above a horizontal plane may be said to have h
  ft. of elevation head relatively to that datum plane. Further, since
  the pressure p at the orifice which produces outflow is connected with
  h by the relation p/G = h, the quantity p/G may be termed the pressure
  head at the orifice. Lastly, the velocity v is connected with h by the
  relation v²/2g = h, so that v²/2g may be termed the head due to the
  velocity v.

  § 17. _Coefficients of Velocity and Resistance._--As the actual
  velocity of discharge differs from [root]2gh by a small quantity, let
  the actual velocity

    = v_a = c_v [root](2gh),   (3)

  where c_v is a coefficient to be determined by experiment, called the
  _coefficient of velocity_. This coefficient is found to be tolerably
  constant for different heads with well-formed simple orifices, and it
  very often has the value 0.97.

  The difference between the velocity of discharge and the velocity due
  to the head may be reckoned in another way. The total height h causing
  outflow consists of two parts--one part h_e expended effectively in
  producing the velocity of outflow, another h_r in overcoming the
  resistances due to viscosity and friction. Let

    h_r = c_r h_e,

  where c{r} is a coefficient determined by experiment, and called the
  _coefficient of resistance_ of the orifice. It is tolerably constant
  for different heads with well-formed orifices. Then

    v_a = [root](2gh_e) = [root]{2gh/(1 + c_r)}.   (4)

  The relation between c_v and c_r for any orifice is easily found:--

    v_a = c_v[root](2gh) = [root]{2gh/(1 + c_r)}

    c_v = [root]{1/(1 + c_r)}   (5)

    c_r = 1/c_v² - 1   (5a)

  Thus if c_v = 0.97, then c_r = 0.0628. That is, for such an orifice
  about 6¼% of the head is expended in overcoming frictional resistances
  to flow.

  [Illustration: FIG. 15.]

  _Coefficient of Contraction--Sharp-edged Orifices in Plane
  Surfaces._--When a jet issues from an aperture in a vessel, it may
  either spring clear from the inner edge of the orifice as at a or b
  (fig. 15), or it may adhere to the sides of the orifice as at c. The
  former condition will be found if the orifice is bevelled outwards as
  at a, so as to be sharp edged, and it will also occur generally for a
  prismatic aperture like b, provided the thickness of the plate in
  which the aperture is formed is less than the diameter of the jet. But
  if the thickness is greater the condition shown at c will occur.

  When the discharge occurs as at a or b, the filaments converging
  towards the orifice continue to converge beyond it, so that the
  section of the jet where the filaments have become parallel is smaller
  than the section of the orifice. The inertia of the filaments opposes
  sudden change of direction of motion at the edge of the orifice, and
  the convergence continues for a distance of about half the diameter of
  the orifice beyond it. Let [omega] be the area of the orifice, and
  c_c[omega] the area of the jet at the point where convergence ceases;
  then c_c is a coefficient to be determined experimentally for each
  kind of orifice, called the _coefficient of contraction_. When the
  orifice is a sharp-edged orifice in a plane surface, the value of c_c
  is on the average 0.64, or the section of the jet is very nearly
  five-eighths of the area of the orifice.

  _Coefficient of Discharge._--In applying the general formula Q =
  [omega]v to a stream, it is assumed that the filaments have a common
  velocity v normal to the section [omega]. But if the jet contracts, it
  is at the contracted section of the jet that the direction of motion
  is normal to a transverse section of the jet. Hence the actual
  discharge when contraction occurs is

    Q_a = c_vv × c_c[omega] = c_c c_v[omega][root](2gh),

  or simply, if c = c_vc_c,

    Q_a = c[omega][root](2gh),

  where c is called the _coefficient of discharge_. Thus for a
  sharp-edged plane orifice c = 0.97 × 0.64 = 0.62.

  [Illustration: FIG. 16.]

  § 18. _Experimental Determination of c_v, c_c, and c._--The
  coefficient of contraction c_c is directly determined by measuring the
  dimensions of the jet. For this purpose fixed screws of fine pitch
  (fig. 16) are convenient. These are set to touch the jet, and then the
  distance between them can be measured at leisure.

  The coefficient of velocity is determined directly by measuring the
  parabolic path of a horizontal jet.

  Let OX, OY (fig. 17) be horizontal and vertical axes, the origin being
  at the orifice. Let h be the head, and x, y the coordinates of a point
  A on the parabolic path of the jet. If v_a is the velocity at the
  orifice, and t the time in which a particle moves from O to A, then

    x = v_a t; y = ½gt².

  Eliminating t,

    v_a = [root](gx²/2y).

  Then

    c_v = v_a [root](2gh) = [root](x²/4yh).

  In the case of large orifices such as weirs, the velocity can be
  directly determined by using a Pitot tube (§ 144).

  [Illustration: FIG. 17.]

  The coefficient of discharge, which for practical purposes is the most
  important of the three coefficients, is best determined by tank
  measurement of the flow from the given orifice in a suitable time. If
  Q is the discharge measured in the tank per second, then

    c = Q/[omega][root](2gh).

  Measurements of this kind though simple in principle are not free from
  some practical difficulties, and require much care. In fig. 18 is
  shown an arrangement of measuring tank. The orifice is fixed in the
  wall of the cistern A and discharges either into the waste channel BB,
  or into the measuring tank. There is a short trough on rollers C which
  when run under the jet directs the discharge into the tank, and when
  run back again allows the discharge to drop into the waste channel. D
  is a stilling screen to prevent agitation of the surface at the
  measuring point, E, and F is a discharge valve for emptying the
  measuring tank. The rise of level in the tank, the time of the flow
  and the head over the orifice at that time must be exactly observed.

  [Illustration: FIG. 18.]

  For well made sharp-edged orifices, small relatively to the water
  surface in the supply reservoir, the coefficients under different
  conditions of head are pretty exactly known. Suppose the same quantity
  of water is made to flow in succession through such an orifice and
  through another orifice of which the coefficient is required, and when
  the rate of flow is constant the heads over each orifice are noted.
  Let h1, h2 be the heads, [omega]1, [omega]2 the areas of the orifices,
  c1, c2 the coefficients. Then since the flow through each orifice is
  the same

    Q = c1[omega]1 [root](2gh1) = c2[omega]2 [root](2gh2).

    c2 = c1([omega]1/[omega]2) [root](h1/h2).

  [Illustration: FIG. 19.]

  § 19. _Coefficients for Bellmouths and Bellmouthed Orifices._--If an
  orifice is furnished with a mouthpiece exactly of the form of the
  contracted vein, then the whole of the contraction occurs within the
  mouthpiece, and if the area of the orifice is measured at the smaller
  end, c_c must be put = 1. It is often desirable to bellmouth the ends
  of pipes, to avoid the loss of head which occurs if this is not
  done; and such a bellmouth may also have the form of the contracted
  jet. Fig. 19 shows the proportions of such a bellmouth or bell-mouthed
  orifice, which approximates to the form of the contracted jet
  sufficiently for any practical purpose.

  For such an orifice L. J. Weisbach found the following values of the
  coefficients with different heads.

    +--------------------------------+------+------+------+------+-------+
    | Head over orifice, in ft. = h  | .66  | 1.64 |11.48 |55.77 |337.93 |
    +--------------------------------+------+------+------+------+-------+
    | Coefficient of velocity = c_v  | .959 | .967 | .975 | .994 |  .994 |
    | Coefficient of resistance = c_r| .087 | .069 | .052 | .012 |  .012 |
    +--------------------------------+------+------+------+------+-------+

  As there is no contraction after the jet issues from the orifice, c_c
  = 1, c = c_v; and therefore

    Q = c(v)[omega][root](2gh) = [omega][root]{2gh/(1 + c_r}.

  § 20. _Coefficients for Sharp-edged or virtually Sharp-edged
  Orifices._--There are a very large number of measurements of discharge
  from sharp-edged orifices under different conditions of head. An
  account of these and a very careful tabulation of the average values
  of the coefficients will be found in the _Hydraulics_ of the late
  Hamilton Smith (Wiley & Sons, New York, 1886). The following short
  table abstracted from a larger one will give a fair notion of how the
  coefficient varies according to the most trustworthy of the
  experiments.

    _Coefficient of Discharge for Vertical Circular Orifices, Sharp-edged,
    with free Discharge into the Air._ Q = c[omega][root](2gh).

    +-----------+------------------------------------------------+
    |   Head    |            Diameters of Orifice.               |
    |measured to+------+------+------+------+------+------+------+
    | Centre of | .02  | .04  | .10  | .20  | .40  | .60  | 1.0  |
    |  Orifice. +------+------+------+------+------+------+------+
    |           |                  Values of C.                  |
    +-----------+------+------+------+------+------+------+------+
    |    0.3    |  ..  | ..   | .621 |  ..  |  ..  |  ..  |  ..  |
    |    0.4    |  ..  | .637 | .618 |  ..  |  ..  |  ..  |  ..  |
    |    0.6    | .655 | .630 | .613 | .601 | .596 | .588 |  ..  |
    |    0.8    | .648 | .626 | .610 | .601 | .597 | .594 | .583 |
    |    1.0    | .644 | .623 | .608 | .600 | .598 | .595 | .591 |
    |    2.0    | .632 | .614 | .604 | .599 | .599 | .597 | .595 |
    |    4.0    | .623 | .609 | .602 | .599 | .598 | .597 | .596 |
    |    8.0    | .614 | .605 | .600 | .598 | .597 | .596 | .596 |
    |   20.0    | .601 | .599 | .596 | .596 | .596 | .596 | .594 |
    +-----------+------+------+------+------+------+------+------+

  At the same time it must be observed that differences of sharpness in
  the edge of the orifice and some other circumstances affect the
  results, so that the values found by different careful experimenters
  are not a little discrepant. When exact measurement of flow has to be
  made by a sharp-edged orifice it is desirable that the coefficient for
  the particular orifice should be directly determined.

  The following results were obtained by Dr H. T. Bovey in the
  laboratory of McGill University.

    _Coefficient of Discharge for Sharp-edged Orifices._

    +----+------------------------------------------------------------------+
    |    |                         Form of Orifice.                         |
    |    +------+----------------+-----------------+-----------------+------+
    |    |      |     Square.    |Rectangular Ratio|Rectangular Ratio|      |
    |Head|      |                |  of Sides 4:1   |  of Sides 16:1  |      |
    | in | Cir- +------+---------+---------+-------+---------+-------+ Tri- |
    | ft.|cular.|Sides |         |  Long   | Long  |  Long   | Long  |angu- |
    |    |      |Verti-|Diagonal |  Sides  | Sides |  Sides  | Sides | lar. |
    |    |      | cal. |Vertical.|Vertical.| hori- |Vertical.| Hori- |      |
    |    |      |      |         |         |zontal.|         |zontal.|      |
    +----+------+------+---------+---------+-------+---------+-------+------+
    |  1 | .620 | .627 |  .628   |   .642  |  .643 |   .663  |  .664 | .636 |
    |  2 | .613 | .620 |  .628   |   .634  |  .636 |   .650  |  .651 | .628 |
    |  4 | .608 | .616 |  .618   |   .628  |  .629 |   .641  |  .642 | .623 |
    |  6 | .607 | .614 |  .616   |   .626  |  .627 |   .637  |  .637 | .620 |
    |  8 | .606 | .613 |  .614   |   .623  |  .625 |   .634  |  .635 | .619 |
    | 10 | .605 | .612 |  .613   |   .622  |  .624 |   .632  |  .633 | .618 |
    | 12 | .604 | .611 |  .612   |   .622  |  .623 |   .631  |  .631 | .618 |
    | 14 | .604 | .610 |  .612   |   .621  |  .622 |   .630  |  .630 | .618 |
    | 16 | .603 | .610 |  .611   |   .620  |  .622 |   .630  |  .630 | .617 |
    | 18 | .603 | .610 |  .611   |   .620  |  .621 |   .630  |  .629 | .616 |
    | 20 | .603 | .609 |  .611   |   .620  |  .621 |   .629  |  .628 | .616 |
    +----+------+------+---------+---------+-------+---------+-------+------+

  The orifice was 0.196 sq. in. area and the reductions were made with g
  = 32.176 the value for Montreal. The value of the coefficient appears
  to increase as (perimeter) / (area) increases. It decreases as the
  head increases. It decreases a little as the size of the orifice is
  greater.

  Very careful experiments by J. G. Mair (_Proc. Inst. Civ. Eng._
  lxxxiv.) on the discharge from circular orifices gave the results
  shown on top of next column.

  The edges of the orifices were got up with scrapers to a sharp square
  edge. The coefficients generally fall as the head increases and as the
  diameter increases. Professor W. C. Unwin found that the results agree
  with the formula

    c = 0.6075 + 0.0098/[root]h - 0.0037d,

  where h is in feet and d in inches.

    _Coefficients of Discharge from Circular Orifices. Temperature 51° to
    55°._

    +-------+--------------------------------------------------------------+
    |Head in|             Diameters of Orifices in Inches (d).             |
    | feet  +------+------+------+------+------+------+------+------+------+
    |   h.  |   1  |  1¼  |  1½  |  1¾  |   2  |  2¼  |  2½  |  2¾  |   3  |
    +-------+------+------+------+------+------+------+------+------+------+
    |       |                       Coefficients (c).                      |
    |       +------+------+------+------+------+------+------+------+------+
    |   .75 | .616 | .614 | .616 | .610 | .616 | .612 | .607 | .607 | .609 |
    |  1.0  | .613 | .612 | .612 | .611 | .612 | .611 | .604 | .608 | .609 |
    |  1.25 | .613 | .614 | .610 | .608 | .612 | .608 | .605 | .605 | .606 |
    |  1.50 | .610 | .612 | .611 | .606 | .610 | .607 | .603 | .607 | .605 |
    |  1.75 | .612 | .611 | .611 | .605 | .611 | .605 | .604 | .607 | .605 |
    |  2.00 | .609 | .613 | .609 | .606 | .609 | .606 | .604 | .604 | .605 |
    +-------+------+------+------+------+------+------+------+------+------+

  The following table, compiled by J. T. Fanning (_Treatise on Water
  Supply Engineering_), gives values for rectangular orifices in
  vertical plane surfaces, the head being measured, not immediately over
  the orifice, where the surface is depressed, but to the still-water
  surface at some distance from the orifice. The values were obtained by
  graphic interpolation, all the most reliable experiments being plotted
  and curves drawn so as to average the discrepancies.

    _Coefficients of Discharge for Rectangular Orifices, Sharp-edged, in
    Vertical Plane Surfaces._

    +--------+----------------------------------------------------------------+
    |  Head  |                  Ratio of Height to Width.                     |
    |   to   |                                                                |
    | Centre +------+------+------+------+--------+--------+--------+---------+
    |   of   |      |      |      |      |        |        |        |         |
    |Orifice.|  4   |  2   |  1½  |  1   |   ¾    |    ½   |    ¼   |   1/8   |
    +--------+------+------+------+------+--------+--------+--------+---------+
    |        | 4 ft.| 2 ft.|1½ ft.| 1 ft.|0.75 ft.|0.50 ft.|0.25 ft.|0.125 ft.|
    |        | high.| high.| high.| high.|  high. |  high. |  high. |  high.  |
    |  Feet. |      |      |      |      |        |        |        |         |
    |        | 1 ft.| 1 ft.| 1 ft.| 1 ft.|  1 ft. |  1 ft. |  1 ft. |  1 ft.  |
    |        | wide.| wide.| wide.| wide.|  wide. |  wide. |  wide. |  wide.  |
    +--------+------+------+------+------+--------+--------+--------+---------+
    |   0.2  |  ..  |  ..  |  ..  |  ..  |   ..   |   ..   |   ..   |  .6333  |
    |    .3  |  ..  |  ..  |  ..  |  ..  |   ..   |   ..   | .6293  |  .6334  |
    |    .4  |  ..  |  ..  |  ..  |  ..  |   ..   | .6140  | .6306  |  .6334  |
    |    .5  |  ..  |  ..  |  ..  |  ..  | .6050  | .6150  | .6313  |  .6333  |
    |    .6  |  ..  |  ..  |  ..  |.5984 | .6063  | .6156  | .6317  |  .6332  |
    |    .7  |  ..  |  ..  |  ..  |.5994 | .6074  | .6162  | .6319  |  .6328  |
    |    .8  |  ..  |  ..  |.6130 |.6000 | .6082  | .6165  | .6322  |  .6326  |
    |    .9  |  ..  |  ..  |.6134 |.6006 | .6086  | .6168  | .6323  |  .6324  |
    |   1.0  |  ..  |  ..  |.6135 |.6010 | .6090  | .6172  | .6320  |  .6320  |
    |   1.25 |  ..  |.6188 |.6140 |.6018 | .6095  | .6173  | .6317  |  .6312  |
    |   1.50 |  ..  |.6187 |.6144 |.6026 | .6100  | .6172  | .6313  |  .6303  |
    |   1.75 |  ..  |.6186 |.6145 |.6033 | .6103  | .6168  | .6307  |  .6296  |
    |   2    |  ..  |.6183 |.6144 |.6036 | .6104  | .6166  | .6302  |  .6291  |
    |   2.25 |  ..  |.6180 |.6143 |.6029 | .6103  | .6163  | .6293  |  .6286  |
    |   2.50 |.6290 |.6176 |.6139 |.6043 | .6102  | .6157  | .6282  |  .6278  |
    |   2.75 |.6280 |.6173 |.6136 |.6046 | .6101  | .6155  | .6274  |  .6273  |
    |   3    |.6273 |.6170 |.6132 |.6048 | .6100  | .6153  | .6267  |  .6267  |
    |   3.5  |.6250 |.6160 |.6123 |.6050 | .6094  | .6146  | .6254  |  .6254  |
    |   4    |.6245 |.6150 |.6110 |.6047 | .6085  | .6136  | .6236  |  .6236  |
    |   4.5  |.6226 |.6138 |.6100 |.6044 | .6074  | .6125  | .6222  |  .6222  |
    |   5    |.6208 |.6124 |.6088 |.6038 | .6063  | .6114  | .6202  |  .6202  |
    |   6    |.6158 |.6094 |.6063 |.6020 | .6044  | .6087  | .6154  |  .6154  |
    |   7    |.6124 |.6064 |.6038 |.6011 | .6032  | .6058  | .6110  |  .6114  |
    |   8    |.6090 |.6036 |.6022 |.6010 | .6022  | .6033  | .6073  |  .6087  |
    |   9    |.6060 |.6020 |.6014 |.6010 | .6015  | .6020  | .6045  |  .6070  |
    |  10    |.6035 |.6015 |.6010 |.6010 | .6010  | .6010  | .6030  |  .6060  |
    |  15    |.6040 |.6018 |.6010 |.6011 | .6012  | .6013  | .6033  |  .6066  |
    |  20    |.6045 |.6024 |.6012 |.6012 | .6014  | .6018  | .6036  |  .6074  |
    |  25    |.6048 |.6028 |.6014 |.6012 | .6016  | .6022  | .6040  |  .6083  |
    |  30    |.6054 |.6034 |.6017 |.6013 | .6018  | .6027  | .6044  |  .6092  |
    |  35    |.6060 |.6039 |.6021 |.6014 | .6022  | .6032  | .6049  |  .6103  |
    |  40    |.6066 |.6045 |.6025 |.6015 | .6026  | .6037  | .6055  |  .6114  |
    |  45    |.6054 |.6052 |.6029 |.6016 | .6030  | .6043  | .6062  |  .6125  |
    |  50    |.6086 |.6060 |.6034 |.6018 | .6035  | .6050  | .6070  |  .6140  |
    +--------+------+------+------+------+--------+--------+--------+---------+

  § 21. _Orifices with Edges of Sensible Thickness._--When the edges of
  the orifice are not bevelled outwards, but have a sensible thickness,
  the coefficient of discharge is somewhat altered. The following table
  gives values of the coefficient of discharge for the arrangements of
  the orifice shown in vertical section at P, Q, R (fig. 20). The plan
  of all the orifices is shown at S. The planks forming the orifice and
  sluice were each 2 in. thick, and the orifices were all 24 in. wide.
  The heads were measured immediately over the orifice. In this case,

    Q = cb(H - h) [root]{2g(H + h)/2}.

  § 22. _Partially Suppressed Contraction._--Since the contraction of
  the jet is due to the convergence towards the orifice of the issuing
  streams, it will be diminished if for any portion of the edge of the
  orifice the convergence is prevented. Thus, if an internal rim or
  border is applied to part of the edge of the orifice (fig. 21), the
  convergence for so much of the edge is suppressed. For such cases G.
  Bidone found the following empirical formulae applicable:--

    _Table of Coefficients of Discharge for Rectangular Vertical Orifices
    in Fig. 20._

    +--------+-----------------------------------------------------------------------------------------------+
    |Head h  |                                                                                               |
    |above   |                                Height of Orifice, H - h, in feet                              |
    |upper   +-----------------------+-----------------------+-----------------------+-----------------------+
    |edge of |          1.31         |          0.66         |          0.16         |          0.10         |
    |Orifice +-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
    |in feet.|   P   |   Q   |   R   |   P   |   Q   |   R   |   P   |   Q   |   R   |   P   |   Q   |   R   |
    +--------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
    |  0.328 | 0.598 | 0.644 | 0.648 | 0.634 | 0.665 | 0.668 | 0.691 | 0.664 | 0.666 | 0.710 | 0.694 | 0.696 |
    |   .656 | 0.609 | 0.653 | 0.657 | 0.640 | 0.672 | 0.675 | 0.685 | 0.687 | 0.688 | 0.696 | 0.704 | 0.706 |
    |   .787 | 0.612 | 0.655 | 0.659 | 0.641 | 0.674 | 0.677 | 0.684 | 0.690 | 0.692 | 0.694 | 0.706 | 0.708 |
    |   .984 | 0.616 | 0.656 | 0.660 | 0.641 | 0.675 | 0.678 | 0.683 | 0.693 | 0.695 | 0.692 | 0.709 | 0.711 |
    |  1.968 | 0.618 | 0.649 | 0.653 | 0.640 | 0.676 | 0.679 | 0.678 | 0.695 | 0.697 | 0.688 | 0.710 | 0.712 |
    |  3.28  | 0.608 | 0.632 | 0.634 | 0.638 | 0.674 | 0.676 | 0.673 | 0.694 | 0.695 | 0.680 | 0.704 | 0.705 |
    |  4.27  | 0.602 | 0.624 | 0.626 | 0.637 | 0.673 | 0.675 | 0.672 | 0.693 | 0.694 | 0.678 | 0.701 | 0.702 |
    |  4.92  | 0.598 | 0.620 | 0.622 | 0.637 | 0.673 | 0.674 | 0.672 | 0.692 | 0.693 | 0.676 | 0.699 | 0.699 |
    |  5.58  | 0.596 | 0.618 | 0.620 | 0.637 | 0.672 | 0.673 | 0.672 | 0.692 | 0.693 | 0.676 | 0.698 | 0.698 |
    |  6.56  | 0.595 | 0.615 | 0.617 | 0.636 | 0.671 | 0.672 | 0.671 | 0.691 | 0.692 | 0.675 | 0.696 | 0.696 |
    |  9.84  | 0.592 | 0.611 | 0.612 | 0.634 | 0.669 | 0.670 | 0.668 | 0.689 | 0.690 | 0.672 | 0.693 | 0.693 |
    +--------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+

  For rectangular orifices,

    C_c = 0.62(1 + 0.152n/p);

  and for circular orifices,

    C_c = 0.62(1 + 0.128n/p);

  when n is the length of the edge of the orifice over which the border
  extends, and p is the whole length of edge or perimeter of the
  orifice. The following are the values of c_c, when the border extends
  over ¼, ½, or ¾ of the whole perimeter:--

    +--------+-----------------------+--------------------+
    |        |          C_c          |        C_c         |
    |   n/p  | Rectangular Orifices. | Circular Orifices. |
    +--------+-----------------------+--------------------+
    |  0.25  |         0.643         |        .640        |
    |  0.50  |         0.667         |        .660        |
    |  0.75  |         0.691         |        .680        |
    +--------+-----------------------+--------------------+

  [Illustration: FIG. 20.]

  [Illustration: FIG. 21.]

  For larger values of n/p the formulae are not applicable. C. R.
  Bornemann has shown, however, that these formulae for suppressed
  contraction are not reliable.

  § 23. _Imperfect Contraction._--If the sides of the vessel approach
  near to the edge of the orifice, they interfere with the convergence
  of the streams to which the contraction is due, and the contraction is
  then modified. It is generally stated that the influence of the sides
  begins to be felt if their distance from the edge of the orifice is
  less than 2.7 times the corresponding width of the orifice. The
  coefficients of contraction for this case are imperfectly known.

  [Illustration: FIG. 22.]

  § 24. _Orifices Furnished with Channels of Discharge._--These external
  borders to an orifice also modify the contraction.

  The following coefficients of discharge were obtained with openings 8
  in. wide, and small in proportion to the channel of approach (fig. 22,
  A, B, C).

    +-----------+-------------------------------------------------------+
    |   h2--h1  |                       h1 in feet.                     |
    |  in feet  |------+-----+-----+-----+------+-----+-----+-----+-----+
    |           |.0656 |.164 |.328 |.656 |1.640 |3.28 |4.92 |6.56 |9.84 |
    +-----------+------+-----+-----+-----+------+-----+-----+-----+-----+
    | A\        | .480 |.511 |.542 |.574 | .599 |.601 |.601 |.601 |.601 |
    | B > 0.656 | .480 |.510 |.538 |.506 | .592 |.600 |.602 |.602 |.601 |
    | C/        | .527 |.553 |.574 |.592 | .607 |.610 |.610 |.609 |.608 |
    |           |      |     |     |     |      |     |     |     |     |
    | A\        | .488 |.577 |.624 |.631 | .625 |.624 |.619 |.613 |.606 |
    | B > 0.164 | .487 |.571 |.606 |.617 | .626 |.628 |.627 |.623 |.618 |
    | C/        | .585 |.614 |.633 |.645 | .652 |.651 |.650 |.650 |.649 |
    +-----------+------+-----+-----+-----+------+-----+-----+-----+-----+

  [Illustration: FIG. 23.]

  § 25. _Inversion of the Jet._--When a jet issues from a horizontal
  orifice, or is of small size compared with the head, it presents no
  marked peculiarity of form. But if the orifice is in a vertical
  surface, and if its dimensions are not small compared with the head,
  it undergoes a series of singular changes of form after leaving
  the orifice. These were first investigated by G. Bidone (1781-1839);
  subsequently H. G. Magnus (1802-1870) measured jets from different
  orifices; and later Lord Rayleigh (_Proc. Roy. Soc._ xxix. 71)
  investigated them anew.

  Fig. 23 shows some forms, the upper figure giving the shape of the
  orifices, and the others sections of the jet. The jet first contracts
  as described above, in consequence of the convergence of the fluid
  streams within the vessel, retaining, however, a form similar to that
  of the orifice. Afterwards it expands into sheets in planes
  perpendicular to the sides of the orifice. Thus the jet from a
  triangular orifice expands into three sheets, in planes bisecting at
  right angles the three sides of the triangle. Generally a jet from an
  orifice, in the form of a regular polygon of n sides, forms n sheets
  in planes perpendicular to the sides of the polygon.

  Bidone explains this by reference to the simpler case of meeting
  streams. If two equal streams having the same axis, but moving in
  opposite directions, meet, they spread out into a thin disk normal to
  the common axis of the streams. If the directions of two streams
  intersect obliquely they spread into a symmetrical sheet perpendicular
  to the plane of the streams.

  [Illustration: FIG. 24.]

  Let a1, a2 (fig. 24) be two points in an orifice at depths h1, h2 from
  the free surface. The filaments issuing at a1, a2 will have the
  different velocities [root](2gh1) and [root](2gh2). Consequently they
  will tend to describe parabolic paths a1cb1 and a2cb2 of different
  horizontal range, and intersecting in the point c. But since two
  filaments cannot simultaneously flow through the same point, they must
  exercise mutual pressure, and will be deflected out of the paths they
  tend to describe. It is this mutual pressure which causes the
  expansion of the jet into sheets.

  Lord Rayleigh pointed out that, when the orifices are small and the
  head is not great, the expansion of the sheets in directions
  perpendicular to the direction of flow reaches a limit. Sections taken
  at greater distance from the orifice show a contraction of the sheets
  until a compact form is reached similar to that at the first
  contraction. Beyond this point, if the jet retains its coherence,
  sheets are thrown out again, but in directions bisecting the angles
  between the previous sheets. Lord Rayleigh accepts an explanation of
  this contraction first suggested by H. Buff (1805-1878), namely, that
  it is due to surface tension.

  § 26. _Influence of Temperature on Discharge of Orifices._--Professor
  VV. C. Unwin found (_Phil. Mag._, October 1878, p. 281) that for
  sharp-edged orifices temperature has a very small influence on the
  discharge. For an orifice 1 cm. in diameter with heads of about 1 to
  1½ ft. the coefficients were:--

    Temperature F.      C.
        205°          .594
         62°          .598

  For a conoidal or bell-mouthed orifice 1 cm. diameter the effect of
  temperature was greater:--

    Temperature F.      C.
        190°          0.987
        130°          0.974
         60°          0.942

  an increase in velocity of discharge of 4% when the temperature
  increased 130°.

  J. G. Mair repeated these experiments on a much larger scale (_Proc.
  Inst. Civ. Eng._ lxxxiv.). For a sharp-edged orifice 2½ in. diameter,
  with a head of 1.75 ft., the coefficient was 0.604 at 57° and 0.607 at
  179° F., a very small difference. With a conoidal orifice the
  coefficient was 0.961 at 55° and 0.98l at 170° F. The corresponding
  coefficients of resistance are 0.0828 and 0.0391, showing that the
  resistance decreases to about half at the higher temperature.

  § 27. _Fire Hose Nozzles._--Experiments have been made by J. R.
  Freeman on the coefficient of discharge from smooth cone nozzles used
  for fire purposes. The coefficient was found to be 0.983 for ¾-in.
  nozzle; 0.982 for 7/8 in.; 0.972 for 1 in.; 0.976 for 1(1/8) in.;
  and 0.971 for 1¼ in. The nozzles were fixed on a taper play-pipe, and
  the coefficient includes the resistance of this pipe (_Amer. Soc. Civ.
  Eng._ xxi., 1889). Other forms of nozzle were tried such as ring
  nozzles for which the coefficient was smaller.


  IV. THEORY OF THE STEADY MOTION OF FLUIDS.

  § 28. The general equation of the steady motion of a fluid given under
  Hydrodynamics furnishes immediately three results as to the
  distribution of pressure in a stream which may here be assumed.

  (a) If the motion is rectilinear and uniform, the variation of
  pressure is the same as in a fluid at rest. In a stream flowing in an
  open channel, for instance, when the effect of eddies produced by the
  roughness of the sides is neglected, the pressure at each point is
  simply the hydrostatic pressure due to the depth below the free
  surface.

  (b) If the velocity of the fluid is very small, the distribution of
  pressure is approximately the same as in a fluid at rest.

  (c) If the fluid molecules take precisely the accelerations which they
  would have if independent and submitted only to the external forces,
  the pressure is uniform. Thus in a jet falling freely in the air the
  pressure throughout any cross section is uniform and equal to the
  atmospheric pressure.

  (d) In any bounded plane section traversed normally by streams which
  are rectilinear for a certain distance on either side of the section,
  the distribution of pressure is the same as in a fluid at rest.


  DISTRIBUTION OF ENERGY IN INCOMPRESSIBLE FLUIDS.

  § 29. _Application of the Principle of the Conservation of Energy to
  Cases of Stream Line Motion._--The external and internal work done on
  a mass is equal to the change of kinetic energy produced. In many
  hydraulic questions this principle is difficult to apply, because from
  the complicated nature of the motion produced it is difficult to
  estimate the total kinetic energy generated, and because in some cases
  the internal work done in overcoming frictional or viscous resistances
  cannot be ascertained; but in the case of stream line motion it
  furnishes a simple and important result known as Bernoulli's theorem.

  [Illustration: FIG. 25.]

  Let AB (fig. 25) be any one elementary stream, in a steadily moving
  fluid mass. Then, from the steadiness of the motion, AB is a fixed
  path in space through which a stream of fluid is constantly flowing.
  Let OO be the free surface and XX any horizontal datum line. Let
  [omega] be the area of a normal cross section, v the velocity, p the
  intensity of pressure, and z the elevation above XX, of the elementary
  stream AB at A, and [omega]1, p1, v1, z1 the same quantities at B.
  Suppose that in a short time t the mass of fluid initially occupying
  AB comes to A´B´. Then AA´, BB´ are equal to vt, v1t, and the volumes
  of fluid AA´, BB´ are the equal inflow and outflow = Qt = [omega]vt =
  [omega]1v1t, in the given time. If we suppose the filament AB
  surrounded by other filaments moving with not very different
  velocities, the frictional or viscous resistance on its surface will
  be small enough to be neglected, and if the fluid is incompressible no
  internal work is done in change of volume. Then the work done by
  external forces will be equal to the kinetic energy produced in the
  time considered.

  The normal pressures on the surface of the mass (excluding the ends A,
  B) are at each point normal to the direction of motion, and do no
  work. Hence the only external forces to be reckoned are gravity and
  the pressures on the ends of the stream.

  The work of gravity when AB falls to A´B´ is the same as that of
  transferring AA´ to BB´; that is, GQt(z - z1). The work of the
  pressures on the ends, reckoning that at B negative, because it is
  opposite to the direction of motion, is (p[omega] × vt) - (p1[omega]1
  × v1t) = Qt(p - p1). The change of kinetic energy in the time t is the
  difference of the kinetic energy originally possessed by AA´ and that
  finally acquired by BB´, for in the intermediate part A´B there is no
  change of kinetic energy, in consequence of the steadiness of the
  motion. But the mass of AA´ and BB´ is GQt/g, and the change of
  kinetic energy is therefore (GQt/g) (v1²/2 - v²/2). Equating this to
  the work done on the mass AB,

    GQt(z - z1) + Qt(p - p1) = (GQt/g)(v1²/2 - v²/2).

  Dividing by GQt and rearranging the terms,

    v²/2g + p/G + z = v1²/2g + p1/G + z1;   (1)

  or, as A and B are any two points,

    v²/2g + p/G + z = constant = H.   (2)

  Now v²/2g is the head due to the velocity v, p/G is the head
  equivalent to the pressure, and z is the elevation above the datum
  (see § 16). Hence the terms on the left are the total head due to
  velocity, pressure, and elevation at a given cross section of the
  filament, z is easily seen to be the work in foot-pounds which would
  be done by 1 lb. of fluid falling to the datum line, and similarly p/G
  and v²/2g are the quantities of work which would be done by 1 lb. of
  fluid due to the pressure p and velocity v. The expression on the left
  of the equation is, therefore, the total energy of the stream at the
  section considered, per lb. of fluid, estimated with reference to the
  datum line XX. Hence we see that in stream line motion, under
  the restrictions named above, the total energy per lb. of fluid is
  uniformly distributed along the stream line. If the free surface of
  the fluid OO is taken as the datum, and -h, -h1 are the depths of A
  and B measured down from the free surface, the equation takes the form

    v²/2g + p/G - h = v1²/2g + p1/G - h1;   (3)

  or generally

    v²/2g + p/G - h = constant.   (3a)

  [Illustration: FIG. 26.]

  § 30. _Second Form of the Theorem of Bernoulli._--Suppose at the two
  sections A, B (fig. 26) of an elementary stream small vertical pipes
  are introduced, which may be termed pressure columns (§ 8), having
  their lower ends accurately parallel to the direction of flow. In such
  tubes the water will rise to heights corresponding to the pressures at
  A and B. Hence b = p/G, and b´ = p1/G. Consequently the tops of the
  pressure columns A´ and B´ will be at total heights b + c = p/G + z
  and b´ + c´ = p1/G + z1 above the datum line XX. The difference of
  level of the pressure column tops, or the fall of free surface level
  between A and B, is therefore

    [xi] = (p - p1)/G + (z - z1);

  and this by equation (1), § 29 is (v1² - v²)/2g. That is, the fall of
  free, surface level between two sections is equal to the difference of
  the heights due to the velocities at the sections. The line A´B´ is
  sometimes called the line of hydraulic gradient, though this term is
  also used in cases where friction needs to be taken into account. It
  is the line the height of which above datum is the sum of the
  elevation and pressure head at that point, and it falls below a
  horizontal line A´´B´´ drawn at H ft. above XX by the quantities a =
  v²/2g and a´ = v1²/2g, when friction is absent.

  § 31. _Illustrations of the Theorem of Bernoulli._ In a lecture to the
  mechanical section of the British Association in 1875, W. Froude gave
  some experimental illustrations of the principle of Bernoulli. He
  remarked that it was a common but erroneous impression that a fluid
  exercises in a contracting pipe A (fig. 27) an excess of pressure
  against the entire converging surface which it meets, and that,
  conversely, as it enters an enlargement B, a relief of pressure is
  experienced by the entire diverging surface of the pipe. Further it is
  commonly assumed that when passing through a contraction C, there is
  in the narrow neck an excess of pressure due to the squeezing together
  of the liquid at that point. These impressions are in no respect
  correct; the pressure is smaller as the section of the pipe is smaller
  and conversely.

  [Illustration: FIG. 27.]

  Fig. 28 shows a pipe so formed that a contraction is followed by an
  enlargement, and fig. 29 one in which an enlargement is followed by a
  contraction. The vertical pressure columns show the decrease of
  pressure at the contraction and increase of pressure at the
  enlargement. The line abc in both figures shows the variation of free
  surface level, supposing the pipe frictionless. In actual pipes,
  however, work is expended in friction against the pipe; the total head
  diminishes in proceeding along the pipe, and the free surface level is
  a line such as ab1c1, falling below abc.

  Froude further pointed out that, if a pipe contracts and enlarges
  again to the same size, the resultant pressure on the converging part
  exactly balances the resultant pressure on the diverging part so that
  there is no tendency to move the pipe bodily when water flows through
  it. Thus the conical part AB (fig. 30) presents the same projected
  surface as HI, and the pressures parallel to the axis of the pipe,
  normal to these projected surfaces, balance each other. Similarly the
  pressures on BC, CD balance those on GH, EG. In the same way, in any
  combination of enlargements and contractions, a balance of pressures,
  due to the flow of liquid parallel to the axis of the pipe, will be
  found, provided the sectional area and direction of the ends are the
  same.

  [Illustration: FIG. 28.]

  [Illustration: FIG. 29.]

  The following experiment is interesting. Two cisterns provided with
  converging pipes were placed so that the jet from one was exactly
  opposite the entrance to the other. The cisterns being filled very
  nearly to the same level, the jet from the left-hand cistern A entered
  the right-hand cistern B (fig. 31), shooting across the free space
  between them without any waste, except that due to indirectness of aim
  and want of exact correspondence in the form of the orifices. In the
  actual experiment there was 18 in. of head in the right and 20½ in. of
  head in the left-hand cistern, so that about 2½ in. were wasted in
  friction. It will be seen that in the open space between the orifices
  there was no pressure, except the atmospheric pressure acting
  uniformly throughout the system.

  [Illustration: FIG. 30.]

  [Illustration: FIG. 31.]

  § 32. _Venturi Meter._--An ingenious application of the variation of
  pressure and velocity in a converging and diverging pipe has been made
  by Clemens Herschel in the construction of what he terms a Venturi
  Meter for measuring the flow in water mains. Suppose that, as in fig.
  32, a contraction is made in a water main, the change of section being
  gradual to avoid the production of eddies. The ratio [rho] of the
  cross sections at A and B, that is at inlet and throat, is in actual
  meters 5 to 1 to 20 to 1, and is very carefully determined by the
  maker of the meter. Then, if v and u are the velocities at A and B, u
  = [rho]v. Let pressure pipes be introduced at A, B and C, and let H1,
  H, H2 be the pressure heads at those points. Since the velocity at B
  is greater than at A the pressure will be less. Neglecting friction

    H1 + v²/2g = H + u²/2g,

    H1 - H = (u² - v²)/2g = ([rho]² - 1)v²/2g.

  Let h = H1 - H be termed the Venturi head, then

    u = [root]{[rho]²·2gh/([rho]² - 1)},

  from which the velocity through the throat and the discharge of the
  main can be calculated if the areas at A and B are known and h
  observed. Thus if the diameters at A and B are 4 and 12 in., the areas
  are 12.57 and 113.1 sq. in., and [rho] = 9,

    u = [root]81/80 [root](2gh) = 1.007 [root](2gh).

  If the observed Venturi head is 12 ft.,

    u = 28 ft. per sec.,

  and the discharge of the main is

    28 × 12.57 = 351 cub. ft. per sec.

  [Illustration: FIG. 32.]

  Hence by a simple observation of pressure difference, the flow in the
  main at any moment can be determined. Notice that the pressure height
  at C will be the same as at A except for a small loss h_f due to
  friction and eddying between A and B. To get the pressure at the
  throat very exactly Herschel surrounds it by an annular passage
  communicating with the throat by several small holes, sometimes formed
  in vulcanite to prevent corrosion. Though constructed to prevent
  eddying as much as possible there is some eddy loss. The main effect
  of this is to cause a loss of head between A and C which may vary from
  a fraction of a foot to perhaps 5 ft. at the highest velocities at
  which a meter can be used. The eddying also affects a little the
  Venturi head h. Consequently an experimental coefficient must be
  determined for each meter by tank measurement. The range of this
  coefficient is, however, surprisingly small. If to allow for friction,
  u = k[root]{[rho]²/([rho]² - 1)}[root](2gh), then Herschel found
  values of k from 0.97 to 1.0 for throat velocities varying from 8 to
  28 ft. per sec. The meter is extremely convenient. At Staines
  reservoirs there are two meters of this type on mains 94 in. in
  diameter. Herschel contrived a recording arrangement which records the
  variation of flow from hour to hour and also the total flow in any
  given time. In Great Britain the meter is constructed by G. Kent, who
  has made improvements in the recording arrangement.

  [Illustration: FIG. 33.]

  In the Deacon Waste Water Meter (fig. 33) a different principle is
  used. A disk D, partly counter-balanced by a weight, is suspended in
  the water flowing through the main in a conical chamber. The
  unbalanced weight of the disk is supported by the impact of the water.
  If the discharge of the main increases the disk rises, but as it rises
  its position in the chamber is such that in consequence of the larger
  area the velocity is less. It finds, therefore, a new position of
  equilibrium. A pencil P records on a drum moved by clockwork the
  position of the disk, and from this the variation of flow is inferred.

  § 33. _Pressure, Velocity and Energy in Different Stream Lines._--The
  equation of Bernoulli gives the variation of pressure and velocity
  from point to point along a stream line, and shows that the total
  energy of the flow across any two sections is the same. Two other
  directions may be defined, one normal to the stream line and in the
  plane containing its radius of curvature at any point, the other
  normal to the stream line and the radius of curvature. For the
  problems most practically useful it will be sufficient to consider the
  stream lines as parallel to a vertical or horizontal plane. If the
  motion is in a vertical plane, the action of gravity must be taken
  into the reckoning; if the motion is in a horizontal plane, the terms
  expressing variation of elevation of the filament will disappear.[3]

  [Illustration: FIG. 34.]

  Let AB, CD (fig. 34) be two consecutive stream lines, at present
  assumed to be in a vertical plane, and PQ a normal to these lines
  making an angle [phi] with the vertical. Let P, Q be two particles
  moving along these lines at a distance PQ = ds, and let z be the
  height of Q above the horizontal plane with reference to which the
  energy is measured, v its velocity, and p its pressure. Then, if H is
  the total energy at Q per unit of weight of fluid,

    H = z + p/G + v²/2g.

  Differentiating, we get

    dH = dz + dp/G + vdv/g,   (1)

  for the increment of energy between Q and P. But

    dz = PQ cos [phi] = ds cos [phi];

    .: dH = dp/G + v dv/g + ds cos [phi],   (1a)

  where the last term disappears if the motion is in a horizontal plane.

  Now imagine a small cylinder of section [omega] described round PQ as
  an axis. This will be in equilibrium under the action of its
  centrifugal force, its weight and the pressure on its ends. But its
  volume is [omega] ds and its weight G[omega]ds. Hence, taking the
  components of the forces parallel to PQ--

    [omega]dp = Gv²[omega] ds/g[rho] - G[omega] cos [phi] ds,

  where [rho] is the radius of curvature of the stream line at Q.
  Consequently, introducing these values in (1),

    dH = v² ds/g[rho] + v dv/g = (v/g)(v/[rho] + dv/ds) ds.   (2)


  CURRENTS

  § 34. _Rectilinear Current._--Suppose the motion is in parallel
  straight stream lines (fig. 35) in a vertical plane. Then [rho] is
  infinite, and from eq. (2), § 33,

     dH = v dv/g.

  Comparing this with (1) we see that

    dz + dp/G = 0;

    .: z + p/G = constant; (3)

  or the pressure varies hydrostatically as in a fluid at rest. For two
  stream lines in a horizontal plane, z is constant, and therefore p is
  constant.

  [Illustration: FIG. 35.]

  _Radiating Current._--Suppose water flowing radially between
  horizontal parallel planes, at a distance apart = [delta]. Conceive
  two cylindrical sections of the current at radii r1 and r2, where the
  velocities are v1 and v2, and the pressures p1 and p2. Since the flow
  across each cylindrical section of the current is the same,

    Q = 2[pi]r1[delta]v1 = 2[pi]r2[delta]v2

    r1v1 = r2v2

    r1/r2 = v2/v1.   (4)

  The velocity would be infinite at radius 0, if the current could be
  conceived to extend to the axis. Now, if the motion is steady,

    H = p1/G + v1²/2g = p2/G + v2²/2g;
      = p2/G + r1² + v1²/r2²2g;

    (p2- p1)/G = v1²(1 - r1²/r2²)/2g;   (5)

    p2/G = H - r1²v1²/r2²2g.   (6)

  Hence the pressure increases from the interior outwards, in a way
  indicated by the pressure columns in fig. 36, the curve through the
  free surfaces of the pressure columns being, in a radial section, the
  quasi-hyperbola of the form xy² = c³. This curve is asymptotic to a
  horizontal line, H ft. above the line from which the pressures are
  measured, and to the axis of the current.

  [Illustration: FIG. 36.]

  _Free Circular Vortex._--A free circular vortex is a revolving mass of
  water, in which the stream lines are concentric circles, and in which
  the total head for each stream line is the same. Hence, if by any slow
  radial motion portions of the water strayed from one stream line to
  another, they would take freely the velocities proper to their new
  positions under the action of the existing fluid pressures only.

  For such a current, the motion being horizontal, we have for all the
  circular elementary streams

    H = p/G + v²/2g = constant;

    .: dH = dp/G + v dv/g = 0.   (7)

  Consider two stream lines at radii r and r + dr (fig. 36). Then in
  (2), § 33, [rho] = r and ds = dr,

    v² dr/gr + v dv/g = 0,

    dv/v = -dr/r,

    v [oo] 1/r,   (8)

  precisely as in a radiating current; and hence the distribution of
  pressure is the same, and formulae 5 and 6 are applicable to this
  case.

  _Free Spiral Vortex._--As in a radiating and circular current the
  equations of motion are the same, they will also apply to a vortex in
  which the motion is compounded of these motions in any proportions,
  provided the radial component of the motion varies inversely as the
  radius as in a radial current, and the tangential component varies
  inversely as the radius as in a free vortex. Then the whole velocity
  at any point will be inversely proportional to the radius of the
  point, and the fluid will describe stream lines having a constant
  inclination to the radius drawn to the axis of the current. That is,
  the stream lines will be logarithmic spirals. When water is delivered
  from the circumference of a centrifugal pump or turbine into a
  chamber, it forms a free vortex of this kind. The water flows spirally
  outwards, its velocity diminishing and its pressure increasing
  according to the law stated above, and the head along each spiral
  stream line is constant.

  § 35. _Forced Vortex._--If the law of motion in a rotating current is
  different from that in a free vortex, some force must be applied to
  cause the variation of velocity. The simplest case is that of a
  rotating current in which all the particles have equal angular
  velocity, as for instance when they are driven round by radiating
  paddles revolving uniformly. Then in equation (2), § 33, considering
  two circular stream lines of radii r and r + dr (fig. 37), we have
  [rho] = r, ds = dr. If the angular velocity is [alpha], then v =
  [alpha]r and dv = [alpha]dr. Hence

    dH = [alpha]²r dr/g + [alpha]²r dr/g = 2[alpha]²r dr/g.

  Comparing this with (1), § 33, and putting dz = 0, because the motion
  is horizontal,

    dp/G + [alpha]²r dr/g = 2[alpha]²r dr/g,

    dp/G = [alpha]²rdr/g,

    p/G = [alpha]²/2g + constant.   (9)

  Let p1, r1, v1 be the pressure, radius and velocity of one cylindrical
  section, p2, r2, v2 those of another; then

    p1/G - [alpha]²r1²/2g = p2/G - [alpha]²r2²/2g;

    (p2 - p1)/G = [alpha]²(r2² - r1²)/2g = (v2² - v1²)/2g.   (10)

  That is, the pressure increases from within outwards in a curve which
  in radial sections is a parabola, and surfaces of equal pressure are
  paraboloids of revolution (fig. 37).

  [Illustration: FIG. 37.]


  DISSIPATION OF HEAD IN SHOCK

  § 36. _Relation of Pressure and Velocity in a Stream in Steady Motion
  when the Changes of Section of the Stream are Abrupt._--When a stream
  changes section abruptly, rotating eddies are formed which dissipate
  energy. The energy absorbed in producing rotation is at once
  abstracted from that effective in causing the flow, and sooner or
  later it is wasted by frictional resistances due to the rapid relative
  motion of the eddying parts of the fluid. In such cases the work thus
  expended internally in the fluid is too important to be neglected, and
  the energy thus lost is commonly termed energy lost in shock. Suppose
  fig. 38 to represent a stream having such an abrupt change of section.
  Let AB, CD be normal sections at points where ordinary stream line
  motion has not been disturbed and where it has been re-established.
  Let [omega], p, v be the area of section, pressure and velocity at AB,
  and [omega]1, p1, v1 corresponding quantities at CD. Then if no work
  were expended internally, and assuming the stream horizontal, we
  should have

    p/G + v²/2g = p1/G + v1²/2g.   (1)

  But if work is expended in producing irregular eddying motion, the
  head at the section CD will be diminished.

  Suppose the mass ABCD comes in a short time t to A´B´C´D´. The
  resultant force parallel to the axis of the stream is

    p[omega] + p0([omega]1 - [omega]) - p1[omega]1,

  where p0 is put for the unknown pressure on the annular space between
  AB and EF. The impulse of that force is

    {p[omega] + p0([omega]1 - [omega]) - p1[omega]1} t.

  [Illustration: FIG. 38.]

  The horizontal change of momentum in the same time is the difference
  of the momenta of CDC´D´ and ABA´B´, because the amount of momentum
  between A´B´ and CD remains unchanged if the motion is steady. The
  volume of ABA´B´ or CDC´D´, being the inflow and outflow in the time
  t, is Qt = [omega]vt = [omega]1v1t, and the momentum of these masses
  is (G/g)Qvt and (G/g)Qv1t. The change of momentum is therefore
  (G/g)Qt(v1 - v). Equating this to the impulse,

    {p[omega] + p0([omega]1 - [omega]) - p1[omega]1}t = (G/g)Qt(v1 - v).

  Assume that p0 = p, the pressure at AB extending unchanged through the
  portions of fluid in contact with AE, BF which lie out of the path of
  the stream. Then (since Q = [omega]1v1)

    (p - p1) = (G/g) v1 (v1 - v);

    p/G - p1/G = v1 (v1 - v)/g;   (2)

    p/G + v²/2g = p1/G + v1²/2g + (v - v1)²/2g.   (3)

  This differs from the expression (1), § 29, obtained for cases where
  no sensible internal work is done, by the last term on the right. That
  is, (v - v1)²/2g has to be added to the total head at CD, which is
  p1/G + v1²/2g, to make it equal to the total head at AB, or (v -
  v1)²/2g is the head lost in shock at the abrupt change of section. But
  (v - v1) is the relative velocity of the two parts of the stream.
  Hence, when an abrupt change of section occurs, the head due to the
  relative velocity is lost in shock, or (v - v1)²/2g foot-pounds of
  energy is wasted for each pound of fluid. Experiment verifies this
  result, so that the assumption that p0 = p appears to be admissible.

  If there is no shock,

    p1/G = p/G + (v² - v1²)/2g.

  If there is shock,

    p1/G = p/G - v1(v1 - v)/g.

  Hence the pressure head at CD in the second case is less than in the
  former by the quantity (v - v1)²/2g, or, putting [omega]1v1 =
  [omega]v, by the quantity

    (v²/2g)(1 - [omega]/[omega]1)².   (4)


  V. THEORY OF THE DISCHARGE FROM ORIFICES AND MOUTHPIECES

  [Illustration: FIG. 39.]

  § 37. _Minimum Coefficient of Contraction. Re-entrant Mouthpiece of
  Borda._--In one special case the coefficient of contraction can be
  determined theoretically, and, as it is the case where the convergence
  of the streams approaching the orifice takes place through the
  greatest possible angle, the coefficient thus determined is the
  minimum coefficient.

  Let fig. 39 represent a vessel with vertical sides, OO being the free
  water surface, at which the pressure is p_a. Suppose the liquid issues
  by a horizontal mouthpiece, which is re-entrant and of the greatest
  length which permits the jet to spring clear from the inner end of the
  orifice, without adhering to its sides. With such an orifice the
  velocity near the points CD is negligible, and the pressure at those
  points may be taken equal to the hydrostatic pressure due to the depth
  from the free surface. Let [Omega] be the area of the mouthpiece AB,
  [omega] that of the contracted jet aa Suppose that in a short time t,
  the mass OOaa comes to the position O´O´ a´a´; the impulse of the
  horizontal external forces acting on the mass during that time is
  equal to the horizontal change of momentum.

  The pressure on the side OC of the mass will be balanced by the
  pressure on the opposite side OE, and so for all other portions of the
  vertical surfaces of the mass, excepting the portion EF opposite the
  mouthpiece and the surface AaaB of the jet. On EF the pressure is
  simply the hydrostatic pressure due to the depth, that is, (p_a + Gh).
  On the surface and section AaaB of the jet, the horizontal resultant
  of the pressure is equal to the atmospheric pressure p_a acting on the
  vertical projection AB of the jet; that is, the resultant pressure is
  -p_a[Omega]. Hence the resultant horizontal force for the whole mass
  OOaa is (p_a + Gh)[Omega] - p_a[Omega] = Gh[Omega]. Its impulse in the
  time t is Gh[Omega]t. Since the motion is steady there is no change of
  momentum between O´O´ and aa. The change of horizontal momentum is,
  therefore, the difference of the horizontal momentum lost in the space
  OOO´O´ and gained in the space aaa´a´. In the former space there is no
  horizontal momentum.

  The volume of the space aaa´a´ is [omega]vt; the mass of liquid in
  that space is (G/g)[omega]vt; its momentum is (G/g)[omega]v²t.
  Equating impulse to momentum gained,

    Gh[Omega] = (G/g)[omega]v²t;

    .: [omega]/[Omega] = gh/v²

  But

    v² = 2gh, and [omega]/[Omega] = c_c;

    .: [omega]/[Omega] = ½ = c_c;

  a result confirmed by experiment with mouthpieces of this kind. A
  similar theoretical investigation is not possible for orifices in
  plane surfaces, because the velocity along the sides of the vessel in
  the neighbourhood of the orifice is not so small that it can be
  neglected. The resultant horizontal pressure is therefore greater than
  Gh[Omega], and the contraction is less. The experimental values of the
  coefficient of discharge for a re-entrant mouthpiece are 0.5149
  (Borda), 0.5547 (Bidone), 0.5324 (Weisbach), values which differ
  little from the theoretical value, 0.5, given above.

  [Illustration: FIG. 40.]

  § 38. _Velocity of Filaments issuing in a Jet._--A jet is composed of
  fluid filaments or elementary streams, which start into motion at some
  point in the interior of the vessel from which the fluid is
  discharged, and gradually acquire the velocity of the jet. Let Mm,
  fig. 40 be such a filament, the point M being taken where the velocity
  is insensibly small, and m at the most contracted section of the jet,
  where the filaments have become parallel and exercise uniform mutual
  pressure. Take the free surface AB for datum line, and let p1, v1, h1,
  be the pressure, velocity and depth below datum at M; p, v, h, the
  corresponding quantities at m. Then § 29, eq. (3a),

    v1²/2g + p1/G - h1 = v²/2g + p/G - h   (1)

  But at M, since the velocity is insensible, the pressure is the
  hydrostatic pressure due to the depth; that is v1 = 0, p1 = p_a + Gh1.
  At m, p = p_a, the atmospheric pressure round the jet. Hence,
  inserting these values,

    0 + p_a/G + h1 - h1 = v²/2g + p_a/G - h;

    v²/2g = h;   (2)

    or v = [root](2gh) = 8.025V [root]h.   (2a)

  [Illustration: FIG. 41.]

  That is, neglecting the viscosity of the fluid, the velocity of
  filaments at the contracted section of the jet is simply the velocity
  due to the difference of level of the free surface in the reservoir
  and the orifice. If the orifice is small in dimensions compared with
  h, the filaments will all have nearly the same velocity, and if h is
  measured to the centre of the orifice, the equation above gives the
  mean velocity of the jet.

  _Case of a Submerged Orifice._--Let the orifice discharge below the
  level of the tail water. Then using the notation shown in fig. 41, we
  have at M, v1 = 0, p1 = Gh; + p_a at m, p = Gh3 + p_a. Inserting these
  values in (3), § 29,

    0 + h1 + p_a/G - h1 = v²/2g + h3 - h2 + p_a/G;

    v²/2g = h2 - h3 = h,   (3)

  where h is the difference of level of the head and tail water, and may
  be termed the _effective head_ producing flow.

  [Illustration: FIG. 42.]

  _Case where the Pressures are different on the Free Surface and at the
  Orifice._--Let the fluid flow from a vessel in which the pressure is
  p0 into a vessel in which the pressure is p, fig. 42. The pressure p0
  will produce the same effect as a layer of fluid of thickness p0/G
  added to the head water; and the pressure p, will produce the same
  effect as a layer of thickness p/G added to the tail water. Hence the
  effective difference of level, or effective head producing flow, will
  be

    h = h0 + p0/G - p/G;

  and the velocity of discharge will be

    v = [root][2g {h0 + (p0 - p)/G}].   (4)

  We may express this result by saying that differences of pressure at
  the free surface and at the orifice are to be reckoned as part of the
  effective head.

  Hence in all cases thus far treated the velocity of the jet is the
  velocity due to the effective head, and the discharge, allowing for
  contraction of the jet, is

    Q = c[omega]v = c[omega] [root](2gh),   (5)

  where [omega] is the area of the orifice, c[omega] the area of the
  contracted section of the jet, and h the effective head measured to
  the centre of the orifice. If h and [omega] are taken in feet, Q is in
  cubic feet per second.

  It is obvious, however, that this formula assumes that all the
  filaments have sensibly the same velocity. That will be true for
  horizontal orifices, and very approximately true in other cases, if
  the dimensions of the orifice are not large compared with the head h.
  In large orifices in say a vertical surface, the value of h is
  different for different filaments, and then the velocity of different
  filaments is not sensibly the same.


  SIMPLE ORIFICES--HEAD CONSTANT

  [Illustration: FIG. 43.]

  § 39. _Large Rectangular Jets from Orifices in Vertical Plane
  Surfaces._--Let an orifice in a vertical plane surface be so formed
  that it produces a jet having a rectangular contracted section with
  vertical and horizontal sides. Let b (fig. 43) be the breadth of the
  jet, h1 and h2 the depths below the free surface of its upper and
  lower surfaces. Consider a lamina of the jet between the depths h and
  h + dh. Its normal section is bdh, and the velocity of discharge
  [root](2gh). The discharge per second in this lamina is therefore
  b[root](2gh) dh, and that of the whole jet is therefore
         _
        /h2
    Q = |   b [root](2gh) dh
       _/h1

    = 2/3 b[root](2g) {h2^(3/2) - h1^(3/2)},   (6)

  where the first factor on the right is a coefficient depending on the
  form of the orifice.

  Now an orifice producing a rectangular jet must itself be very
  approximately rectangular. Let B be the breadth, H1, H2, the depths to
  the upper and lower edges of the orifice. Put

    b [h2^(3/2) - h1^(3/2)] / B [H2^(3/2) - H1^(3/2)] = c.   (7)

  Then the discharge, in terms of the dimensions of the orifice, instead
  of those of the jet, is

    Q = (2/3)cB [root](2g) [H2^(3/2) - H1^(3/2)],   (8)

  the formula commonly given for the discharge of rectangular orifices.
  The coefficient c is not, however, simply the coefficient of
  contraction, the value of which is

    b(h2 - h1)/B(H2 - H1),

  and not that given in (7). It cannot be assumed, therefore, that c in
  equation (8) is constant, and in fact it is found to vary for
  different values of B/H2 and B/H1, and must be ascertained
  experimentally.

  _Relation between the Expressions (5) and (8)._--For a rectangular
  orifice the area of the orifice is [omega] = B(H2 - H1), and the
  depth measured to its centre is ½(H2 + H1). Putting these values in
  (5),

    Q1 = cB(H2 - H1) [root]{g(H2 + H1)}.

  From (8) the discharge is

    Q2 = (2/3)cB [root](2g) [H2^(3/2) - H1^(3/2)].

  Hence, for the same value of c in the two cases,

    Q2/Q1 = (2/3)[H2^(3/2) - H1^(3/2)] / [(H2 - H1)[root]{(H2 + H1)/2}].

  Let H1/H2 = [sigma], then

    Q2/Q1 = 0.9427(1 - [sigma]^(3/2)) /
      {1 - [sigma] [root]{(1 + [sigma])}}.   (9)

  If H1 varies from 0 to [infinity], [sigma]( = H1/H2) varies from 0 to
  1. The following table gives values of the two estimates of the
  discharge for different values of [sigma]:--

    +------------------+--------+------------------+--------+
    | H1/H2 = [sigma]. | Q2/Q1. | H1/H2 = [sigma]. | Q2/Q1. |
    +------------------+--------+------------------+--------+
    |       0.0        |  .943  |       0.8        |   .999 |
    |       0.2        |  .979  |       0.9        |   .999 |
    |       0.5        |  .995  |       1.0        |  1.000 |
    |       0.7        |  .998  |                  |        |
    +------------------+--------+------------------+--------+

  Hence it is obvious that, except for very small values of [sigma], the
  simpler equation (5) gives values sensibly identical with those of
  (8). When [sigma]<0.5 it is better to use equation (8) with values of
  c determined experimentally for the particular proportions of orifice
  which are in question.

  [Illustration: FIG. 44.]

  § 40. _Large Jets having a Circular Section from Orifices in a
  Vertical Plane Surface._--Let fig. 44 represent the section of the
  jet, OO being the free surface level in the reservoir. The discharge
  through the horizontal strip aabb, of breadth aa = b, between the
  depths h1 + y and h1 + y + dy, is

    dQ = b [root]{2g(h1 + y)} dy.

  The whole discharge of the jet is
         _
        /d
    Q = |  b [root]{2g(h1 + y)} dy.
       _/0

  But b = d sin [phi]; y = ½d(1 - cos [phi]); dy = ½d sin [phi] d[phi].
  Let [epsilon] = d/(2h1 + d), then
                                  _
                                 /[pi]
    Q = ½d² [root]{2g(h1 + d/2)} |  sin² [phi][root]{1 - [epsilon] cos [phi]} d[phi].
                                _/0

  From eq. (5), putting [omega] = [pi]d²/4, h = h1 + d/2, c = 1 when d
  is the diameter of the jet and not that of the orifice,

    Q1 = ¼[pi]d² [root]{2g (h1 + d/2)},
                    _
                   /[pi]
    Q/Q1 = 2/[pi]  |   sin² [phi] [root]{1 - [epsilon] cos [phi]} d[phi].
                  _/0

  For

    h1 = [infinity], [epsilon] = 0 and Q/Q1 = 1;

  and for

    h1 = 0, [epsilon] = 1 and Q/Q1 = 0.96.

  So that in this case also the difference between the simple formula
  (5) and the formula above, in which the variation of head at different
  parts of the orifice is taken into account, is very small.


  NOTCHES AND WEIRS

  § 41. _Notches, Weirs and Byewashes._--A notch is an orifice extending
  up to the free surface level in the reservoir from which the discharge
  takes place. A weir is a structure over which the water flows, the
  discharge being in the same conditions as for a notch. The formula of
  discharge for an orifice of this kind is ordinarily deduced by putting
  H1 = 0 in the formula for the corresponding orifice, obtained as in
  the preceding section. Thus for a rectangular notch, put H1 = 0 in
  (8). Then

    Q = (2/3)cB [root](2g) H^(3/2),   (11)

  where H is put for the depth to the crest of the weir or the bottom of
  the notch. Fig. 45 shows the mode in which the discharge occurs in the
  case of a rectangular notch or weir with a level crest. As, the free
  surface level falls very sensibly near the notch, the head H should be
  measured at some distance back from the notch, at a point where the
  velocity of the water is very small.

  Since the area of the notch opening is BH, the above formula is of the
  form

    Q = c × BH × k [root](2gH),

  where k is a factor depending on the form of the notch and expressing
  the ratio of the mean velocity of discharge to the velocity due to the
  depth H.

  § 42. _Francis's Formula for Rectangular Notches._--The jet discharged
  through a rectangular notch has a section smaller than BH, (a) because
  of the fall of the water surface from the point where H is measured
  towards the weir, (b) in consequence of the crest contraction, (c) in
  consequence of the end contractions. It may be pointed out that while
  the diminution of the section of the jet due to the surface fall and
  to the crest contraction is proportional to the length of the weir,
  the end contractions have nearly the same effect whether the weir is
  wide or narrow.

  [Illustration: FIG. 45.]

  J. B. Francis's experiments showed that a perfect end contraction,
  when the heads varied from 3 to 24 in., and the length of the weir was
  not less than three times the head, diminished the effective length of
  the weir by an amount approximately equal to one-tenth of the head.
  Hence, if l is the length of the notch or weir, and H the head
  measured behind the weir where the water is nearly still, then the
  width of the jet passing through the notch would be l - 0.2H, allowing
  for two end contractions. In a weir divided by posts there may be more
  than two end contractions. Hence, generally, the width of the jet is l
  - 0.1nH, where n is the number of end contractions of the stream. The
  contractions due to the fall of surface and to the crest contraction
  are proportional to the width of the jet. Hence, if cH is the
  thickness of the stream over the weir, measured at the contracted
  section, the section of the jet will be c(l - 0.1nH)H and (§ 41) the
  mean velocity will be 2/3 [root](2gH). Consequently the discharge
  will be given by an equation of the form

    Q = (2/3)c (l - 0.1nH)H [root](2gH)
      = 5.35c (l - 0.1nH) H^(3/2).

  This is Francis's formula, in which the coefficient of discharge c is
  much more nearly constant for different values of l and h than in the
  ordinary formula. Francis found for c the mean value 0.622, the weir
  being sharp-edged.

  § 43. _Triangular Notch_ (fig. 46).--Consider a lamina issuing between
  the depths h and h + dh. Its area, neglecting contraction, will be
  bdh, and the velocity at that depth is [root](2gh). Hence the
  discharge for this lamina is

    b[root](2gh) dh.

  But

    B/b = H/(H - h); b = B(H - h)/H.

  Hence discharge of lamina

    = B(H - h) [root](2gh) dh/H;

  and total discharge of notch
                       _
                      /H
    = Q = B[root](2g) |  (H - h)h^(½) dh/H
                     _/0

        = (4/15) B[root](2g)H^(3/2).

  or, introducing a coefficient to allow for contraction,

    Q = (4/15)cB [root](2g) H^(½),

  [Illustration: FIG. 46.]

  When a notch is used to gauge a stream of varying flow, the ratio B/H
  varies if the notch is rectangular, but is constant if the notch is
  triangular. This led Professor James Thomson to suspect that the
  coefficient of discharge, c, would be much more constant with
  different values of H in a triangular than in a rectangular notch, and
  this has been experimentally shown to be the case. Hence a triangular
  notch is more suitable for accurate gaugings than a rectangular notch.
  For a sharp-edged triangular notch Professor J. Thomson found c =
  0.617. It will be seen, as in § 41, that since ½BH is the area of
  section of the stream through the notch, the formula is again of the
  form

    Q = c × ½BH × k[root](2gH),

  where k = 8/15 is the ratio of the mean velocity in the notch to the
  velocity at the depth H. It may easily be shown that for all notches
  the discharge can be expressed in this form.

    _Coefficients for the Discharge over Weirs, derived from the
    Experiments of T. E. Blackwell. When more than one experiment was
    made with the same head, and the results were pretty uniform, the
    resulting coefficients are marked with an (*). The effect of the
    converging wing-boards is very strongly marked._

    +----------+-------------+---------------------------------+-----------------------------------------+
    |          |             |       Planks 2 in. thick,       |                                         |
    | Heads in | Sharp Edge. |        square on Crest.         |            Crests 3 ft. wide.           |
    |  inches  +------+------+-----+-----+-------+-------------+------+------+------+------+------+------+
    | measured |      |      |     |     |       |10 ft. long, | 3 ft.| 3 ft.| 3 ft.| 6 ft.|10 ft.|10 ft.|
    |from still| 3 ft.|10 ft.|3 ft.|6 ft.| 10 ft.| wing-boards | long,| long,| long,| long,| long,| long,|
    | Water in | long.| long.|long.|long.| long. |  making an  |level.|fall 1|fall 1|level.|level.|fall 1|
    |Reservoir.|      |      |     |     |       |angle of 60°.|      |in 18.|in 12.|      |      |in 18.|
    +----------+------+------+-----+-----+-------+-------------+------+------+------+------+------+------+
    |     1    | .677 | .809 |.467 |.459 |.435[4]|    .754     | .452 | .545 | .467 |  ..  | .381 | .467 |
    |     2    | .675 | .803 |.509*|.561 |.585*  |    .675     | .482 | .546 | .533 |  ..  | .479*| .495*|
    |     3    | .630 | .642*|.563*|.597*|.569*  |     ..      | .441 | .537 | .539 | .492*|  ..  |  ..  |
    |     4    | .617 | .656 |.549 |.575 |.602*  |    .656     | .419 | .431 | .455 | .497*|  ..  | .515 |
    |     5    | .602 | .650*|.588 |.601*|.609*  |    .671     | .479 | .516 |  ..  |  ..  | .518 |  ..  |
    |     6    | .593 |  ..  |.593*|.608*|.576*  |     ..      | .501*|  ..  | .531 | .507 | .513 | .543 |
    |     7    |  ..  |  ..  |.617*|.608*|.576*  |     ..      | .488 | .513 | .527 | .497 |  ..  |  ..  |
    |     8    |  ..  | .581 |.606*|.590*|.548*  |     ..      | .470 | .491 |  ..  |  ..  | .468 | .507 |
    |     9    |  ..  | .530 |.600 |.569*|.558*  |     ..      | .476 | .492*| .498 | .480*| .486 |  ..  |
    |    10    |  ..  |  ..  |.614*|.539 |.534*  |     ..      |  ..  |  ..  |  ..  | .465*| .455 |  ..  |
    |    12    |  ..  |  ..  | ..  |.525 |.534*  |     ..      |  ..  |  ..  |  ..  | .467*|  ..  |  ..  |
    |    14    |  ..  |  ..  | ..  |.549*| ..    |     ..      |  ..  |  ..  |  ..  |  ..  |  ..  |  ..  |
    +----------+------+------+-----+-----+-------+-------------+------+------+------+------+------+------+

  [Illustration: FIG. 47.]

  § 44. _Weir with a Broad Sloping Crest._--Suppose a weir formed with a
  broad crest so sloped that the streams flowing over it have a movement
  sensibly rectilinear and uniform (fig. 47). Let the inner edge be so
  rounded as to prevent a crest contraction. Consider a filament aa´,
  the point a being so far back from the weir that the velocity of
  approach is negligible. Let OO be the surface level in the reservoir,
  and let a be at a height h´´ below OO, and h´ above a´. Let h be the
  distance from OO to the weir crest and e the thickness of the stream
  upon it. Neglecting atmospheric pressure, which has no influence, the
  pressure at a is Gh´´; at a´ it is Gz. If v be the velocity at a´,

    v²/2g = h´ + h´´ - z = h - e;

    Q = be [root]{2g(h - e)}.

  Theory does not furnish a value for e, but Q = 0 for e = 0 and for e =
  h. Q has therefore a maximum for a value of e between 0 and h,
  obtained by equating dQ/de to zero. This gives e = (2/3)h, and,
  inserting this value,

    Q = 0.385 bh [root](2gh),

  as a maximum value of the discharge with the conditions assigned.
  Experiment shows that the actual discharge is very approximately equal
  to this maximum, and the formula is more legitimately applicable to
  the discharge over broad-crested weirs and to cases such as the
  discharge with free upper surface through large masonry sluice
  openings than the ordinary weir formula for sharp-edged weirs. It
  should be remembered, however, that the friction on the sides and
  crest of the weir has been neglected, and that this tends to reduce a
  little the discharge. The formula is equivalent to the ordinary weir
  formula with c = 0.577.


  SPECIAL CASES OF DISCHARGE FROM ORIFICES

  § 45. _Cases in which the Velocity of Approach needs to be taken into
  Account. Rectangular Orifices and Notches._--In finding the velocity
  at the orifice in the preceding investigations, it has been assumed
  that the head h has been measured from the free surface of still water
  above the orifice. In many cases which occur in practice the channel
  of approach to an orifice or notch is not so large, relatively to the
  stream through the orifice or notch, that the velocity in it can be
  disregarded.

  [Illustration: FIG. 48.]

  Let h1, h2 (fig. 48) be the heads measured from the free surface to
  the top and bottom edges of a rectangular orifice, at a point in the
  channel of approach where the velocity is u. It is obvious that a fall
  of the free surface,

    [h] = u²/2g

  has been somewhere expended in producing the velocity u, and hence the
  true heads measured in still water would have been h1 + [h] and h2 +
  [h]. Consequently the discharge, allowing for the velocity of
  approach, is

    Q = (2/3)cb [root](2g) {(h2 + [h])^(3/2) - (h1 + [h])^(3/2)}.   (1)

  And for a rectangular notch for which h1 = 0, the discharge is

    Q = (2/3)cb [root](2g) {(h2 + [h])^(3/2) - [h]^(3/2)}.   (2)

  In cases where u can be directly determined, these formulae give the
  discharge quite simply. When, however, u is only known as a function
  of the section of the stream in the channel of approach, they become
  complicated. Let [Omega] be the sectional area of the channel where h1
  and h2 are measured. Then u = Q/[Omega] and [h] = Q²/2g [Omega]².

  This value introduced in the equations above would render them
  excessively cumbrous. In cases therefore where [Omega] only is known,
  it is best to proceed by approximation. Calculate an approximate value
  Q´ of Q by the equation

    Q´ = (2/3)cb [root](2g) {h2^(3/2) - h1^(3/2)}.

  Then [h] = Q´²/2g[Omega]² nearly. This value of [h] introduced in the
  equations above will give a second and much more approximate value of
  Q.

  [Illustration: FIG. 49.]

  § 46. _Partially Submerged Rectangular Orifices and Notches._--When
  the tail water is above the lower but below the upper edge of the
  orifice, the flow in the two parts of the orifice, into which it is
  divided by the surface of the tail water, takes place under different
  conditions. A filament M1m1 (fig. 49) in the upper part of the orifice
  issues with a head h´ which may have any value between h1 and h. But a
  filament M2m2 issuing in the lower part of the orifice has a velocity
  due to h´´ - h´´´, or h, simply. In the upper part of the orifice the
  head is variable, in the lower constant. If Q1, Q2 are the discharges
  from the upper and lower parts of the orifice, b the width of the
  orifice, then

    Q1 = (2/3)cb [root](2g) {h^(3/2) - h1^(3/2)}
                                                  (3)
    Q1 = cb (h2 - h) [root](2gh).

  In the case of a rectangular notch or weir, h1 = 0. Inserting this
  value, and adding the two portions of the discharge together, we get
  for a drowned weir

    Q = cb[root](2gh) (h2 - h/3),   (4)

  where h is the difference of level of the head and tail water, and h2
  is the head from the free surface above the weir to the weir crest
  (fig. 50).

  From some experiments by Messrs A. Fteley and F.P. Stearns (_Trans.
  Am. Soc. C.E._, 1883, p. 102) some values of the coefficient c can be
  reduced

    h3/h2     c     h3/h2      c

     0.1    0.629    0.7     0.578
     0.2    0.614    0.8     0.583
     0.3    0.600    0.9     0.596
     0.4    0.590    0.95    0.607
     0.5    0.582    1.00    0.628
     0.6    0.578

  If velocity of approach is taken into account, let [h] be the
  head due to that velocity; then, adding [h] to each of the
  heads in the equations (3), and reducing, we get for a weir

    Q = cb [root]{2g} [(h2 + [h]) (h + [h])^(½) - (1/3)(h + [h])^(3/2)
      - (2/3)[h]^(3/2)];   (5)

  an equation which may be useful in estimating flood discharges.

  [Illustration: FIG. 50.]

  _Bridge Piers and other Obstructions in Streams._--When the piers of a
  bridge are erected in a stream they create an obstruction to the flow
  of the stream, which causes a difference of surface-level above and
  below the pier (fig. 51). If it is necessary to estimate this
  difference of level, the flow between the piers may be treated as if
  it occurred over a drowned weir. But the value of c in this case is
  imperfectly known.

  § 47. _Bazin's Researches on Weirs._--H. Bazin has executed a long
  series of researches on the flow over weirs, so systematic and
  complete that they almost supersede other observations. The account of
  them is contained in a series of papers in the _Annales des Ponts et
  Chaussées_ (October 1888, January 1890, November 1891, February 1894,
  December 1896, 2nd trimestre 1898). Only a very abbreviated account
  can be given here. The general plan of the experiments was to
  establish first the coefficients of discharge for a standard weir
  without end contractions; next to establish weirs of other types in
  series with the standard weir on a channel with steady flow, to
  compare the observed heads on the different weirs and to determine
  their coefficients from the discharge computed at the standard weir. A
  channel was constructed parallel to the Canal de Bourgogne, taking
  water from it through three sluices 0.3 × 1.0 metres. The water enters
  a masonry chamber 15 metres long by 4 metres wide where it is stilled
  and passes into the canal at the end of which is the standard weir.
  The canal has a length of 15 metres, a width of 2 metres and a depth
  of 0.6 metres. From this extends a channel 200 metres in length with a
  slope of 1 mm. per metre. The channel is 2 metres wide with vertical
  sides. The channels were constructed of concrete rendered with cement.
  The water levels were taken in chambers constructed near the canal, by
  floats actuating an index on a dial. Hook gauges were used in
  determining the heads on the weirs.

  [Illustration: FIG. 51.]

  _Standard Weir._--The weir crest was 3.72 ft. above the bottom of the
  canal and formed by a plate ¼ in. thick. It was sharp-edged with free
  overfall. It was as wide as the canal so that end contractions were
  suppressed, and enlargements were formed below the crest to admit air
  under the water sheet. The channel below the weir was used as a
  gauging tank. Gaugings were made with the weir 2 metres in length and
  afterwards with the weir reduced to 1 metre and 0.5 metre in length,
  the end contractions being suppressed in all cases. Assuming the
  general formula

    Q = mlh [root](2gh),   (1)

  Bazin arrives at the following values of _m_:--

    _Coefficients of Discharge of Standard Weir._

    +----------------+--------------+--------+
    | Head h metres. | Head h feet. |    m   |
    +----------------+--------------+--------+
    |      0.05      |     .164     | 0.4485 |
    |      0.10      |     .328     | 0.4336 |
    |      0.15      |     .492     | 0.4284 |
    |      0.20      |     .656     | 0.4262 |
    |      0.25      |     .820     | 0.4259 |
    |      0.30      |     .984     | 0.4266 |
    |      0.35      |    1.148     | 0.4275 |
    |      0.40      |    1.312     | 0.4286 |
    |      0.45      |    1.476     | 0.4299 |
    |      0.50      |    1.640     | 0.4313 |
    |      0.55      |    1.804     | 0.4327 |
    |      0.60      |    1.968     | 0.4341 |
    +----------------+--------------+--------+

  Bazin compares his results with those of Fteley and Stearns in 1877
  and 1879, correcting for a different velocity of approach, and finds a
  close agreement.

  _Influence of Velocity of Approach._--To take account of the velocity
  of approach u it is usual to replace h in the formula by h + au²/2g
  where [alpha] is a coefficient not very well ascertained. Then

    Q = [mu]l (h + [alpha]u²/2g) [root]{2g(h + [alpha]u²/2g)}
      = [mu]lh [root](2gh)(1 + [alpha]u²/2gh)^(3/2).   (2)

  The original simple equation can be used if

    m = [mu](1 + [alpha]u²/2gh)^(3/2)

  or very approximately, since u²/2gh is small,

    m = [mu](1 + (3/2)[alpha]u²/2gh).   (3)

  [Illustration: FIG. 52.]

  Now if p is the height of the weir crest above the bottom of the canal
  (fig. 52), u = Q/l(p + h). Replacing Q by its value in (1)

    u²/2gh = Q²/{2ghl²(p + h)²} = m²{h/(p + h)}²,   (4)

  so that (3) may be written

    m = [mu][1 + k{h/(p + h)}²].   (5)

  Gaugings were made with weirs of 0.75, 0.50, 0.35, and 0.24 metres
  height above the canal bottom and the results compared with those of
  the standard weir taken at the same time. The discussion of the
  results leads to the following values of m in the general equation
  (1):--

    m = [mu](1 + 2.5u²/2gh)
      = [mu][1 + 0.55 {h/(p + h)}²].

  Values of [mu]--

    +----------------+--------------+--------+
    | Head h metres. | Head h feet. |  [mu]  |
    +----------------+--------------+--------+
    |      0.05      |     .164     | 0.4481 |
    |      0.10      |     .328     | 0.4322 |
    |      0.20      |     .656     | 0.4215 |
    |      0.30      |     .984     | 0.4174 |
    |      0.40      |    1.312     | 0.4144 |
    |      0.50      |    1.640     | 0.4118 |
    |      0.60      |    1.968     | 0.4092 |
    +----------------+--------------+--------+

  An approximate formula for [mu] is:

    [mu] = 0.405 + 0.003/h (h in metres)

    [mu] = 0.405 + 0.01/h (h in feet).

  _Inclined Weirs._---Experiments were made in which the plank weir was
  inclined up or down stream, the crest being sharp and the end
  contraction suppressed. The following are coefficients by which the
  discharge of a vertical weir should be multiplied to obtain the
  discharge of the inclined weir.

                                      Coefficient.
    Inclination up stream      1 to 1     0.93
         "           "         3 to 2     0.94
         "           "         3 to 1     0.96
    Vertical weir                         1.00
    Inclination down stream    3 to 1     1.04
         "           "         3 to 2     1.07
         "           "         1 to 1     1.10
         "           "         1 to 2     1.12
         "           "         1 to 4     1.09

  The coefficient varies appreciably, if h/p approaches unity, which
  case should be avoided.

  In all the preceding cases the sheet passing over the weir is detached
  completely from the weir and its under-surface is subject to
  atmospheric pressure. These conditions permit the most exact
  determination of the coefficient of discharge. If the sides of the
  canal below the weir are not so arranged as to permit the access of
  air under the sheet, the phenomena are more complicated. So long as
  the head does not exceed a certain limit the sheet is detached from
  the weir, but encloses a volume of air which is at less than
  atmospheric pressure, and the tail water rises under the sheet. The
  discharge is a little greater than for free overfall. At greater head
  the air disappears from below the sheet and the sheet is said to be
  "drowned." The drowned sheet may be independent of the tail water
  level or influenced by it. In the former case the fall is followed by
  a rapid, terminating in a standing wave. In the latter case when the
  foot of the sheet is drowned the level of the tail water influences
  the discharge even if it is below the weir crest.

  [Illustration: FIG. 53.]

  [Illustration: FIG. 54.]

  _Weirs with Flat Crests._--The water sheet may spring clear from the
  upstream edge or may adhere to the flat crest falling free beyond the
  down-stream edge. In the former case the condition is that of a
  sharp-edged weir and it is realized when the head is at least double
  the width of crest. It may arise if the head is at least 1½ the width
  of crest. Between these limits the condition of the sheet is unstable.
  When the sheet is adherent the coefficient m depends on the ratio of
  the head h to the width of crest c (fig. 53), and is given by the
  equation m = m1 [0.70 + 0.185h/c], where m1 is the coefficient for a
  sharp-edged weir in similar conditions. Rounding the upstream edge
  even to a small extent modifies the discharge. If R is the radius of
  the rounding the coefficient m is increased in the ratio 1 to 1 + R/h
  nearly. The results are limited to R less than ½ in.

  _Drowned Weirs._--Let h (fig. 54) be the height of head water and h1
  that of tail water above the weir crest. Then Bazin obtains as the
  approximate formula for the coefficient of discharge

    m = 1.05m1 [1 + (1/5)h1/p] [root 3]{(h - h1)/h},

  where as before m1 is the coefficient for a sharp-edged weir in
  similar conditions, that is, when the sheet is free and the weir of
  the same height.

  [Illustration: FIG. 55.]

  [Illustration: FIG. 56.]

  § 48. _Separating Weirs._--Many towns derive their water-supply from
  streams in high moorland districts, in which the flow is extremely
  variable. The water is collected in large storage reservoirs, from
  which an uniform supply can be sent to the town. In such cases it is
  desirable to separate the coloured water which comes down the streams
  in high floods from the purer water of ordinary flow. The latter is
  sent into the reservoirs; the former is allowed to flow away down the
  original stream channel, or is stored in separate reservoirs and used
  as compensation water. To accomplish the separation of the flood and
  ordinary water, advantage is taken of the different horizontal range
  of the parabolic path of the water falling over a weir, as the depth
  on the weir and, consequently, the velocity change. Fig. 55 shows one
  of these separating weirs in the form in which they were first
  introduced on the Manchester Waterworks; fig. 56 a more modern weir of
  the same kind designed by Sir A. Binnie for the Bradford Waterworks.
  When the quantity of water coming down the stream is not excessive, it
  drops over the weir into a transverse channel leading to the
  reservoirs. In flood, the water springs over the mouth of this channel
  and is led into a waste channel.

  It may be assumed, probably with accuracy enough for practical
  purposes, that the particles describe the parabolas due to the mean
  velocity of the water passing over the weir, that is, to a velocity

    (2/3)[root](2gh),

  where h is the head above the crest of the weir.

  Let cb = x be the width of the orifice and ac = y the difference of
  level of its edges (fig. 57). Then, if a particle passes from a to b
  in t seconds,

    y = ½gt², x = (2/3)[root](2gh) t;

    .: y = (9/16)x²/h,

  which gives the width x for any given difference of level y and head
  h, which the jet will just pass over the orifice. Set off ad
  vertically and equal to ½g on any scale; af horizontally and equal to
  2/3 [root](gh). Divide af, fe into an equal number of equal parts.
  Join a with the divisions on ef. The intersections of these lines with
  verticals from the divisions on af give the parabolic path of the jet.

  [Illustration: FIG. 57.]


  MOUTHPIECES--HEAD CONSTANT

  § 49. _Cylindrical Mouthpieces._--When water issues from a short
  cylindrical pipe or mouthpiece of a length at least equal to l½ times
  its smallest transverse dimension, the stream, after contraction
  within the mouthpiece, expands to fill it and issues full bore, or
  without contraction, at the point of discharge. The discharge is found
  to be about one-third greater than that from a simple orifice of the
  same size. On the other hand, the energy of the fluid per unit of
  weight is less than that of the stream from a simple orifice with the
  same head, because part of the energy is wasted in eddies produced at
  the point where the stream expands to fill the mouthpiece, the action
  being something like that which occurs at an abrupt change of section.

  Let fig. 58 represent a vessel discharging through a cylindrical
  mouthpiece at the depth h from the free surface, and let the axis of
  the jet XX be taken as the datum with reference to which the head is
  estimated. Let [Omega] be the area of the mouthpiece, [omega] the area
  of the stream at the contracted section EF. Let v, p be the velocity
  and pressure at EF, and v1, p1 the same quantities at GH. If the
  discharge is into the air, p1 is equal to the atmospheric pressure
  p_a.

  The total head of any filament which goes to form the jet, taken at a
  point where its velocity is sensibly zero, is h + p_a/G; at EF the
  total head is v²/2g + p/G; at GH it is v1²/2g + p1/G.

  Between EF and GH there is a loss of head due to abrupt change of
  velocity, which from eq. (3), § 36, may have the value

    (v - v1)²/2g.

  Adding this head lost to the head at GH, before equating it to the
  heads at EF and at the point where the filaments start into motion,--

    h + p_a/G = v²/2g + p/G = v1²/2g + p1/G + (v - v1)²/2g.

  But [omega]v = [Omega]v1, and [omega] = c_c[Omega], if c_c is the
  coefficient of contraction within the mouthpiece. Hence

    v = [Omega]v1/[omega] = v1/c_c.

  Supposing the discharge into the air, so that p1 = p_a,

    h + p_a/G = v1²/2g + p_a/G + (v1²/2g)(1/c_c - 1)²;

    (v1/2g){1 + (1/c_c - 1)²} = h;

    .: v1 = [root](2gh)/[root]{1 + (1/c_c - 1)²};   (1)

  [Illustration: FIG. 58.]

  where the coefficient on the right is evidently the coefficient of
  velocity for the cylindrical mouthpiece in terms of the coefficient of
  contraction at EF. Let c_c = 0.64, the value for simple orifices, then
  the coefficient of velocity is

    c_v = 1/[root]{1 + (1/c_c - 1)²} = 0.87   (2)

  The actual value of c_v, found by experiment is 0.82, which does not
  differ more from the theoretical value than might be expected if the
  friction of the mouthpiece is allowed for. Hence, for mouthpieces of
  this kind, and for the section at GH,

    c_v = 0.82 c_c = 1.00 c = 0.82,

    Q = 0.82[Omega] [root](2gh).

  It is easy to see from the equations that the pressure p at EF is less
  than atmospheric pressure. Eliminating v1, we get

    (p_a - p)/G = ¾h nearly;   (3)

  or

    p = p_a - ¾Gh lb. per sq. ft.

  If a pipe connected with a reservoir on a lower level is introduced
  into the mouthpiece at the part where the contraction is formed (fig.
  59), the water will rise in this pipe to a height

    KL = (p_a - p)/G = ¾h nearly.

  If the distance X is less than this, the water from the lower
  reservoir will be forced continuously into the jet by the atmospheric
  pressure, and discharged with it. This is the crudest form of a kind
  of pump known as the jet pump.

  § 50. _Convergent Mouthpieces._--With convergent mouthpieces there is
  a contraction within the mouthpiece causing a loss of head, and a
  diminution of the velocity of discharge, as with cylindrical
  mouthpieces. There is also a second contraction of the stream outside
  the mouthpiece. Hence the discharge is given by an equation of the
  form

    Q = c_v c_c[Omega] [root](2gh),   (4)

  where [Omega] is the area of the external end of the mouthpiece, and
  c_c[Omega] the section of the contracted jet beyond the mouthpiece.

    _Convergent Mouthpieces (Castel's Experiments).--Smallest diameter of
    orifice = 0.05085 ft. Length of mouthpiece = 2.6 Diameters._

    +------------+--------------+--------------+--------------+
    |            |Coefficient of|Coefficient of|Coefficient of|
    |  Angle of  | Contraction, |   Velocity,  |  Discharge,  |
    |Convergence.|     c_c      |      c_v     |       c      |
    +------------+--------------+--------------+--------------+
    |    0°  0´  |     .999     |     .830     |     .829     |
    |    1° 36´  |    1.000     |     .866     |     .866     |
    |    3° 10´  |    1.001     |     .894     |     .895     |
    |    4° 10´  |    1.002     |     .910     |     .912     |
    |    5° 26´  |    1.004     |     .920     |     .924     |
    |    7° 52´  |     .998     |     .931     |     .929     |
    |    8° 58´  |     .992     |     .942     |     .934     |
    |   10° 20´  |     .987     |     .950     |     .938     |
    |   12°  4´  |     .986     |     .955     |     .942     |
    |   13° 24´  |     .983     |     .962     |     .946     |
    |   14° 28´  |     .979     |     .966     |     .941     |
    |   16° 36´  |     .969     |     .971     |     .938     |
    |   19° 28´  |     .953     |     .970     |     .924     |
    |   21°  0´  |     .945     |     .971     |     .918     |
    |   23°  0´  |     .937     |     .974     |     .913     |
    |   29° 58´  |     .919     |     .975     |     .896     |
    |   40° 20´  |     .887     |     .980     |     .869     |
    |   48° 50´  |     .861     |     .984     |     .847     |
    +------------+--------------+--------------+--------------+

  The maximum coefficient of discharge is that for a mouthpiece with a
  convergence of 13°24´.

  The values of c_v and c_c must here be determined by experiment. The
  above table gives values sufficient for practical purposes. Since the
  contraction beyond the mouthpiece increases with the convergence, or,
  what is the same thing, c_c diminishes, and on the other hand the loss
  of energy diminishes, so that c_v increases with the convergence,
  there is an angle for which the product c_c c_v, and consequently the
  discharge, is a maximum.

  [Illustration: FIG. 59.]

  § 51. _Divergent Conoidal Mouthpiece._--Suppose a mouthpiece so
  designed that there is no abrupt change in the section or velocity of
  the stream passing through it. It may have a form at the inner end
  approximately the same as that of a simple contracted vein, and may
  then enlarge gradually, as shown in fig. 60. Suppose that at EF it
  becomes cylindrical, so that the jet may be taken to be of the
  diameter EF. Let [omega], v, p be the section, velocity and pressure
  at CD, and [Omega], v1, p1 the same quantities at EF, p_a being as
  usual the atmospheric pressure, or pressure on the free surface AB.
  Then, since there is no loss of energy, except the small frictional
  resistance of the surface of the mouthpiece,

    h + p_a/G = v²/2g + p/G = v1²/2g + p1/G.

  If the jet discharges into the air, p1 = p_a; and

    v1²/2g = h;

    v1 = [root](2gh);

  or, if a coefficient is introduced to allow for friction,

    v1 = c_v [root](2gh);

  where c_v is about 0.97 if the mouthpiece is smooth and well formed.

    Q = [Omega] v1 = c_v [Omega] [root](2gh).

  [Illustration: FIG. 60.]

  Hence the discharge depends on the area of the stream at EF, and not
  at all on that at CD, and the latter may be made as small as we please
  without affecting the amount of water discharged.

  There is, however, a limit to this. As the velocity at CD is greater
  than at EF the pressure is less, and therefore less than atmospheric
  pressure, if the discharge is into the air. If CD is so contracted
  that p = 0, the continuity of flow is impossible. In fact the stream
  disengages itself from the mouthpiece for some value of p greater than
  0 (fig. 61).

  [Illustration: FIG. 61.]

  From the equations,

    p/G = p_a/G = (v² - v1²)/2g.

  Let [Omega]/[omega] = m. Then

    v = v1m;

    p/G = p_a/G - v1²(m² - 1)/2g
        = p_a/G - (m² - 1)h;

  whence we find that p/G will become zero or negative if

    [Omega]/[omega] >= [root]{(h + p_a/G)/h}
                     = [root]{1 + p_a/Gh};

  or, putting p_a/G = 34 ft., if

    [Omega]/[omega] >= [root]{(h + 34)/h}.

  In practice there will be an interruption of the full bore flow with a
  less ratio of [Omega]/[omega], because of the disengagement of air
  from the water. But, supposing this does not occur, the maximum
  discharge of a mouthpiece of this kind is

    Q = [omega] [root]{2g(h + p_a/G)};

  that is, the discharge is the same as for a well-bell-mouthed
  mouthpiece of area [omega], and without the expanding part,
  discharging into a vacuum.

  § 52. _Jet Pump._--A divergent mouthpiece may be arranged to act as a
  pump, as shown in fig. 62. The water which supplies the energy
  required for pumping enters at A. The water to be pumped enters at B.
  The streams combine at DD where the velocity is greatest and the
  pressure least. Beyond DD the stream enlarges in section, and its
  pressure increases, till it is sufficient to balance the head due to
  the height of the lift, and the water flows away by the discharge pipe
  C.

  [Illustration: FIG. 62.]

  Fig. 63 shows the whole arrangement in a diagrammatic way. A is the
  reservoir which supplies the water that effects the pumping; B is the
  reservoir of water to be pumped; C is the reservoir into which the
  water is pumped.

  [Illustration: FIG. 63.]


  DISCHARGE WITH VARYING HEAD

  § 53. _Flow from a Vessel when the Effective Head varies with the
  Time._--Various useful problems arise relating to the time of emptying
  and filling vessels, reservoirs, lock chambers, &c., where the flow is
  dependent on a head which increases or diminishes during the
  operation. The simplest of these problems is the case of filling or
  emptying a vessel of constant horizontal section.

  [Illustration: FIG. 64.]

  _Time of Emptying or Filling a Vertical-sided Lock Chamber._--Suppose
  the lock chamber, which has a water surface of [Omega] square ft., is
  emptied through a sluice in the tail gates, of area [omega], placed
  below the tail-water level. Then the effective head producing flow
  through the sluice is the difference of level in the chamber and tail
  bay. Let H (fig. 64) be the initial difference of level, h the
  difference of level after t seconds. Let -dh be the fall of level in
  the chamber during an interval dt. Then in the time dt the volume in
  the chamber is altered by the amount -[Omega]dh, and the outflow from
  the sluice in the same time is c[omega][root](2gh)dt. Hence the
  differential equation connecting h and t is

    c[omega] [root](2gh) dt + [Omega]h = 0.

  For the time t, during which the initial head H diminishes to any
  other value h,
                                    _               _
                                   /h              /t
    -{[Omega]/(c[omega] [root]2g)} |  dh/[root]h = |  dt.
                                  _/H             _/0

    .: t = 2[Omega]([root]H - [root]h) / {c[omega] [root](2g)}
         = ([Omega]/c[omega]){[root](2H/g) - [root](2h/g)}.

  For the whole time of emptying, during which h diminishes from H to 0,

    T = ([Omega]/c[omega]) [root](2H/g).

  Comparing this with the equation for flow under a constant head, it
  will be seen that the time is double that required for the discharge
  of an equal volume under a constant head.

  The time of filling the lock through a sluice in the head gates is
  exactly the same, if the sluice is below the tail-water level. But if
  the sluice is above the tail-water level, then the head is constant
  till the level of the sluice is reached, and afterwards it diminishes
  with the time.


  PRACTICAL USE OF ORIFICES IN GAUGING WATER

  § 54. If the water to be measured is passed through a known orifice
  under an arrangement by which the constancy of the head is ensured,
  the amount which passes in a given time can be ascertained by the
  formulae already given. It will obviously be best to make the orifices
  of the forms for which the coefficients are most accurately
  determined; hence sharp-edged orifices or notches are most commonly
  used.

  _Water Inch._--For measuring small quantities of water circular
  sharp-edged orifices have been used. The discharge from a circular
  orifice one French inch in diameter, with a head of one line above the
  top edge, was termed by the older hydraulic writers a water-inch. A
  common estimate of its value was 14 pints per minute, or 677 English
  cub. ft. in 24 hours. An experiment by C. Bossut gave 634 cub. ft. in
  24 hours (see Navier's edition of _Belidor's Arch. Hydr._, p. 212).

  L. J. Weisbach points out that measurements of this kind would be made
  more accurately with a greater head over the orifice, and he proposes
  that the head should be equal to the diameter of the orifice. Several
  equal orifices may be used for larger discharges.

  [Illustration: FIG. 65.]

  _Pin Ferrules or Measuring Cocks._--To give a tolerably definite
  supply of water to houses, without the expense of a meter, a ferrule
  with an orifice of a definite size, or a cock, is introduced in the
  service-pipe. If the head in the water main is constant, then a
  definite quantity of water would be delivered in a given time. The
  arrangement is not a very satisfactory one, and acts chiefly as a
  check on extravagant use of water. It is interesting here chiefly as
  an example of regulation of discharge by means of an orifice. Fig. 65
  shows a cock of this kind used at Zurich. It consists of three cocks,
  the middle one having the orifice of the predetermined size in a small
  circular plate, protected by wire gauze from stoppage by impurities in
  the water. The cock on the right hand can be used by the consumer for
  emptying the pipes. The one on the left and the measuring cock are
  connected by a key which can be locked by a padlock, which is under
  the control of the water company.

  § 55. _Measurement of the Flow in Streams._--To determine the quantity
  of water flowing off the ground in small streams, which is available
  for water supply or for obtaining water power, small temporary weirs
  are often used. These may be formed of planks supported by piles and
  puddled to prevent leakage. The measurement of the head may be made by
  a thin-edged scale at a short distance behind the weir, where the
  water surface has not begun to slope down to the weir and where the
  velocity of approach is not high. The measurements are conveniently
  made from a short pile driven into the bed of the river, accurately
  level with the crest of the weir (fig. 66). Then if at any moment the
  head is h, the discharge is, for a rectangular notch of breadth b,

    Q = (2/3)cbh [root](2gh)

  where c = 0.62; or, better, the formula in § 42 may be used.

  Gauging weirs are most commonly in the form of rectangular notches;
  and care should be taken that the crest is accurately horizontal, and
  that the weir is normal to the direction of flow of the stream. If the
  planks are thick, they should be bevelled (fig. 67), and then the edge
  may be protected by a metal plate about (1/10)th in. thick to secure
  the requisite accuracy of form and sharpness of edge. In permanent
  gauging weirs, a cast steel plate is sometimes used to form the edge
  of the weir crest. The weir should be large enough to discharge the
  maximum volume flowing in the stream, and at the same time it is
  desirable that the minimum head should not be too small (say half a
  foot) to decrease the effects of errors of measurement. The section of
  the jet over the weir should not exceed one-fifth the section of the
  stream behind the weir, or the velocity of approach will need to be
  taken into account. A triangular notch is very suitable for
  measurements of this kind.

  [Illustration: FIG. 66.]

  If the flow is variable, the head h must be recorded at equidistant
  intervals of time, say twice daily, and then for each 12-hour period
  the discharge must be calculated for the mean of the heads at the
  beginning and end of the time. As this involves a good deal of
  troublesome calculation, E. Sang proposed to use a scale so graduated
  as to read off the discharge in cubic feet per second. The lengths of
  the principal graduations of such a scale are easily calculated by
  putting Q = 1, 2, 3 ... in the ordinary formulae for notches; the
  intermediate graduations may be taken accurately enough by subdividing
  equally the distances between the principal graduations.

  [Illustration: FIG. 67.]

  [Illustration: FIG. 68.]

  The accurate measurement of the discharge of a stream by means of a
  weir is, however, in practice, rather more difficult than might be
  inferred from the simplicity of the principle of the operation. Apart
  from the difficulty of selecting a suitable coefficient of discharge,
  which need not be serious if the form of the weir and the nature of
  its crest are properly attended to, other difficulties of measurement
  arise. The length of the weir should be very accurately determined,
  and if the weir is rectangular its deviations from exactness of level
  should be tested. Then the agitation of the water, the ripple on its
  surface, and the adhesion of the water to the scale on which the head
  is measured, are liable to introduce errors. Upon a weir 10 ft. long,
  with 1 ft. depth of water flowing over, an error of 1-1000th of a foot
  in measuring the head, or an error of 1-100th of a foot in measuring
  the length of the weir, would cause an error in computing the
  discharge of 2 cub. ft. per minute.

  _Hook Gauge._--For the determination of the surface level of water,
  the most accurate instrument is the hook gauge used first by U. Boyden
  of Boston, in 1840. It consists of a fixed frame with scale and
  vernier. In the instrument in fig. 68 the vernier is fixed to the
  frame, and the scale slides vertically. The scale carries at its lower
  end a hook with a fine point, and the scale can be raised or lowered
  by a fine pitched screw. If the hook is depressed below the water
  surface and then raised by the screw, the moment of its reaching the
  water surface will be very distinctly marked, by the reflection from a
  small capillary elevation of the water surface over the point of the
  hook. In ordinary light, differences of level of the water of .001 of
  a foot are easily detected by the hook gauge. If such a gauge is used
  to determine the heads at a weir, the hook should first be set
  accurately level with the weir crest, and a reading taken. Then the
  difference of the reading at the water surface and that for the weir
  crest will be the head at the weir.

  § 56. _Modules used in Irrigation._--In distributing water for
  irrigation, the charge for the water may be simply assessed on the
  area of the land irrigated for each consumer, a method followed in
  India; or a regulated quantity of water may be given to each consumer,
  and the charge may be made proportional to the quantity of water
  supplied, a method employed for a long time in Italy and other parts
  of Europe. To deliver a regulated quantity of water from the
  irrigation channel, arrangements termed modules are used. These are
  constructions intended to maintain a constant or approximately
  constant head above an orifice of fixed size, or to regulate the size
  of the orifice so as to give a constant discharge, notwithstanding the
  variation of level in the irrigating channel.

  [Illustration: FIG. 69.]

  § 57. _Italian Module._--The Italian modules are masonry
  constructions, consisting of a regulating chamber, to which water is
  admitted by an adjustable sluice from the canal. At the other end of
  the chamber is an orifice in a thin flagstone of fixed size. By means
  of the adjustable sluice a tolerably constant head above the fixed
  orifice is maintained, and therefore there is a nearly constant
  discharge of ascertainable amount through the orifice, into the
  channel leading to the fields which are to be irrigated.

  [Illustration: FIG. 70.--Scale 1/100.]

  In fig. 69, A is the adjustable sluice by which water is admitted to
  the regulating chamber, B is the fixed orifice through which the water
  is discharged. The sluice A is adjusted from time to time by the canal
  officers, so as to bring the level of the water in the regulating
  chamber to a fixed level marked on the wall of the chamber. When
  adjusted it is locked. Let [omega]1 be the area of the orifice through
  the sluice at A, and [omega]2 that of the fixed orifice at B; let h1
  be the difference of level between the surface of the water in the
  canal and regulating chamber; h2 the head above the centre of the
  discharging orifice, when the sluice has been adjusted and the flow
  has become steady; Q the normal discharge in cubic feet per second.
  Then, since the flow through the orifices at A and B is the same,

    Q = c1[omega]1 [root](2gh1) = c2[omega]2 [root](2gh2),

  where c1 and c2 are the coefficients of discharge suitable for the two
  orifices. Hence

    c1[omega]1/c2[omega]2 = [root](h2/h1).

  If the orifice at B opened directly into the canal without any
  intermediate regulating chamber, the discharge would increase for a
  given change of level in the canal in exactly the same ratio.
  Consequently the Italian module in no way moderates the fluctuations
  of discharge, except so far as it affords means of easy adjustment
  from time to time. It has further the advantage that the cultivator,
  by observing the level of the water in the chamber, can always see
  whether or not he is receiving the proper quantity of water.

  On each canal the orifices are of the same height, and intended to
  work with the same normal head, the width of the orifices being varied
  to suit the demand for water. The unit of discharge varies on
  different canals, being fixed in each case by legal arrangements. Thus
  on the Canal Lodi the unit of discharge or one module of water is the
  discharge through an orifice 1.12 ft. high, 0.12416 ft. wide, with a
  head of 0.32 ft. above the top edge of the orifice, or .88 ft. above
  the centre. This corresponds to a discharge of about 0.6165 cub. ft.
  per second.

  [Illustration: FIG. 71.]

  In the most elaborate Italian modules the regulating chamber is arched
  over, and its dimensions are very exactly prescribed. Thus in the
  modules of the Naviglio Grande of Milan, shown in fig. 70, the
  measuring orifice is cut in a thin stone slab, and so placed that the
  discharge is into the air with free contraction on all sides. The
  adjusting sluice is placed with its sill flush with the bottom of the
  canal, and is provided with a rack and lever and locking arrangement.
  The covered regulating chamber is about 20 ft. long, with a breadth
  1.64 ft. greater than that of the discharging orifice. At precisely
  the normal level of the water in the regulating chamber, there is a
  ceiling of planks intended to still the agitation of the water. A
  block of stone serves to indicate the normal level of the water in the
  chamber. The water is discharged into an open channel 0.655 ft. wider
  than the orifice, splaying out till it is 1.637 ft. wider than the
  orifice, and about 18 ft. in length.

  § 58. _Spanish Module._--On the canal of Isabella II., which supplies
  water to Madrid, a module much more perfect in principle than the
  Italian module is employed. Part of the water is supplied for
  irrigation, and as it is very valuable its strict measurement is
  essential. The module (fig. 72) consists of two chambers one above the
  other, the upper chamber being in free communication with the
  irrigation canal, and the lower chamber discharging by a culvert to
  the fields. In the arched roof between the chambers there is a
  circular sharp-edged orifice in a bronze plate. Hanging in this there
  is a bronze plug of variable diameter suspended from a hollow brass
  float. If the water level in the canal lowers, the plug descends and
  gives an enlarged opening, and conversely. Thus a perfectly constant
  discharge with a varying head can be obtained, provided no clogging or
  silting of the chambers prevents the free discharge of the water or
  the rise and fall of the float. The theory of the module is very
  simple. Let R (fig. 71) be the radius of the fixed opening, r the
  radius of the plug at a distance h from the plane of flotation of the
  float, and Q the required discharge of the module. Then

    Q = c[pi](R² - r²) [root](2gh).

  Taking c = 0.63,

    Q = 15.88(R² - r²) [root]h;

    r = [root]{R² - Q/15.88 [root]h}.

  Choosing a value for R, successive values of r can be found for
  different values of h, and from these the curve of the plug can be
  drawn. The module shown in fig. 72 will discharge 1 cubic metre per
  second. The fixed opening is 0.2 metre diameter, and the greatest head
  above the fixed orifice is 1 metre. The use of this module involves a
  great sacrifice of level between the canal and the fields. The module
  is described in Sir C. Scott-Moncrieff's _Irrigation in Southern
  Europe_.

  § 59. _Reservoir Gauging Basins._--In obtaining the power to store the
  water of streams in reservoirs, it is usual to concede to riparian
  owners below the reservoirs a right to a regulated supply throughout
  the year. This compensation water requires to be measured in such a
  way that the millowners and others interested in the matter can assure
  themselves that they are receiving a proper quantity, and they are
  generally allowed a certain amount of control as to the times during
  which the daily supply is discharged into the stream.

  [Illustration: FIG. 72.]

  Fig. 74 shows an arrangement designed for the Manchester water works.
  The water enters from the reservoir at chamber A, the object of which
  is to still the irregular motion of the water. The admission is
  regulated by sluices at b, b, b. The water is discharged by orifices
  or notches at a, a, over which a tolerably constant head is maintained
  by adjusting the sluices at b, b, b. At any time the millowners can
  see whether the discharge is given and whether the proper head is
  maintained over the orifices. To test at any time the discharge of the
  orifices, a gauging basin B is provided. The water ordinarily flows
  over this, without entering it, on a floor of cast-iron plates. If the
  discharge is to be tested, the water is turned for a definite time
  into the gauging basin, by suddenly opening and closing a sluice at c.
  The volume of flow can be ascertained from the depth in the gauging
  chamber. A mechanical arrangement (fig. 73) was designed for securing
  an absolutely constant head over the orifices at a, a. The orifices
  were formed in a cast-iron plate capable of sliding up and down,
  without sensible leakage, on the face of the wall of the chamber. The
  orifice plate was attached by a link to a lever, one end of which
  rested on the wall and the other on floats f in the chamber A. The
  floats rose and fell with the changes of level in the chamber, and
  raised and lowered the orifice plate at the same time. This mechanical
  arrangement was not finally adopted, careful watching of the sluices
  at b, b, b, being sufficient to secure a regular discharge. The
  arrangement is then equivalent to an Italian module, but on a large
  scale.

  [Illustration: FIG. 73.--Scale 1/120.]

  [Illustration: FIG. 74.--Scale 1/500.]

  § 60. _Professor Fleeming Jenkin's Constant Flow Valve._--In the
  modules thus far described constant discharge is obtained by varying
  the area of the orifice through which the water flows. Professor F.
  Jenkin has contrived a valve in which a constant pressure head is
  obtained, so that the orifice need not be varied (_Roy. Scot. Society_
  _of Arts_, 1876). Fig. 75 shows a valve of this kind suitable for a
  6-in. water main. The water arriving by the main C passes through an
  equilibrium valve D into the chamber A, and thence through a sluice O,
  which can be set for any required area of opening, into the
  discharging main B. The object of the arrangement is to secure a
  constant difference of pressure between the chambers A and B, so that
  a constant discharge flows through the stop valve O. The equilibrium
  valve D is rigidly connected with a plunger P loosely fitted in a
  diaphragm, separating A from a chamber B2 connected by a pipe B1 with
  the discharging main B. Any increase of the difference of pressure in
  A and B will drive the plunger up and close the equilibrium valve, and
  conversely a decrease of the difference of pressure will cause the
  descent of the plunger and open the equilibrium valve wider. Thus a
  constant difference of pressure is obtained in the chambers A and B.
  Let [omega] be the area of the plunger in square feet, p the
  difference of pressure in the chambers A and B in pounds per square
  foot, w the weight of the plunger and valve. Then if at any moment
  p[omega] exceeds w the plunger will rise, and if it is less than w the
  plunger will descend. Apart from friction, and assuming the valve D to
  be strictly an equilibrium valve, since [omega] and w are constant, p
  must be constant also, and equal to w/[omega]. By making w small and
  [omega] large, the difference of pressure required to ensure the
  working of the apparatus may be made very small. Valves working with a
  difference of pressure of ½ in. of water have been constructed.

  [Illustration: FIG. 75.--Scale 1/24.]


  VI. STEADY FLOW OF COMPRESSIBLE FLUIDS.

  [Illustration: FIG. 76.]

  § 61. _External Work during the Expansion of Air._--If air expands
  without doing any external work, its temperature remains constant.
  This result was first experimentally demonstrated by J. P. Joule. It
  leads to the conclusion that, however air changes its state, the
  internal work done is proportional to the change of temperature. When,
  in expanding, air does work against an external resistance, either
  heat must be supplied or the temperature falls.

  To fix the conditions, suppose 1 lb. of air confined behind a piston
  of 1 sq. ft. area (fig. 76). Let the initial pressure be p1 and the
  volume of the air v1, and suppose this to expand to the pressure p2
  and volume v2. If p and v are the corresponding pressure and volume at
  any intermediate point in the expansion, the work done on the piston
  during the expansion from v to v + dv is pdv, and the whole work
  during the expansion from v1 to v2, represented by the area abcd, is
      _
     /v2
     |   p dv.
    _/v1

  Amongst possible cases two may be selected.

  _Case 1._--So much heat is supplied to the air during expansion that
  the temperature remains constant. Hyperbolic expansion.

  Then

    pv = p1v1.

  Work done during expansion per pound of air
        _                _
       /v2              /v2
    =  |   p dv  = p1v1 |   dv/v
      _/v1             _/v1

    = p1v1 log_[epsilon] v2v1 = p1v1 log_[epsilon] p1p2.   (1)

  Since the weight per cubic foot is the reciprocal of the volume per
  pound, this may be written

    (p1/G1) log_[epsilon] G1/G2.   (1a)

  Then the expansion curve ab is a common hyperbola.

  _Case 2._--No heat is supplied to the air during expansion. Then the
  air loses an amount of heat equivalent to the external work done and
  the temperature falls. Adiabatic expansion.

  In this case it can be shown that

    pv^[gamma] = p1v1^[gamma],

  where [gamma] is the ratio of the specific heats of air at constant
  pressure and volume. Its value for air is 1.408, and for dry steam
  1.135.

  Work done during expansion per pound of air.

        _                         _
       /v2                       /v2
    =  |   p dv   = p1v1^[gamma] |   dv/v^[gamma]
      _/v1                      _/v1

    = - {p1v1^[gamma]/([gamma] - 1)} {1/v2^([gamma] - 1) -  1/v1^([gamma] - 1)}

    = {p1v1^[gamma]/([gamma] - 1)} {1/v1^([gamma] - 1) -  1/v2^([gamma] - 1)}

    = {p1v1/([gamma] - 1)} {1 - (v1/v2)^([gamma] - 1)}.   (2)

  The value of p1v1 for any given temperature can be found from the data
  already given.

  As before, substituting the weights G1, G2 per cubic foot for the
  volumes per pound, we get for the work of expansion

    (p1/G1){1/([gamma] - 1)} {1 - (G2/G1)^([gamma] - 1)},   (2a)

    = p1v1{1/([gamma] - 1)} {1 - (p2/p1)^([gamma] - 1)/[gamma]}.   (2b)

  [Illustration: FIG. 77.]

  § 62. _Modification of the Theorem of Bernoulli for the Case of a
  Compressible Fluid._--In the application of the principle of work to a
  filament of compressible fluid, the internal work done by the
  expansion of the fluid, or absorbed in its compression, must be taken
  into account. Suppose, as before, that AB (fig. 77) comes to A´B´ in a
  short time t. Let p1, [omega]1, v1, G1 be the pressure, sectional area
  of stream, velocity and weight of a cubic foot at A, and p2, [omega]2,
  v2, G2 the same quantities at B. Then, from the steadiness of motion,
  the weight of fluid passing A in any given time must be equal to the
  weight passing B:

    G1[omega]1v1t = G2[omega]2v2t.

  Let z1, z2 be the heights of the sections A and B above any given
  datum. Then the work of gravity on the mass AB in t seconds is

    G1[omega]1v1t(z1 - z2) = W(z1 - z2)t,

  where W is the weight of gas passing A or B per second. As in the case
  of an incompressible fluid, the work of the pressures on the ends of
  the mass AB is

    p1[omega]1v1t - p2[omega]2v2t,
    = (p1/G1 - p2/G2)Wt.

  The work done by expansion of Wt lb. of fluid between A and B is Wt
  [int][v1 to v2] p dv. The change of kinetic energy as before is (W/2g)
  (v2² - v1²)t. Hence, equating work to change of kinetic energy,

                                       _
                                      /v2
    W(z1 - z2)t + (p1/G1 - p2/G2)Wt + |   p dv = (W/2g)(v2² - v1²)t;
                                     _/v1
                                                    _
                                                   /v2                                                                        /
    .: z1 + p1/G1 + v1²/2g = z2 + p²/G2 + v2²/2g - |   p dv.   (1)
                                                  _/v1

  Now the work of expansion per pound of fluid has already been given.
  If the temperature is constant, we get (eq. 1a, § 61)

    z1 + p1/G1 + v1²/2g
      = z2 + p²/G2 + v2²/2g - (p1/G1) log_[epsilon] (G1/G2).

  But at constant temperature p1/G1 = p2/G2;

    .: z1 + v1²/2g = z2 + v2²/2g - (p1/G1) log_[epsilon] (p1/p2),   (2)

  or, neglecting the difference of level,

    (v2² - v1²)/2g = (p1/G1) log_[epsilon] (p1/p2).   (2a)

  Similarly, if the expansion is adiabatic (eq. 2a, § 61),

    z1 + p1/G1 + v1²/2g = z2 + p2/G2 + v2²/2g
      - (p1/G1){1/([gamma] - 1)} {1 - (p2/p1)^([gamma] - 1)/[gamma]};   (3)

  or, neglecting the difference of level,

    (v2² - v1²)/2g =
      (p1/G1)[1 + 1/([gamma] - 1){1 - (p2/p1)^([gamma]-1)/[gamma]}] - p2/G2.   (3a)

  It will be seen hereafter that there is a limit in the ratio p1/p2
  beyond which these expressions cease to be true.

  § 63. _Discharge of Air from an Orifice._--The form of the equation of
  work for a steady stream of compressible fluid is

    z1 + p1/G1 + v1²/2g = z2 + p2/G2 + v2²/2g -
      (p1/G1){1/([gamma] - 1)} {1 - (p2/p1^([gamma] - 1)/[gamma]},

  the expansion being adiabatic, because in the flow of the streams of
  air through an orifice no sensible amount of heat can be communicated
  from outside.

  Suppose the air flows from a vessel, where the pressure is p1 and the
  velocity sensibly zero, through an orifice, into a space where the
  pressure is p2. Let v2 be the velocity of the jet at a point where the
  convergence of the streams has ceased, so that the pressure in the jet
  is also p2. As air is light, the work of gravity will be small
  compared with that of the pressures and expansion, so that z1z2 may be
  neglected. Putting these values in the equation above--

    p1/G1 = p2/G2 + v2²/2g - (p1/G1){1/([gamma] - 1)}
      {1 - (p2/p1)^([gamma] - 1)/[gamma];

    v2²/2g = p1/G1 - p2/G2 + (p1/G1){1/([gamma] - 1)}
      {1 - (p2/p1)^([gamma] - 1)/[gamma]}

    = (p1/G1){[gamma]/([gamma] - 1) - (p2/p1)^([gamma] - 1)/[gamma]/([gamma] - 1)} - p2/G2.

  But

    p1/G1^([gamma]) = p2/G2^([gamma])
    .: p2/G2 = (p1/G1)(p2/p1)^([gamma] - 1)/[gamma]

    v2²/2g = (p1/G1){[gamma]/([gamma] - 1)} {1 - (p2/p1)^(([gamma] - 1)/[gamma]};   (1)

  or

    v2²/2g = {[gamma]/([gamma] - 1)} {(p1/G1) - (p2/G2)};

  an equation commonly ascribed to L. J. Weisbach (_Civilingenieur_,
  1856), though it appears to have been given earlier by A. J. C. Barre
  de Saint Venant and L. Wantzel.

  It has already (§ 9, eq. 4a) been seen that

    p1/G1 = (p0/G0) ([tau]1/[tau]0)

  where for air p0 = 2116.8, G0 = .08075 and [tau]0 = 492.6.

    v2²/2g = {p0[tau]1[gamma]/G0[tau]0([gamma] - 1)}
      {1 - (p2/p1)^([gamma] - 1)/[gamma]};   (2)

  or, inserting numerical values,

    v2²/2g = 183.6[tau]1 {1 - (p2/p1)^(0.29)};   (2a)

  which gives the velocity of discharge v2 in terms of the pressure and
  absolute temperature, p1, [tau]1, in the vessel from which the air
  flows, and the pressure p2 in the vessel into which it flows.

  Proceeding now as for liquids, and putting [omega] for the area of the
  orifice and c for the coefficient of discharge, the volume of air
  discharged per second at the pressure p2 and temperature [tau]2 is

    Q2 = c[omega]v2 = c[omega] [root][(2g[gamma]p1/([gamma] - 1)G1)
      (1 - (p2/p1)^([gamma] - 1)/[gamma])]

    = 108.7c[omega] [root][[tau]1 {1 - (p2/p1)^(0.29)}].   (3)

  If the volume discharged is measured at the pressure p1 and absolute
  temperature [tau]1 in the vessel from which the air flows, let Q1 be
  that volume; then

    p1Q1^[gamma] = p2Q2^[gamma];

    Q1 = (p2/p1)^(1/[gamma]) Q2;

    Q1 = c[omega] [root][{2g[gamma]p1/([gamma] - 1)G1}
      {(p2/p1)^(2/[gamma]) - (p2/p1)^([gamma] + 1)/[gamma]}].

  Let

    (p2/p1)^(2/[gamma]) - (p2/p1)^([gamma] - 1)/[gamma] =
      (p2/p1)^(1.41) - (p2/p1)^(1.7) = [psi]; then

    Q1 = c[omega] [root][2g[gamma]p1[psi]/([gamma] - 1)G1]
       = 108.7c[omega] [root]([tau]1[psi]).   (4)

  The weight of air at pressure p1 and temperature [tau]1 is

    G1 = p1/53.2[tau]1 lb. per cubic foot.

  Hence the weight of air discharged is

    W = G1Q1 = c[omega] [root][2g[gamma]p1G1[psi]/([gamma] - 1)]
      = 2.043c[omega]p1 [root]([psi]/[tau]1).   (5)

  Weisbach found the following values of the coefficient of discharge
  c:--

    Conoidal mouthpieces of the form of the \
      contracted vein with effective         >       c =
      pressures of .23 to 1.1 atmosphere    /   0.97 to 0.99
    Circular sharp-edged orifices               0.563 " 0.788
    Short cylindrical mouthpieces               0.81  " 0.84
    The same rounded at the inner end           0.92  " 0.93
    Conical converging mouthpieces              0.90  " 0.99

  § 64. _Limit to the Application of the above Formulae._--In the
  formulae above it is assumed that the fluid issuing from the orifice
  expands from the pressure p1 to the pressure p2, while passing from
  the vessel to the section of the jet considered in estimating the area
  [omega]. Hence p2 is strictly the pressure in the jet at the plane of
  the external orifice in the case of mouthpieces, or at the plane of
  the contracted section in the case of simple orifices. Till recently
  it was tacitly assumed that this pressure p2 was identical with the
  general pressure external to the orifice. R. D. Napier first
  discovered that, when the ratio p2/p1 exceeded a value which does not
  greatly differ from 0.5, this was no longer true. In that case the
  expansion of the fluid down to the external pressure is not completed
  at the time it reaches the plane of the contracted section, and the
  pressure there is greater than the general external pressure; or, what
  amounts to the same thing, the section of the jet where the expansion
  is completed is a section which is greater than the area c_c[omega] of
  the contracted section of the jet, and may be greater than the area
  [omega] of the orifice. Napier made experiments with steam which
  showed that, so long as p2/p1 > 0.5, the formulae above were
  trustworthy, when p2 was taken to be the general external pressure,
  but that, if p2/p1 < 0.5, then the pressure at the contracted section
  was independent of the external pressure and equal to 0.5p1. Hence in
  such cases the constant value 0.5 should be substituted in the
  formulae for the ratio of the internal and external pressures p2/p1.

  It is easily deduced from Weisbach's theory that, if the pressure
  external to an orifice is gradually diminished, the weight of air
  discharged per second increases to a maximum for a value of the ratio

    p2/p1 = {2/([gamma] + 1)}^([gamma] - 1/[gamma])
          = 0.527 for air
          = 0.58 for dry steam.

  For a further decrease of external pressure the discharge
  diminishes,--a result no doubt improbable. The new view of Weisbach's
  formula is that from the point where the maximum is reached, or not
  greatly differing from it, the pressure at the contracted section
  ceases to diminish.

  A. F. Fliegner showed (_Civilingenieur_ xx., 1874) that for air
  flowing from well-rounded mouthpieces there is no discontinuity of the
  law of flow, as Napier's hypothesis implies, but the curve of flow
  bends so sharply that Napier's rule may be taken to be a good
  approximation to the true law. The limiting value of the ratio p2/p1,
  for which Weisbach's formula, as originally understood, ceases to
  apply, is for air 0.5767; and this is the number to be substituted for
  p2/p1 in the formulae when p2/p1 falls below that value. For later
  researches on the flow of air, reference may be made to G. A. Zeuner's
  paper (_Civilingenieur_, 1871), and Fliegner's papers (_ibid._, 1877,
  1878).


  VII. FRICTION OF LIQUIDS.

  § 65. When a stream of fluid flows over a solid surface, or conversely
  when a solid moves in still fluid, a resistance to the motion is
  generated, commonly termed fluid friction. It is due to the viscosity
  of the fluid, but generally the laws of fluid friction are very
  different from those of simple viscous resistance. It would appear
  that at all speeds, except the slowest, rotating eddies are formed by
  the roughness of the solid surface, or by abrupt changes of velocity
  distributed throughout the fluid; and the energy expended in producing
  these eddying motions is gradually lost in overcoming the viscosity of
  the fluid in regions more or less distant from that where they are
  first produced.

  The laws of fluid friction are generally stated thus:--

  1. The frictional resistance is independent of the pressure between
  the fluid and the solid against which it flows. This may be verified
  by a simple direct experiment. C. H. Coulomb, for instance, oscillated
  a disk under water, first with atmospheric pressure acting on the
  water surface, afterwards with the atmospheric pressure removed. No
  difference in the rate of decrease of the oscillations was observed.
  The chief proof that the friction is independent of the pressure is
  that no difference of resistance has been observed in water mains and
  in other cases, where water flows over solid surfaces under widely
  different pressures.

  2. The frictional resistance of large surfaces is proportional to the
  area of the surface.

  3. At low velocities of not more than 1 in. per second for water, the
  frictional resistance increases directly as the relative velocity of
  the fluid and the surface against which it flows. At velocities of ½
  ft. per second and greater velocities, the frictional resistance is
  more nearly proportional to the square of the relative velocity.

  In many treatises on hydraulics it is stated that the frictional
  resistance is independent of the nature of the solid surface. The
  explanation of this was supposed to be that a film of fluid remained
  attached to the solid surface, the resistance being generated between
  this fluid layer and layers more distant from the surface. At
  extremely low velocities the solid surface does not seem to have much
  influence on the friction. In Coulomb's experiments a metal surface
  covered with tallow, and oscillated in water, had exactly the same
  resistance as a clean metal surface, and when sand was scattered over
  the tallow the resistance was only very slightly increased. The
  earlier calculations of the resistance of water at higher velocities
  in iron and wood pipes and earthen channels seemed to give a similar
  result. These, however, were erroneous, and it is now well understood
  that differences of roughness of the solid surface very greatly
  influence the friction, at such velocities as are common in
  engineering practice. H. P. G. Darcy's experiments, for instance,
  showed that in old and incrusted water mains the resistance was twice
  or sometimes thrice as great as in new and clean mains.

  § 66. _Ordinary Expressions for Fluid Friction at Velocities not
  Extremely Small._--Let f be the frictional resistance estimated in
  pounds per square foot of surface at a velocity of 1 ft. per second;
  [omega] the area of the surface in square feet; and v its velocity in
  feet per second relatively to the water in which it is immersed. Then,
  in accordance with the laws stated above, the total resistance of the
  surface is

    R = f[omega]v²   (1)

  where f is a quantity approximately constant for any given surface. If

    [xi] = 2gf/G,

    R = [xi]G[omega]v²/2g,   (2)

  where [xi] is, like f, nearly constant for a given surface, and is
  termed the coefficient of friction.

  The following are average values of the coefficient of friction for
  water, obtained from experiments on large plane surfaces, moved in an
  indefinitely large mass of water.

    +------------------------------------+--------------+-----------------+
    |                                    | Coefficient  |    Frictional   |
    |                                    | of Friction, |  Resistance in  |
    |                                    |     [xi]     | lb. per sq. ft. |
    |                                    |              |        f        |
    +------------------------------------+--------------+-----------------+
    |                                    |              |                 |
    | New well-painted iron plate        |    .00489    |     .00473      |
    | Painted and planed plank (Beaufoy) |    .00350    |     .00339      |
    | Surface of iron ships (Rankine)    |    .00362    |     .00351      |
    | Varnished surface (Froude)         |    .00258    |     .00250      |
    | Fine sand surface     "            |    .00418    |     .00405      |
    | Coarser sand surface  "            |    .00503    |     .00488      |
    +------------------------------------+--------------+-----------------+

  The distance through which the frictional resistance is overcome is v
  ft. per second. The work expended in fluid friction is therefore given
  by the equation--

    Work expended = f[omega]v³ foot-pounds per second \   (3).
                  = [xi]G[omega]v³/2g   "        "    /

  The coefficient of friction and the friction per square foot of
  surface can be indirectly obtained from observations of the discharge
  of pipes and canals. In obtaining them, however, some assumptions as
  to the motion of the water must be made, and it will be better
  therefore to discuss these values in connexion with the cases to which
  they are related.

  Many attempts have been made to express the coefficient of friction in
  a form applicable to low as well as high velocities. The older
  hydraulic writers considered the resistance termed fluid friction to
  be made up of two parts,--a part due directly to the distortion of the
  mass of water and proportional to the velocity of the water relatively
  to the solid surface, and another part due to kinetic energy imparted
  to the water striking the roughnesses of the solid surface and
  proportional to the square of the velocity. Hence they proposed to
  take

    [xi] = [alpha] + [beta]/v

  in which expression the second term is of greatest importance at very
  low velocities, and of comparatively little importance at velocities
  over about ½ ft. per second. Values of [xi] expressed in this and
  similar forms will be given in connexion with pipes and canals.

  All these expressions must at present be regarded as merely empirical
  expressions serving practical purposes.

  The frictional resistance will be seen to vary through wider limits
  than these expressions allow, and to depend on circumstances of which
  they do not take account.

  § 67. _Coulomb's Experiments._--The first direct experiments on fluid
  friction were made by Coulomb, who employed a circular disk suspended
  by a thin brass wire and oscillated in its own plane. His experiments
  were chiefly made at very low velocities. When the disk is rotated to
  any given angle, it oscillates under the action of its inertia and the
  torsion of the wire. The oscillations diminish gradually in
  consequence of the work done in overcoming the friction of the disk.
  The diminution furnishes a means of determining the friction.

  [Illustration: FIG. 78.]

  Fig. 78 shows Coulomb's apparatus. LK supports the wire and disk: ag
  is the brass wire, the torsion of which causes the oscillations; DS is
  a graduated disk serving to measure the angles through which the
  apparatus oscillates. To this the friction disk is rigidly attached
  hanging in a vessel of water. The friction disks were from 4.7 to 7.7
  in. diameter, and they generally made one oscillation in from 20 to 30
  seconds, through angles varying from 360° to 6°. When the velocity of
  the circumference of the disk was less than 6 in. per second, the
  resistance was sensibly proportional to the velocity.

  _Beaufoy's Experiments._--Towards the end of the 18th century Colonel
  Mark Beaufoy (1764-1827) made an immense mass of experiments on the
  resistance of bodies moved through water (_Nautical and Hydraulic
  Experiments_, London, 1834). Of these the only ones directly bearing
  on surface friction were some made in 1796 and 1798. Smooth painted
  planks were drawn through water and the resistance measured. For two
  planks differing in area by 46 sq. ft., at a velocity of 10 ft. per
  second, the difference of resistance, measured on the difference of
  area, was 0.339 lb. per square foot. Also the resistance varied as the
  1.949th power of the velocity.

  [Illustration: FIG. 79.]

  § 68. _Froude's Experiments._--The most important direct experiments
  on fluid friction at ordinary velocities are those made by William
  Froude (1810-1879) at Torquay. The method adopted in these experiments
  was to tow a board in a still water canal, the velocity and the
  resistance being registered by very ingenious recording arrangements.
  The general arrangement of the apparatus is shown in fig. 79. AA is
  the board the resistance of which is to be determined. B is a cutwater
  giving a fine entrance to the plane surfaces of the board. CC is a bar
  to which the board AA is attached, and which is suspended by a
  parallel motion from a carriage running on rails above the still water
  canal. G is a link by which the resistance of the board is transmitted
  to a spiral spring H. A bar I rigidly connects the other end of the
  spring to the carriage. The dotted lines K, L indicate the position of
  a couple of levers by which the extension of the spring is caused to
  move a pen M, which records the extension on a greatly increased
  scale, by a line drawn on the paper cylinder N. This cylinder revolves
  at a speed proportionate to that of the carriage, its motion being
  obtained from the axle of the carriage wheels. A second pen O,
  receiving jerks at every second and a quarter from a clock P, records
  time on the paper cylinder. The scale for the line of resistance is
  ascertained by stretching the spiral spring by known weights. The
  boards used for the experiment were 3/16 in. thick, 19 in. deep, and
  from 1 to 50 ft. in length, cutwater included. A lead keel
  counteracted the buoyancy of the board. The boards were covered with
  various substances, such as paint, varnish, Hay's composition,
  tinfoil, &c., so as to try the effect of different degrees of
  roughness of surface. The results obtained by Froude may be summarized
  as follows:--

  1. The friction per square foot of surface varies very greatly for
  different surfaces, being generally greater as the sensible roughness
  of the surface is greater. Thus, when the surface of the board was
  covered as mentioned below, the resistance for boards 50 ft. long, at
  10 ft. per second, was--

    Tinfoil or varnish   0.25 lb. per sq. ft.
    Calico               0.47     "      "
    Fine sand            0.405    "      "
    Coarser sand         0.488    "      "

  2. The power of the velocity to which the friction is proportional
  varies for different surfaces. Thus, with short boards 2 ft. long,

    For tinfoil the resistance varied as v^(2.16).
    For other surfaces the resistance varied as v^(2.00).

  With boards 50 ft. long,

    For varnish or tinfoil the resistance varied as v^(1.83).
    For sand the resistance varied as v^(2.00).

  3. The average resistance per square foot of surface was much greater
  for short than for long boards; or, what is the same thing, the
  resistance per square foot at the forward part of the board was
  greater than the friction per square foot of portions more sternward.
  Thus,

                                      Mean Resistance in
                                       lb. per sq. ft.
    Varnished surface     2 ft. long        0.41
                         50    "            0.25
    Fine sand surface     2    "            0.81
                         50    "            0.405

  This remarkable result is explained thus by Froude: "The portion of
  surface that goes first in the line of motion, in experiencing
  resistance from the water, must in turn communicate motion to the
  water, in the direction in which it is itself travelling. Consequently
  the portion of surface which succeeds the first will be rubbing,
  not against stationary water, but against water partially moving in
  its own direction, and cannot therefore experience so much resistance
  from it."

  § 69. The following table gives a general statement of Froude's
  results. In all the experiments in this table, the boards had a fine
  cutwater and a fine stern end or run, so that the resistance was
  entirely due to the surface. The table gives the resistances per
  square foot in pounds, at the standard speed of 600 feet per minute,
  and the power of the speed to which the friction is proportional, so
  that the resistance at other speeds is easily calculated.

    +------------+---------------------------------------------------------------------------+
    |            |           Length of Surface, or Distance from Cutwater, in feet.          |
    |            +------------------+------------------+------------------+------------------+
    |            |       2 ft.      |       8 ft.      |       20 ft.     |      50 ft.      |
    |            +------+-----+-----+------+-----+-----+------+-----+-----+------+-----+-----+
    |            |   A  |  B  |  C  |   A  |  B  |  C  |   A  |  B  |  C  |   A  |  B  |  C  |
    +------------+------+-----+-----+------+-----+-----+------+-----+-----+------+-----+-----+
    | Varnish    | 2.00 | .41 |.390 | 1.85 |.325 |.264 | 1.85 |.278 |.240 | 1.83 |.250 |.226 |
    | Paraffin   |  ..  | .38 |.370 | 1.94 |.314 |.260 | 1.93 |.271 |.237 |  ..  |  .. |  .. |
    | Tinfoil    | 2.16 | .30 |.295 | 1.99 |.278 |.263 | 1.90 |.262 |.244 | 1.83 |.246 |.232 |
    | Calico     | 1.93 | .87 |.725 | 1.92 |.626 |.504 | 1.89 |.531 |.447 | 1.87 |.474 |.423 |
    | Fine sand  | 2.00 | .81 |.690 | 2.00 |.583 |.450 | 2.00 |.480 |.384 | 2.06 |.405 |.337 |
    | Medium sand| 2.00 | .90 |.730 | 2.00 |.625 |.488 | 2.00 |.534 |.465 | 2.00 |.488 |.456 |
    | Coarse sand| 2.00 |1.10 |.880 | 2.00 |.714 |.520 | 2.00 |.588 |.490 |  ..  |  .. |  .. |
    +--------- --+------+-----+-----+------+-----+-----+------+-----+-----+------+-----+-----+

  Columns A give the power of the speed to which the resistance is
  approximately proportional.

  Columns B give the mean resistance per square foot of the whole
  surface of a board of the lengths stated in the table.

  Columns C give the resistance in pounds of a square foot of surface at
  the distance sternward from the cutwater stated in the heading.

  Although these experiments do not directly deal with surfaces of
  greater length than 50 ft., they indicate what would be the
  resistances of longer surfaces. For at 50 ft. the decrease of
  resistance for an increase of length is so small that it will make no
  very great difference in the estimate of the friction whether we
  suppose it to continue to diminish at the same rate or not to diminish
  at all. For a varnished surface the friction at 10 ft. per second
  diminishes from 0.41 to 0.32 lb. per square foot when the length is
  increased from 2 to 8 ft., but it only diminishes from 0.278 to 0.250
  lb. per square foot for an increase from 20 ft. to 50 ft.

  If the decrease of friction sternwards is due to the generation of a
  current accompanying the moving plane, there is not at first sight any
  reason why the decrease should not be greater than that shown by the
  experiments. The current accompanying the board might be assumed to
  gain in volume and velocity sternwards, till the velocity was nearly
  the same as that of the moving plane and the friction per square foot
  nearly zero. That this does not happen appears to be due to the mixing
  up of the current with the still water surrounding it. Part of the
  water in contact with the board at any point, and receiving energy of
  motion from it, passes afterwards to distant regions of still water,
  and portions of still water are fed in towards the board to take its
  place. In the forward part of the board more kinetic energy is given
  to the current than is diffused into surrounding space, and the
  current gains in velocity. At a greater distance back there is an
  approximate balance between the energy communicated to the water and
  that diffused. The velocity of the current accompanying the board
  becomes constant or nearly constant, and the friction per square foot
  is therefore nearly constant also.

  § 70. _Friction of Rotating Disks._--A rotating disk is virtually a
  surface of unlimited extent and it is convenient for experiments on
  friction with different surfaces at different speeds. Experiments
  carried out by Professor W. C. Unwin (_Proc. Inst. Civ. Eng._ lxxx.)
  are useful both as illustrating the laws of fluid friction and as
  giving data for calculating the resistance of the disks of turbines
  and centrifugal pumps. Disks of 10, 15 and 20 in. diameter fixed on a
  vertical shaft were rotated by a belt driven by an engine. They were
  enclosed in a cistern of water between parallel top and bottom fixed
  surfaces. The cistern was suspended by three fine wires. The friction
  of the disk is equal to the tendency of the cistern to rotate, and
  this was measured by balancing the cistern by a fine silk cord passing
  over a pulley and carrying a scale pan in which weights could be
  placed.

  If [omega] is an element of area on the disk moving with the velocity
  v, the friction on this element is f[omega]v^n, where f and n are
  constant for any given kind of surface. Let [alpha] be the angular
  velocity of rotation, R the radius of the disk. Consider a ring of the
  surface between r and r + dr. Its area is 2[pi]r dr, its velocity
  [alpha]r and the friction of this ring is f2[pi]r dr[alpha]^n r^n. The
  moment of the friction about the axis of rotation is
  2[pi][alpha]^n fr^(n + 2)dr, and the total moment of friction for the
  two sides of the disk is
                          _
                         /R
    M = 4[pi][alpha]^n f | r^(n+2) dr = {4[pi][alpha]^n /(n + 3)}fR^(n+3).      .
                        _/0

  If N is the number of revolutions per sec.,

    M = {2^(n+2) [pi]^(n+1) N^n/(n + 3)} fR^(n+3),

  and the work expended in rotating the disk is

    M[alpha] = {2^(n+3)[pi]^(n+2)N^(n+1)/(n + 3)} fR^(n+3), foot lb. per sec.

  The experiments give directly the values of M for the disks
  corresponding to any speed N. From these the values of f and n can be
  deduced, f being the friction per square foot at unit velocity. For
  comparison with Froude's results it is convenient to calculate the
  resistance at 10 ft. per second, which is F = f10^n.

  The disks were rotated in chambers 22 in. diameter and 3, 6 and 12 in.
  deep. In all cases the friction of the disks increased a little as the
  chamber was made larger. This is probably due to the stilling of the
  eddies against the surface of the chamber and the feeding back of the
  stilled water to the disk. Hence the friction depends not only on the
  surface of the disk but to some extent on the surface of the chamber
  in which it rotates. If the surface of the chamber is made rougher by
  covering with coarse sand there is also an increase of resistance.

  For the smoother surfaces the friction varied as the 1.85th power of
  the velocity. For the rougher surfaces the power of the velocity to
  which the resistance was proportional varied from 1.9 to 2.1. This is
  in agreement with Froude's results.

  Experiments with a bright brass disk showed that the friction
  decreased with increase of temperature. The diminution between 41° and
  130° F. amounted to 18%. In the general equation M = cN^n for any
  given disk,

    c_t = 0.1328(1 - 0.0021t),

  where c_t is the value of c for a bright brass disk 0.85 ft. in
  diameter at a temperature t° F.

  The disks used were either polished or made rougher by varnish or by
  varnish and sand. The following table gives a comparison of the
  results obtained with the disks and Froude's results on planks 50 ft.
  long. The values given are the resistances per square foot at 10 ft.
  per sec.

    _Froude's Experiments._ |          _Disk Experiments._
                            |
    Tinfoil surface  0.232  |   Bright brass      0.202 to 0.229
    Varnish          0.226  |   Varnish           0.220 to 0.233
    Fine sand        0.337  |   Fine sand              0.339
    Medium sand      0.456  |   Very coarse sand  0.587 to 0.715


  VIII. STEADY FLOW OF WATER IN PIPES OF UNIFORM SECTION.

  § 71. The ordinary theory of the flow of water in pipes, on which all
  practical formulae are based, assumes that the variation of velocity
  at different points of any cross section may be neglected. The water
  is considered as moving in plane layers, which are driven through the
  pipe against the frictional resistance, by the difference of pressure
  at or elevation of the ends of the pipe. If the motion is steady the
  velocity at each cross section remains the same from moment to moment,
  and if the cross sectional area is constant the velocity at all
  sections must be the same. Hence the motion is uniform. The most
  important resistance to the motion of the water is the surface
  friction of the pipe, and it is convenient to estimate this
  independently of some smaller resistances which will be accounted for
  presently.

  [Illustration: FIG. 80.]

  In any portion of a uniform pipe, excluding for the present the ends
  of the pipe, the water enters and leaves at the same velocity. For
  that portion therefore the work of the external forces and of the
  surface friction must be equal. Let fig. 80 represent a very short
  portion of the pipe, of length dl, between cross sections at z and z +
  dz ft. above any horizontal datum line xx, the pressures at the cross
  sections being p and p + dp lb. per square foot. Further, let Q be the
  volume of flow or discharge of the pipe per second, [Omega] the area
  of a normal cross section, and [chi] the perimeter of the pipe. The Q
  cubic feet, which flow through the space considered per second, weigh
  GQ lb., and fall through a height -dz ft. The work done by gravity is
  then

    -GQ dz;

  a positive quantity if dz is negative, and vice versa. The resultant
  pressure parallel to the axis of the pipe is p - (p + dp) = -dp lb.
  per square foot of the cross section. The work of this pressure on the
  volume Q is

    -Q dp.

  The only remaining force doing work on the system is the friction
  against the surface of the pipe. The area of that surface is [chi]dl.

  The work expended in overcoming the frictional resistance per second
  is (see § 66, eq. 3)

    -[zeta]G[chi]dlv³/2g,

  or, since Q = [Omega]v,

    -[zeta]G([chi]/[Omega]) Q (v²/2g) dl;

  the negative sign being taken because the work is done against a
  resistance. Adding all these portions of work, and equating the result
  to zero, since the motion is uniform,--

    -GQ dz - Q dp - [zeta]G([chi]/[Omega]) Q (v²/2g) dl = 0.

  Dividing by GQ,

    dz + dp/G + [zeta]([chi]/[Omega])(v²/2g) dl = 0.

  Integrating,

    z + p/G + [zeta]([chi]/[Omega])(v²/2g)l = constant.   (1)

  § 72. Let A and B (fig. 81) be any two sections of the pipe for which
  p, z, l have the values p1, z1, l1, and p2, z2, l2, respectively. Then

    z1 + p1/G + [zeta]([chi]/[Omega])(v²/2g)l1
      = z2 + p2/G + [zeta]([chi]/[Omega])(v²/2g)l2;

  or, if l2 - l1 = L, rearranging the terms,

    [zeta]v²/2g = (1/L){(z1 + p1/G) - (z2 + p2/G)}[Omega]/[chi].   (2)

  Suppose pressure columns introduced at A and B. The water will rise in
  those columns to the heights p1/G and p2/G due to the pressures p1 and
  p2 at A and B. Hence (z1 + p1/G) - (z2 + p2/G) is the quantity
  represented in the figure by DE, the fall of level of the pressure
  columns, or _virtual fall_ of the pipe. If there were no friction in
  the pipe, then by Bernoulli's equation there would be no fall of level
  of the pressure columns, the velocity being the same at A and B. Hence
  DE or h is the head lost in friction in the distance AB. The quantity
  DE/AB = h/L is termed the virtual slope of the pipe or virtual fall
  per foot of length. It is sometimes termed very conveniently the
  relative fall. It will be denoted by the symbol i.

  [Illustration: FIG. 81.]

  The quantity [Omega]/[chi] which appears in many hydraulic equations
  is called the hydraulic mean radius of the pipe. It will be denoted by
  m.

  Introducing these values,

    [zeta]v²/2g = mh/L = mi.   (3)

  For pipes of circular section, and diameter d,

    m = [Omega]/[chi] = ¼[pi]d²/[pi]d = ¼d.

  Then

    [zeta]v²/2g = ¼dh/L = ¼di;   (4)

  or

    h = [zeta](4L/d)(v²/2g);   (4a)

  which shows that the head lost in friction is proportional to the head
  due to the velocity, and is found by multiplying that head by the
  coefficient 4[zeta]L/d. It is assumed above that the atmospheric
  pressure at C and D is the same, and this is usually nearly the case.
  But if C and D are at greatly different levels the excess of
  barometric pressure at C, in feet of water, must be added to p2/G.

  § 73. _Hydraulic Gradient or Line of Virtual Slope._--Join CD. Since
  the head lost in friction is proportional to L, any intermediate
  pressure column between A and B will have its free surface on the line
  CD, and the vertical distance between CD and the pipe at any point
  measures the pressure, exclusive of atmospheric pressure, in the pipe
  at that point. If the pipe were laid along the line CD instead of AB,
  the water would flow at the same velocity by gravity without any
  change of pressure from section to section. Hence CD is termed the
  virtual slope or hydraulic gradient of the pipe. It is the line of
  free surface level for each point of the pipe.

  If an ordinary pipe, connecting reservoirs open to the air, rises at
  any joint above the line of virtual slope, the pressure at that point
  is less than the atmospheric pressure transmitted through the pipe. At
  such a point there is a liability that air may be disengaged from the
  water, and the flow stopped or impeded by the accumulation of air. If
  the pipe rises more than 34 ft. above the line of virtual slope, the
  pressure is negative. But as this is impossible, the continuity of the
  flow will be broken.

  If the pipe is not straight, the line of virtual slope becomes a
  curved line, but since in actual pipes the vertical alterations of
  level are generally small, compared with the length of the pipe,
  distances measured along the pipe are sensibly proportional to
  distances measured along the horizontal projection of the pipe. Hence
  the line of hydraulic gradient may be taken to be a straight line
  without error of practical importance.

  [Illustration: FIG. 82.]

  § 74. _Case of a Uniform Pipe connecting two Reservoirs, when all the
  Resistances are taken into account._--Let h (fig. 82) be the
  difference of level of the reservoirs, and v the velocity, in a pipe
  of length L and diameter d. The whole work done per second is
  virtually the removal of Q cub. ft. of water from the surface of the
  upper reservoir to the surface of the lower reservoir, that is GQh
  foot-pounds. This is expended in three ways. (1) The head v²/2g,
  corresponding to an expenditure of GQv²/2g foot-pounds of work, is
  employed in giving energy of motion to the water. This is ultimately
  wasted in eddying motions in the lower reservoir. (2) A portion of
  head, which experience shows may be expressed in the form
  [zeta]0v²/2g, corresponding to an expenditure of GQ[zeta]0v²/2g
  foot-pounds of work, is employed in overcoming the resistance at the
  entrance to the pipe. (3) As already shown the head expended in
  overcoming the surface friction of the pipe is [zeta](4L/d)(v²/2g)
  corresponding to GQ[zeta](4L/d)(v²/2g) foot-pounds of work. Hence

    GQh = GQv²/2g + GQ[zeta]0v²/2g + GQ[zeta]·4L·v²/d·2g;

    h = (1 + [zeta]0 + [zeta]·4L/d)v²/2g.
                                                      (5)
    v = 8.025 [root][hd/{(1 + [zeta]0)d + 4[zeta]L}].

  If the pipe is bell-mouthed, [zeta]0 is about = .08. If the entrance
  to the pipe is cylindrical, [zeta]0 = 0.505. Hence 1 + [zeta]0 = 1.08
  to 1.505. In general this is so small compared with [zeta]4L/d that,
  for practical calculations, it may be neglected; that is, the losses
  of head other than the loss in surface friction are left out of the
  reckoning. It is only in short pipes and at high velocities that it is
  necessary to take account of the first two terms in the bracket, as
  well as the third. For instance, in pipes for the supply of turbines,
  v is usually limited to 2 ft. per second, and the pipe is bellmouthed.
  Then 1.08v²/2g = 0.067 ft. In pipes for towns' supply v may range from
  2 to 4½ ft. per second, and then 1.5v²/2g = 0.1 to 0.5 ft. In either
  case this amount of head is small compared with the whole virtual fall
  in the cases which most commonly occur.

  When d and v or d and h are given, the equations above are solved
  quite simply. When v and h are given and d is required, it is better
  to proceed by approximation. Find an approximate value of d by
  assuming a probable value for [zeta] as mentioned below. Then from
  that value of d find a corrected value for [zeta] and repeat the
  calculation.

  The equation above may be put in the form

    h = (4[zeta]/d)[{(1 + [zeta]0)d/4[zeta]} + L] v²/2g;   (6)

  from which it is clear that the head expended at the mouthpiece is
  equivalent to that of a length

    (1 + [zeta]0)d/4[zeta]

  of the pipe. Putting 1 + [zeta]0 = 1.505 and [zeta] = 0.01, the length
  of pipe equivalent to the mouthpiece is 37.6d nearly. This may be
  added to the actual length of the pipe to allow for mouthpiece
  resistance in approximate calculations.

  § 75. _Coefficient of Friction for Pipes discharging Water._--From the
  average of a large number of experiments, the value of [zeta] for
  ordinary iron pipes is

    [zeta] = 0.007567.   (7)

  But practical experience shows that no single value can be taken
  applicable to very different cases. The earlier hydraulicians occupied
  themselves chiefly with the dependence of [zeta] on the velocity.
  Having regard to the difference of the law of resistance at very low
  and at ordinary velocities, they assumed that [zeta] might be
  expressed in the form

    [zeta] = a + [beta]/v.

  The following are the best numerical values obtained for [zeta] so
  expressed:--

    +----------------------------------+----------+----------+
    |                                  |  [alpha] |  [beta]  |
    +----------------------------------+----------+----------+
    | R. de Prony (from 51 experiments)| 0.006836 | 0.001116 |
    | J. F. d'Aubuisson de Voisins     | 0.00673  | 0.001211 |
    | J. A. Eytelwein                  | 0.005493 | 0.00143  |
    +----------------------------------+----------+----------+

  Weisbach proposed the formula

    4[zeta] = [alpha] + [beta]/[root]v = 0.003598 + 0.004289/[root]v.   (8)

  § 76. _Darcy's Experiments on Friction in Pipes._--All previous
  experiments on the resistance of pipes were superseded by the
  remarkable researches carried out by H. P. G. Darcy (1803-1858), the
  Inspector-General of the Paris water works. His experiments were
  carried out on a scale, under a variation of conditions, and with a
  degree of accuracy which leaves little to be desired, and the results
  obtained are of very great practical importance. These results may be
  stated thus:--

  1. For new and clean pipes the friction varies considerably with the
  nature and polish of the surface of the pipe. For clean cast iron it
  is about 1½ times as great as for cast iron covered with pitch.

  2. The nature of the surface has less influence when the pipes are old
  and incrusted with deposits, due to the action of the water. Thus old
  and incrusted pipes give twice as great a frictional resistance as new
  and clean pipes. Darcy's coefficients were chiefly determined from
  experiments on new pipes. He doubles these coefficients for old and
  incrusted pipes, in accordance with the results of a very limited
  number of experiments on pipes containing incrustations and deposits.

  3. The coefficient of friction may be expressed in the form [zeta] =
  [alpha] + [beta]/v; but in pipes which have been some time in use it
  is sufficiently accurate to take [zeta] = [alpha]1 simply, where
  [alpha]1 depends on the diameter of the pipe alone, but [alpha] and
  [beta] on the other hand depend both on the diameter of the pipe and
  the nature of its surface. The following are the values of the
  constants.

  For pipes which have been some time in use, neglecting the term
  depending on the velocity;

    [zeta] = [alpha](1 + [beta]/d).   (9)

    +-------------------------------------------------+---------+------+
    |                                                 | [alpha] |[beta]|
    +-------------------------------------------------+---------+------+
    | For drawn wrought-iron or smooth cast-iron pipes| .004973 | .084 |
    | For pipes altered by light incrustations        | .00996  | .084 |
    +-------------------------------------------------+---------+------+

  These coefficients may be put in the following very simple form,
  without sensibly altering their value:--

    For clean pipes              [zeta] = .005(1 + (1/12)d)   (9a)
    For slightly incrusted pipes [zeta] = .01(1 + (1/12)d)

    _Darcy's Value of the Coefficient of Friction [zeta] for Velocities
    not less than 4 in. per second._

    +----------+------------------++----------+------------------+
    | Diameter |     [zeta]       || Diameter |     [zeta]       |
    | of Pipe  +--------+---------|| of Pipe  +------------------+
    |in Inches.|  New   |Incrusted||in Inches.|  New   |Incrusted|
    |          | Pipes. |  Pipes. ||          | Pipes. |  Pipes. |
    +----------+--------+---------++----------+--------+---------+
    |     2    |0.00750 |0.01500  ||    18    | .00528 | .01056  |
    |     3    | .00667 | .01333  ||    21    | .00524 | .01048  |
    |     4    | .00625 | .01250  ||    24    | .00521 | .01042  |
    |     5    | .00600 | .01200  ||    27    | .00519 | .01037  |
    |     6    | .00583 | .01167  ||    30    | .00517 | .01033  |
    |     7    | .00571 | .01143  ||    36    | .00514 | .01028  |
    |     8    | .00563 | .01125  ||    42    | .00512 | .01024  |
    |     9    | .00556 | .01111  ||    48    | .00510 | .01021  |
    |    12    | .00542 | .01083  ||    54    | .00509 | .01019  |
    |    15    | .00533 | .01067  ||          |        |         |
    +----------+--------+---------++----------+--------+---------+

  These values of [zeta] are, however, not exact for widely differing
  velocities. To embrace all cases Darcy proposed the expression

    [zeta] = ([alpha] + [alpha]1/d) + ([beta] + [beta]1/d²)/v;   (10)

  which is a modification of Coulomb's, including terms expressing the
  influence of the diameter and of the velocity. For clean pipes Darcy
  found these values

    [alpha]  = .004346
    [alpha]1 = .0003992
    [beta]   = .0010182
    [beta]1  = .000005205.

  It has become not uncommon to calculate the discharge of pipes by the
  formula of E. Ganguillet and W. R. Kutter, which will be discussed
  under the head of channels. For the value of c in the relation v = c
  [root](mi), Ganguillet and Kutter take

            41.6 + 1.811/n + .00281/i
    c = ----------------------------------
        1 + [(41.6 + .00281/i)(n/[root]m)]

  where n is a coefficient depending only on the roughness of the pipe.
  For pipes uncoated as ordinarily laid n = 0.013. The formula is very
  cumbrous, its form is not rationally justifiable and it is not at all
  clear that it gives more accurate values of the discharge than simpler
  formulae.

  § 77. _Later Investigations on Flow in Pipes._--The foregoing
  statement gives the theory of flow in pipes so far as it can be put in
  a simple rational form. But the conditions of flow are really more
  complicated than can be expressed in any rational form. Taking even
  selected experiments the values of the empirical coefficient [zeta]
  range from 0.16 to 0.0028 in different cases. Hence means of
  discriminating the probable value of [zeta] are necessary in using the
  equations for practical purposes. To a certain extent the knowledge
  that [zeta] decreases with the size of the pipe and increases very
  much with the roughness of its surface is a guide, and Darcy's method
  of dealing with these causes of variation is very helpful. But a
  further difficulty arises from the discordance of the results of
  different experiments. For instance F. P. Stearns and J. M. Gale both
  experimented on clean asphalted cast-iron pipes, 4 ft. in diameter.
  According to one set of gaugings [zeta] = .0051, and according to the
  other [zeta] = .0031. It is impossible in such cases not to suspect
  some error in the observations or some difference in the condition of
  the pipes not noticed by the observers.

  It is not likely that any formula can be found which will give exactly
  the discharge of any given pipe. For one of the chief factors in any
  such formula must express the exact roughness of the pipe surface, and
  there is no scientific measure of roughness. The most that can be done
  is to limit the choice of the coefficient for a pipe within certain
  comparatively narrow limits. The experiments on fluid friction show
  that the power of the velocity to which the resistance is proportional
  is not exactly the square. Also in determining the form of his
  equation for [zeta] Darcy used only eight out of his seventeen series
  of experiments, and there is reason to think that some of these were
  exceptional. Barré de Saint-Venant was the first to propose a formula
  with two constants,

    dh/4l = mV^n,

  where m and n are experimental constants. If this is written in the
  form

    log m + n log v = log (dh/4l),

  we have, as Saint-Venant pointed out, the equation to a straight line,
  of which m is the ordinate at the origin and n the ratio of the slope.
  If a series of experimental values are plotted logarithmically the
  determination of the constants is reduced to finding the straight line
  which most nearly passes through the plotted points. Saint-Venant
  found for n the value of 1.71. In a memoir on the influence of
  temperature on the movement of water in pipes (Berlin, 1854) by G. H.
  L. Hagen (1797-1884) another modification of the Saint-Venant formula
  was given. This is h/l = mv^n/d^x, which involves three experimental
  coefficients. Hagen found n = 1.75; x = 1.25; and m was then nearly
  independent of variations of v and d. But the range of cases examined
  was small. In a remarkable paper in the _Trans. Roy. Soc._, 1883,
  Professor Osborne Reynolds made much clearer the change from regular
  stream line motion at low velocities to the eddying motion, which
  occurs in almost all the cases with which the engineer has to deal.
  Partly by reasoning, partly by induction from the form of
  logarithmically plotted curves of experimental results, he arrived at
  the general equation h/l = c(v^n/d^(3 - n))P^(2 - n), where n = l for
  low velocities and n = 1.7 to 2 for ordinary velocities. P is a
  function of the temperature. Neglecting variations of temperature
  Reynold's formula is identical with Hagen's if x = 3 - n. For
  practical purposes Hagen's form is the more convenient.

    _Values of Index of Velocity._

    +--------------------+---------------+----------+---------------+
    |                    |               | Diameter |               |
    |  Surface of Pipe.  |   Authority.  | of Pipe  |  Values of n. |
    |                    |               |in Metres.|               |
    +--------------------+---------------+----------+---------------+
    | Tin plate          | Bossut        |  /.036   | 1.697 \  1.72 |
    |                    |               |  \.054   | 1.730 /       |
    |                    |               |          |               |
    | Wrought iron (gas  | Hamilton Smith|  /.0159  | 1.756 \  1.75 |
    |   pipe)            |               |  \.0267  | 1.770 /       |
    |                    |               |          |               |
    |                    |               |  /.014   | 1.866 \       |
    | Lead               | Darcy         | < .027   | 1.755  > 1.77 |
    |                    |               |  \.041   | 1.760 /       |
    |                    |               |          |               |
    | Clean brass        | Mair          |   .036   | 1.795    1.795|
    |                    |               |          |               |
    |                  / | Hamilton Smith| / .0266  | 1.760 \       |
    | Asphalted       <  | Lampe.        |<  .4185  | 1.850  > 1.85 |
    |                  | | W. W. Bonn    | | .306   | 1.582 |       |
    |                  \ | Stearns       | \1.219   | 1.880 /       |
    |                    |               |          |               |
    | Riveted wrought \  |               | /.2776   | 1.804 \       |
    |   iron           > | Hamilton Smith|< .3219   | 1.892  > 1.87 |
    |                 /  |               | \.3749   | 1.852 /       |
    |                    |               |          |               |
    | Wrought iron (gas\ |               | /.0122   | 1.900 \       |
    |   pipe)           >| Darcy         |< .0266   | 1.899  > 1.87 |
    |                  / |               | \.0395   | 1.838 /       |
    |                    |               |          |               |
    |                    |               | /.0819   | 1.950 \       |
    | New cast iron      | Darcy         |< .137    | 1.923  > 1.95 |
    |                    |               | |.188    | 1.957 |       |
    |                    |               | \.50     | 1.950 /       |
    |                    |               |          |               |
    |                    |               | /.0364   | 1.835 \       |
    |                    |               | |.0801   | 2.000  > 2.00 |
    | Cleaned cast iron  | Darcy         |< .2447   | 2.000 |       |
    |                    |               | \.397    | 2.07  /       |
    |                    |               |          |               |
    |                    |               | /.0359   | 1.980 \       |
    | Incrusted cast iron| Darcy         |< .0795   | 1.990  > 2.00 |
    |                    |               | \.2432   | 1.990 /       |
    +--------------------+---------------+----------+---------------+

  [Illustration: FIG. 83.]

  In 1886, Professor W. C. Unwin plotted logarithmically all the most
  trustworthy experiments on flow in pipes then available.[5] Fig. 83
  gives one such plotting. The results of measuring the slopes of the
  lines drawn through the plotted points are given in the table.

  It will be seen that the values of the index n range from 1.72 for the
  smoothest and cleanest surface, to 2.00 for the roughest. The numbers
  after the brackets are rounded off numbers.

  The value of n having been thus determined, values of m/d^x were next
  found and averaged for each pipe. These were again plotted
  logarithmically in order to find a value for x. The lines were not
  very regular, but in all cases the slope was greater than 1 to 1, so
  that the value of x must be greater than unity. The following table
  gives the results and a comparison of the value of x and Reynolds's
  value 3 - n.

    +-----------------------+--------+--------+-------+
    |      Kind of Pipe.    |    n    | 3 - n |   x   |
    +-----------------------+--------+--------+-------+
    | Tin plate             |  1.72  |  1.28  | 1.100 |
    | Wrought iron (Smith)  |  1.75  |  1.25  | 1.210 |
    | Asphalted pipes       |  1.85  |  1.15  | 1.127 |
    | Wrought iron (Darcy)  |  1.87  |  1.13  | 1.680 |
    | Riveted wrought iron  |  1.87  |  1.13  | 1.390 |
    | New cast iron         |  1.95  |  1.05  | 1.168 |
    | Cleaned cast iron     |  2.00  |  1.00  | 1.168 |
    | Incrusted cast iron   |  2.00  |  1.00  | 1.160 |
    +-----------------------+--------+--------+-------+

  With the exception of the anomalous values for Darcy's wrought-iron
  pipes, there is no great discrepancy between the values of x and 3 -
  n, but there is no appearance of relation in the two quantities. For
  the present it appears preferable to assume that x is independent of
  n.

  It is now possible to obtain values of the third constant m, using the
  values found for n and x. The following table gives the results, the
  values of m being for metric measures.

  Here, considering the great range of diameters and velocities in the
  experiments, the constancy of m is very satisfactorily close. The
  asphalted pipes give rather variable values. But, as some of these
  were new and some old, the variation is, perhaps, not surprising. The
  incrusted pipes give a value of m quite double that for new pipes but
  that is perfectly consistent with what is known of fluid friction in
  other cases.

    +---------------+----------+-----------+----------+----------------+
    |               | Diameter | Value of  |   Mean   |                |
    | Kind of Pipe. |    in    |     m.    |  Value   |   Authority.   |
    |               |  Metres. |           |   of m.  |                |
    +---------------+----------+-----------+----------+----------------+
    | Tin plate     |  / 0.036 | .01697 \  | .01686   | Bossut         |
    |               |  \ 0.054 | .01676 /  |          |                |
    |               |          |           |          |                |
    | Wrought iron  |  / 0.016 | .01302 \  | .01310   | Hamilton Smith |
    |               |  \ 0.027 | .01319 /  |          |                |
    |               |          |           |          |                |
    |               |  / 0.027 | .01749 \  |        / | Hamilton Smith |
    |               |  | 0.306 | .02058 |  |        | | W. W. Bonn     |
    | Asphalted     | <  0.306 | .02107  > | .01831<  | W. W. Bonn     |
    |   pipes       |  | 0.419 | .01650 |  |        | | Lampe          |
    |               |  | 1.219 | .01317 |  |        | | Stearns        |
    |               |  \ 1.219 | .02107 /  |        \ | Gale           |
    |               |          |           |          |                |
    |               |  / 0.278 | .01370 \  |          |                |
    |               |  | 0.322 | .01440 |  |          |                |
    | Riveted       | <  0.375 | .01390  > | .01403   | Hamilton Smith |
    |   wrought iron|  | 0.432 | .01368 |  |          |                |
    |               |  \ 0.657 | .01448 /  |          |                |
    |               |          |           |          |                |
    |               |  / 0.082 | .01725 \  |          |                |
    | New cast iron | <  0.137 | .01427  > | .01658   | Darcy          |
    |               |  | 0.188 | .01734 |  |          |                |
    |               |  \ 0.500 | .01745 /  |          |                |
    |               |          |           |          |                |
    | Cleaned cast  |  / 0.080 | .01979 \  |          |                |
    |   iron        | <  0.245 | .02091  > | .01994   | Darcy          |
    |               |  \ 0.297 | .01913 /  |          |                |
    |               |          |           |          |                |
    | Incrusted cast|  / 0.036 | .03693 \  |          |                |
    |   iron        | <  0.080 | .03530  > | .03643   | Darcy          |
    |               |  \ 0.243 | .03706 /  |          |                |
    +---------------+----------+-----------+----------+----------------+


  _General Mean Values of Constants._

  The general formula (Hagen's)--h/l = mv^n/d^x.2g--can therefore be
  taken to fit the results with convenient closeness, if the following
  mean values of the coefficients are taken, the unit being a metre:--

    +----------------------+-------+-------+------+
    |     Kind of Pipe.    |   m   |   x   |   n  |
    +----------------------+-------+-------+------+
    | Tin plate            | .0169 | 1.10  | 1.72 |
    | Wrought iron         | .0131 | 1.21  | 1.75 |
    | Asphalted iron       | .0183 | 1.127 | 1.85 |
    | Riveted wrought iron | .0140 | 1.390 | 1.87 |
    | New cast iron        | .0166 | 1.168 | 1.95 |
    | Cleaned cast iron    | .0199 | 1.168 | 2.0  |
    | Incrusted cast iron  | .0364 | 1.160 | 2.0  |
    +----------------------+-------+-------+------+

  The variation of each of these coefficients is within a comparatively
  narrow range, and the selection of the proper coefficient for any
  given case presents no difficulty, if the character of the surface of
  the pipe is known.

  It only remains to give the values of these coefficients when the
  quantities are expressed in English feet. For English measures the
  following are the values of the coefficients:--

    +----------------------+-------+-------+------+
    |     Kind of Pipe.    |   m   |   x   |   n  |
    +----------------------+-------+-------+------+
    | Tin plate            | .0265 | 1.10  | 1.72 |
    | Wrought iron         | .0226 | 1.21  | 1.75 |
    | Asphalted iron       | .0254 | 1.127 | 1.85 |
    | Riveted wrought iron | .0260 | 1.390 | 1.87 |
    | New cast iron        | .0215 | 1.168 | 1.95 |
    | Cleaned cast iron    | .0243 | 1.168 | 2.0  |
    | Incrusted cast iron  | .0440 | 1.160 | 2.0  |
    +----------------------+-------+-------+------+

  § 78. _Distribution of Velocity in the Cross Section of a
  Pipe._--Darcy made experiments with a Pitot tube in 1850 on the
  velocity at different points in the cross section of a pipe. He
  deduced the relation

    V - v = 11.3(r^(3/2)/R) [root]i,

  where V is the velocity at the centre and v the velocity at radius r
  in a pipe of radius R with a hydraulic gradient i. Later Bazin
  repeated the experiments and extended them (_Mém. de l'Académie des
  Sciences_, xxxii. No. 6). The most important result was the ratio of
  mean to central velocity. Let b = Ri/U², where U is the mean velocity
  in the pipe; then V/U = 1 + 9.03 [root]b. A very useful result for
  practical purposes is that at 0.74 of the radius of the pipe the
  velocity is equal to the mean velocity. Fig. 84 gives the velocities
  at different radii as determined by Bazin.

  [Illustration: FIG. 84.]

  § 79. _Influence of Temperature on the Flow through Pipes._--Very
  careful experiments on the flow through a pipe 0.1236 ft. in diameter
  and 25 ft. long, with water at different temperatures, have been made
  by J. G. Mair (_Proc. Inst. Civ. Eng._ lxxxiv.). The loss of head was
  measured from a point 1 ft. from the inlet, so that the loss at entry
  was eliminated. The 1½ in. pipe was made smooth inside and to gauge,
  by drawing a mandril through it. Plotting the results logarithmically,
  it was found that the resistance for all temperatures varied very
  exactly as v^(1.795), the index being less than 2 as in other
  experiments with very smooth surfaces. Taking the ordinary equation of
  flow h = [zeta](4L/D)(v²/2g), then for heads varying from 1 ft. to
  nearly 4 ft., and velocities in the pipe varying from 4 ft. to 9 ft.
  per second, the values of [zeta] were as follows:--

    Temp. F.    [zeta]      |  Temp. F.     [zeta]
       57   .0044 to .0052  |    100    .0039 to .0042
       70   .0042 to .0045  |    110    .0037 to .0041
       80   .0041 to .0045  |    120    .0037 to .0041
       90   .0040 to .0045  |    130    .0035 to .0039
                            |    160    .0035 to .0038

  This shows a marked decrease of resistance as the temperature rises.
  If Professor Osborne Reynolds's equation is assumed h = mLV^n/d^(3 -
  n), and n is taken 1.795, then values of m at each temperature are
  practically constant--

    Temp. F.      m.    |  Temp. F.     m.
       57     0.000276  |    100     0.000244
       70     0.000263  |    110     0.000235
       80     0.000257  |    120     0.000229
       90     0.000250  |    130     0.000225
                        |    160     0.000206

  where again a regular decrease of the coefficient occurs as the
  temperature rises. In experiments on the friction of disks at
  different temperatures Professor W. C. Unwin found that the resistance
  was proportional to constant × (1 - 0.0021t) and the values of m given
  above are expressed almost exactly by the relation

    m = 0.000311(1 - 0.00215 t).

  In tank experiments on ship models for small ordinary variations of
  temperature, it is usual to allow a decrease of 3% of resistance for
  10° F. increase of temperature.

  § 80. _Influence of Deposits in Pipes on the Discharge. Scraping Water
  Mains._--The influence of the condition of the surface of a pipe on
  the friction is shown by various facts known to the engineers of
  waterworks. In pipes which convey certain kinds of water, oxidation
  proceeds rapidly and the discharge is considerably diminished. A main
  laid at Torquay in 1858, 14 m. in length, consists of 10-in., 9-in.
  and 8-in. pipes. It was not protected from corrosion by any coating.
  But it was found to the surprise of the engineer that in eight years
  the discharge had diminished to 51% of the original discharge. J. G.
  Appold suggested an apparatus for scraping the interior of the pipe,
  and this was constructed and used under the direction of William
  Froude (see "Incrustation of Iron Pipes," by W. Ingham, _Proc. Inst.
  Mech. Eng._, 1899). It was found that by scraping the interior of the
  pipe the discharge was increased 56%. The scraping requires to be
  repeated at intervals. After each scraping the discharge diminishes
  rather rapidly to 10% and afterwards more slowly, the diminution in a
  year being about 25%.

  Fig. 85 shows a scraper for water mains, similar to Appold's but
  modified in details, as constructed by the Glenfield Company, at
  Kilmarnock. A is a longitudinal section of the pipe, showing the
  scraper in place; B is an end view of the plungers, and C, D sections
  of the boxes placed at intervals on the main for introducing or
  withdrawing the scraper. The apparatus consists of two plungers,
  packed with leather so as to fit the main pretty closely. On the
  spindle of these plungers are fixed eight steel scraping blades, with
  curved scraping edges fitting the surface of the main. The apparatus
  is placed in the main by removing the cover from one of the boxes
  shown at C, D. The cover is then replaced, water pressure is admitted
  behind the plungers, and the apparatus driven through the main. At
  Lancaster after twice scraping the discharge was increased 56½%, at
  Oswestry 54½%. The increased discharge is due to the diminution of the
  friction of the pipe by removing the roughnesses due to oxidation. The
  scraper can be easily followed when the mains are about 3 ft. deep by
  the noise it makes. The average speed of the scraper at Torquay is
  2(1/3) m. per hour. At Torquay 49% of the deposit is iron rust, the
  rest being silica, lime and organic matter.

  [Illustration: FIG. 85. Scale 1/25.]

  In the opinion of some engineers it is inadvisable to use the scraper.
  The incrustation is only temporarily removed, and if the use of the
  scraper is continued the life of the pipe is reduced. The only
  treatment effective in preventing or retarding the incrustation due to
  corrosion is to coat the pipes when hot with a smooth and perfect
  layer of pitch. With certain waters such as those derived from the
  chalk the incrustation is of a different character, consisting of
  nearly pure calcium carbonate. A deposit of another character which
  has led to trouble in some mains is a black slime containing a good
  deal of iron not derived from the pipes. It appears to be an organic
  growth. Filtration of the water appears to prevent the growth of the
  slime, and its temporary removal may be effected by a kind of brush
  scraper devised by G. F. Deacon (see "Deposits in Pipes," by Professor
  J. C. Campbell Brown, _Proc. Inst. Civ. Eng._, 1903-1904).

  § 81. _Flow of Water through Fire Hose._--The hose pipes used for fire
  purposes are of very varied character, and the roughness of the
  surface varies. Very careful experiments have been made by J. R.
  Freeman (_Am. Soc. Civ. Eng._ xxi., 1889). It was noted that under
  pressure the diameter of the hose increased sufficiently to have a
  marked influence on the discharge. In reducing the results the true
  diameter has been taken. Let v = mean velocity in ft. per sec.; r =
  hydraulic mean radius or one-fourth the diameter in feet; i =
  hydraulic gradient. Then v = n[root](ri).

    +---------------+---------+---------+-------+-------+-------+
    |               | Diameter| Gallons |       |       |       |
    |               |    in   | (United |       |       |       |
    |               | Inches. | States) |   i   |   v   |   n   |
    |               |         | per min.|       |       |       |
    +---------------+---------+---------+-------+-------+-------+
    | Solid rubber  |  2.65   |   215   | .1863 | 12.50 | 123.3 |
    |  hose         |    "    |   344   | .4714 | 20.00 | 124.0 |
    |               |         |         |       |       |       |
    | Woven cotton, |  2.47   |   200   | .2464 | 13.40 | 119.1 |
    |  rubber lined |    "    |   299   | .5269 | 20.00 | 121.5 |
    |               |         |         |       |       |       |
    | Woven cotton, |  2.49   |   200   | .2427 | 13.20 | 117.7 |
    |  rubber lined |    "    |   319   | .5708 | 21.00 | 122.1 |
    |               |         |         |       |       |       |
    | Knit cotton,  |  2.68   |   132   | .0809 |  7.50 | 111.6 |
    |  rubber lined |    "    |   299   | .3931 | 17.00 | 114.8 |
    |               |         |         |       |       |       |
    | Knit cotton,  |  2.69   |   204   | .2357 | 11.50 | 100.1 |
    |  rubber lined |    "    |   319   | .5165 | 18.00 | 105.8 |
    |               |         |         |       |       |       |
    | Woven cotton, |  2.12   |   154   | .3448 | 14.00 | 113.4 |
    |  rubber lined |    "    |   240   | .7673 | 21.81 | 118.4 |
    |               |         |         |       |       |       |
    | Woven cotton, |  2.53   |    54.8 | .0261 |  3.50 |  94.3 |
    |  rubber lined |    "    |   298   | .8264 | 19.00 |  91.0 |
    |               |         |         |       |       |       |
    | Unlined linen |  2.60   |    57.9 | .0414 |  3.50 |  73.9 |
    |  hose         |    "    |   331   |1.1624 | 20.00 |  79.6 |
    +---------------+---------+---------+-------+-------+-------+

  § 82. _Reduction of a Long Pipe of Varying Diameter to an Equivalent
  Pipe of Uniform Diameter. Dupuit's Equation._--Water mains for the
  supply of towns often consist of a series of lengths, the diameter
  being the same for each length, but differing from length to length.
  In approximate calculations of the head lost in such mains, it is
  generally accurate enough to neglect the smaller losses of head and to
  have regard to the pipe friction only, and then the calculations may
  be facilitated by reducing the main to a main of uniform diameter, in
  which there would be the same loss of head. Such a uniform main will
  be termed an equivalent main.

  [Illustration: FIG. 86.]

  In fig. 86 let A be the main of variable diameter, and B the
  equivalent uniform main. In the given main of variable diameter A, let

    l1, l2... be the lengths,
    d1, d2...    the diameters,
    v1, v2...    the velocities,
    i1, i2...    the slopes,

  for the successive portions, and let l, d, v and i be corresponding
  quantities for the equivalent uniform main B. The total loss of head
  in A due to friction is

    h = i1l1 + i2l2 + ...
      = [zeta](v1²·4l1/2gd1) + [zeta](v2²·4l2/2gd2) + ...

  and in the uniform main

    il = [zeta](v²·4l/2gd).

  If the mains are equivalent, as defined above,

    [zeta](v²·4l/2gd) = [zeta](v1²·4l1/2gd1) + [zeta](v2²·4l2/2gd2) + ...

  But, since the discharge is the same for all portions,

    ¼[pi]d²v = ¼[pi]d1²v1 = ¼[pi]d2²v2 = ...

    v1 = vd²/d1²; v2 = vd²/d2² ...

  Also suppose that [zeta] may be treated as constant for all the pipes.
  Then

    l/d = (d^4/d1^4)(l1/d1) + (d^4/d2^4(12/d2) + ...

    l = (d^5/d1^5)l1 + (d^5/d2^5)l2 + ...

  which gives the length of the equivalent uniform main which would have
  the same total loss of head for any given discharge.

  § 83. _Other Losses of Head in Pipes._--Most of the losses of head in
  pipes, other than that due to surface friction against the pipe, are
  due to abrupt changes in the velocity of the stream producing eddies.
  The kinetic energy of these is deducted from the general energy of
  translation, and practically wasted.

  [Illustration: FIG. 87.]

  _Sudden Enlargement of Section._--Suppose a pipe enlarges in section
  from an area [omega]0 to an area [omega]1 (fig. 87); then

    v1/v0 = [omega]0/[omega]1;

  or, if the section is circular,

    v1/v0 = (d0/d1)².

  The head lost at the abrupt change of velocity has already been shown
  to be the head due to the relative velocity of the two parts of the
  stream. Hence head lost

    [h]_e = (v0 - v1)²/2g = ([omega]1/[omega]0 - 1)²v1²/2g
          = {(d1/d0)² - 1}² v1²/2g

  or

    [h]_e = [zeta]_ev1²/2g,   (1)

  if [zeta]_e is put for the expression in brackets.

    +--------------+----+----+----+----+----+----+----+----+----+----+----+-----+-----+-----+-----+
    | [omega]1/    |1.1 |1.2 |1.5 |1.7 |1.8 |1.9 |2.0 |2.5 |3.0 |3.5 |4.0 | 5.0 | 6.0 | 7.0 | 8.0 |
    |   [omega]0 = |    |    |    |    |    |    |    |    |    |    |    |     |     |     |     |
    | d1/d0 =      |1.05|1.10|1.22|1.30|1.34|1.38|1.41|1.58|1.73|1.87|2.00| 2.24| 2.45| 2.65| 2.83|
    |              |    |    |    |    |    |    |    |    |    |    |    |     |     |     |     |
    | [zeta]_e =   | .01| .04| .25| .49| .64| .81|1.00|2.25|4.00|6.25|9.00|16.00|25.00|36.0 |49.0 |
    +--------------+----+----+----+----+----+----+----+----+----+----+----+-----+-----+-----+-----+

  [Illustration: FIG. 88.]

  [Illustration: FIG. 89.]

  _Abrupt Contraction of Section._--When water passes from a larger to a
  smaller section, as in figs. 88, 89, a contraction is formed, and the
  contracted stream abruptly expands to fill the section of the pipe.
  Let [omega] be the section and v the velocity of the stream at bb. At
  aa the section will be c_c[omega], and the velocity
  ([omega]/c_c[omega])v = v/c1, where c_c is the coefficient of
  contraction. Then the head lost is

    [h]_m = (v/c_c - v)²/2g = (1/c_c - 1)²v²/2g;

  and, if c_c is taken 0.64,

    [h]_m = 0.316 v²/2g.   (2)

  The value of the coefficient of contraction for this case is, however,
  not well ascertained, and the result is somewhat modified by friction.
  For water entering a cylindrical, not bell-mouthed, pipe from a
  reservoir of indefinitely large size, experiment gives

    [h]_a = 0.505 v²/2g.   (3)

  If there is a diaphragm at the mouth of the pipe as in fig. 89, let
  [omega]1 be the area of this orifice. Then the area of the contracted
  stream is c_c[omega]1, and the head lost is

    [h]_c = {([omega]/c_c[omega]1) - 1}²v²/2g
          = [zeta]_cv²/2g   (4)

  if [zeta], is put for {([omega]/c_c[omega]1) - 1}². Weisbach has found
  experimentally the following values of the coefficient, when the
  stream approaching the orifice was considerably larger than the
  orifice:--

    +--------------------+-------+------+------+-----+-----+-----+-----+-----+-----+-----+
    | [omega]1/[omega] = |  0.1  |  0.2 |  0.3 | 0.4 | 0.5 | 0.6 |0.7  | 0.8 | 0.9 | 1.0 |
    |                    |       |      |      |     |     |     |     |     |     |     |
    | c_c =              | .616  | .614 | .612 |.610 |.617 |.605 |.603 |.601 |.598 |.596 |
    |                    |       |      |      |     |     |     |     |     |     |     |
    | [zeta]_c =         | 231.7 |50.99 |19.78 |9.612|5.256|3.077|1.876|1.169|0.734|0.480|
    +--------------------+-------+------+------+-----+-----+-----+-----+-----+-----+-----+

  [Illustration: FIG. 90.]

  When a diaphragm was placed in a tube of uniform section (fig. 90) the
  following values were obtained, [omega]1 being the area of the orifice
  and [omega] that of the pipe:--

    +--------------------+-------+------+------+-----+-----+-----+-----+-----+-----+-----+
    | [omega]1/[omega] = |  0.1  | 0.2  | 0.3  | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |
    |                    |       |      |      |     |     |     |     |     |     |     |
    | c_c =              | .624  | .632 | .643 |.659 |.681 |.712 |.755 |.813 |.892 |1.00 |
    |                    |       |      |      |     |     |     |     |     |     |     |
    | [xi]_c =           | 225.9 |47.77 |30.83 |7.801|1.753|1.796|.797 |.290 |.060 |.000 |
    +--------------------+-------+------+------+-----+-----+-----+-----+-----+-----+-----+

  Elbows.--Weisbach considers the loss of head at elbows (fig. 91) to be
  due to a contraction formed by the stream. From experiments with a
  pipe 1¼ in. diameter, he found the loss of head

    [h]_e = [zeta]_e v²/2g;   (5)

    [zeta]_e = 0.9457 sin² ½[phi] + 2.047 sin^4 ½[phi].

    +------------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
    | [phi] =    | 20° | 40° | 60° | 80° | 90° | 100°| 110°| 120°| 130°| 140°|
    |            |     |     |     |     |     |     |     |     |     |     |
    | [zeta]_e = |0.046|0.139|0.364|0.740|0.984|1.260|1.556|1.861|2.158|2.431|
    +------------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+

  Hence at a right-angled elbow the whole head due to the velocity very
  nearly is lost.

  [Illustration: FIG. 91.]

  _Bends._--Weisbach traces the loss of head at curved bends to a
  similar cause to that at elbows, but the coefficients for bends are
  not very satisfactorily ascertained. Weisbach obtained for the loss of
  head at a bend in a pipe of circular section

    [h]_b = [zeta]_b v²/2g;   (6)

    [zeta]_b = 0.131 + 1.847(d/2[rho])^(7/2),

  where d is the diameter of the pipe and [rho] the radius of curvature
  of the bend. The resistance at bends is small and at present very ill
  determined.

  [Illustration: FIG. 92.]

  _Valves, Cocks and Sluices._--These produce a contraction of the
  water-stream, similar to that for an abrupt diminution of section
  already discussed. The loss of head may be taken as before to be

    [h]_v = [zeta]_v v²/2g;   (7)

  where v is the velocity in the pipe beyond the valve and [zeta]_v a
  coefficient determined by experiment. The following are Weisbach's
  results.

  _Sluice in Pipe of Rectangular Section_ (fig. 92). Section at sluice =
  [omega]1 in pipe = [omega].

    +--------------------+----+----+----+----+----+----+----+----+----+-----+
    | [omega]1/[omega] = |1.0 |0.9 |0.8 |0.7 |0.6 |0.5 |0.4 | 0.3| 0.2| 0.1 |
    |                    |    |    |    |    |    |    |    |    |    |     |
    | [zeta]_v =         |0.00|.09 |.39 |.95 |2.08|4.02|8.12|17.8|44.5| 193 |
    +--------------------+----+----+----+----+----+----+----+----+----+-----+

    _Sluice in Cylindrical Pipe_ (fig. 93).

    +-----------------------+----+-----+-----+-----+-----+-----+------+------+
    | Ratio of height of    |    |     |     |     |     |     |      |      |
    |   opening to diameter | 1.0| 7/8 | 3/4 | 5/8 |  ½  | 3/8 |  ¼   | 1/5  |
    |   of pipe             |    |     |     |     |     |     |      |      |
    |  [omega]1/[omega] =   |1.00|0.948|.856 |.740 |.609 |.466 | .315 | .159 |
    |                       |    |     |     |     |     |     |      |      |
    |       [zeta]_v =      |0.00|0.07 |0.26 |0.81 |2.06 |5.52 | 17.0 | 97.8 |
    +-----------------------+----+-----+-----+-----+-----+-----+------+------+

  [Illustration: FIG. 93.]

  [Illustration: FIG. 94.]

    _Cock in a Cylindrical Pipe_ (fig. 94). Angle through which cock is
    turned = [theta].

    +------------+-----+-----+-----+-----+-----+-----+-----+
    | [theta] =  |  5° | 10° | 15° | 20° | 25° | 30° | 35° |
    | Ratio of   |     |     |     |     |     |     |     |
    |   cross    |.926 |.850 |.772 |.692 |.613 |.535 |.458 |
    |   sections |     |     |     |     |     |     |     |
    | [zeta]_v = | .05 | .29 | .75 |1.56 |3.10 |5.47 |9.68 |
    +------------+-----+-----+-----+-----+-----+-----+-----+

    +------------+-----+-----+-----+-----+-----+-----+-----+
    | [theta] =  | 40° | 45° | 50° | 55° | 60° | 65° | 82° |
    | Ratio of   |     |     |     |     |     |     |     |
    |  cross     |.385 |.315 |.250 |.190 |.137 |.091 |  0  |
    |  sections  |     |     |     |     |     |     |     |
    | [zeta]_v = | 17.3| 31.2| 52.6|106  |206  |486  |[oo] |
    +------------+-----+-----+-----+-----+-----+-----+-----+

    _Throttle Valve in a Cylindrical Pip_e (fig. 95)

    +------------+-----+-----+-----+-----+-----+-----+-----+-----+
    | [theta] =  |  5° | 10° | 15° | 20° | 25° | 30° | 35° | 40° |
    |            |     |     |     |     |     |     |     |     |
    | [zeta]_v = | .24 | .52 | .90 | 1.54| 2.51| 3.91| 6.22| 10.8|
    +------------+-----+-----+-----+-----+-----+-----+-----+-----+

    +------------+------+------+------+------+------+------+------+
    | [theta] =  |  45° |  50° |  55° |  60° |  65° |  70° |  90° |
    |            |      |      |      |      |      |      |      |
    | [zeta]_v = | 18.7 | 32.6 | 58.8 |  118 |  256 |  751 | [oo] |
    +------------+------+------+------+------+------+------+------+

  [Illustration: FIG. 95.]

  § 84. _Practical Calculations on the Flow of Water in Pipes._--In the
  following explanations it will be assumed that the pipe is of so great
  a length that only the loss of head in friction against the surface of
  the pipe needs to be considered. In general it is one of the four
  quantities d, i, v or Q which requires to be determined. For since the
  loss of head h is given by the relation h = il, this need not be
  separately considered.

  There are then three equations (see eq. 4, § 72, and 9a, § 76) for the
  solution of such problems as arise:--

    [zeta] = [alpha](1 + 1/12d);   (1)

  where [alpha] = 0.005 for new and = 0.01 for incrusted pipes.

    [zeta]v²/2g = ¼di.   (2)

    Q = ¼[pi]d²v.   (3)

  _Problem 1._ Given the diameter of the pipe and its virtual slope, to
  find the discharge and velocity of flow. Here d and i are given, and Q
  and v are required. Find [zeta] from (1); then v from (2); lastly Q
  from (3). This case presents no difficulty.

  By combining equations (1) and (2), v is obtained directly:--

    v = [root](gdi/2[zeta]) = [root](g/2[alpha]) [root][di/{1 + 1/12d}].   (4)

    For new pipes        [root](g/2[alpha]) = 56.72
    For incrusted pipes  = 40.13

  For pipes not less than 1, or more than 4 ft. in diameter, the mean
  values of [zeta] are

    For new pipes        0.00526
    For incrusted pipes  0.01052.

  Using these values we get the very simple expressions--

    v = 55.31 [root](di) for new pipes
      = 39.11 [root](di) for incrusted pipes.   (4a)

  Within the limits stated, these are accurate enough for practical
  purposes, especially as the precise value of the coefficient [zeta]
  cannot be known for each special case.

  _Problem 2._ Given the diameter of a pipe and the velocity of flow, to
  find the virtual slope and discharge. The discharge is given by (3);
  the proper value of [zeta] by (1); and the virtual slope by (2). This
  also presents no special difficulty.

  _Problem 3._ Given the diameter of the pipe and the discharge, to find
  the virtual slope and velocity. Find v from (3); [zeta] from (1);
  lastly i from (2). If we combine (1) and (2) we get

    i = [zeta](v²/2g) (4/d) = 2a{1 + 1/12d} v²/gd;   (5)

  and, taking the mean values of [zeta] for pipes from 1 to 4 ft.
  diameter, given above, the approximate formulae are

    i = 0.0003268 v²/d for new pipes
      = 0.0006536 v²/d for incrusted pipes.   (5a)

  _Problem 4._ Given the virtual slope and the velocity, to find the
  diameter of the pipe and the discharge. The diameter is obtained from
  equations (2) and (1), which give the quadratic expression

    d² - d(2[alpha]v²/gi) - [alpha]v²/6gi = 0.

    .: d = [alpha]v²/gi + [root]{([alpha]v²/gi) ([alpha]v²/gi + 1/6)}.   (6)

  For practical purposes, the approximate equations

    d = 2[alpha]v²/gi + 1/12   (6a)
      = 0.00031 v²/i + .083 for new pipes
      = 0.00062 v²/i + .083 for incrusted pipes

  are sufficiently accurate.

  _Problem 5._ Given the virtual slope and the discharge, to find the
  diameter of the pipe and velocity of flow. This case, which often
  occurs in designing, is the one which is least easy of direct
  solution. From equations (2) and (3) we get--

    d^5 = 32[zeta]Q²/g[pi]²i.   (7)

  If now the value of [zeta] in (1) is introduced, the equation becomes
  very cumbrous. Various approximate methods of meeting the difficulty
  may be used.

  (a) Taking the mean values of [zeta] given above for pipes of 1 to 4
  ft. diameter we get

    d = [root 5](32[zeta]/g[pi]²) [root 5](Q²/i)   (8)
      = 0.2216 [root 5](Q²/i) for new pipes
      = 0.2541 [root 5](Q²/i) for incrusted pipes;

  equations which are interesting as showing that when the value of
  [zeta] is doubled the diameter of pipe for a given discharge is only
  increased by 13%.

  (b) A second method is to obtain a rough value of d by assuming [zeta]
  = [alpha]. This value is

    d´ = [root 5](32Q²/g[pi]²i) [root 5][alpha]
       = 0.6319 [root 5](Q²/i) [root 5][alpha].

  Then a very approximate value of [zeta] is

    [zeta]´ = [alpha](1 + 1/12d´);

  and a revised value of d, not sensibly differing from the exact value,
  is

    d´´ = [root 5](32Q²/g[pi]²i) [root 5][zeta]´
        = 0.6319 [root 5](Q²/i) [root 5][zeta]´.

  (c) Equation 7 may be put in the form

    d = [root 5](32[alpha]Q²/g[pi]²i) [root 5](1 + 1/12d).   (9)

  Expanding the term in brackets,

    [root 5](1 + 1/12d) = 1 + 1/60d - 1/1800d² ...

  Neglecting the terms after the second,

    d = [root 5](32[alpha]/g[pi]²) [root 5](Q²/i)·{1 + 1/60d}
      = [root 5](32a/g[pi]²) [root 5](Q²/i) + 0.01667;   (9a)

  and

    [root 5](32a/g[pi]²) = 0.219 for new pipes
                         = 0.252 for incrusted pipes.

  [Illustration: FIG. 96.]

  [Illustration: FIG. 97.]

  § 85. _Arrangement of Water Mains for Towns' Supply._--Town mains are
  usually supplied oy gravitation from a service reservoir, which in
  turn is supplied by gravitation from a storage reservoir or by pumping
  from a lower level. The service reservoir should contain three days'
  supply or in important cases much more. Its elevation should be such
  that water is delivered at a pressure of at least about 100 ft. to the
  highest parts of the district. The greatest pressure in the mains is
  usually about 200 ft., the pressure for which ordinary pipes and
  fittings are designed. Hence if the district supplied has great
  variations of level it must be divided into zones of higher and lower
  pressure. Fig. 96 shows a district of two zones each with its service
  reservoir and a range of pressure in the lower district from 100 to
  200 ft. The total supply required is in England about 25 gallons per
  head per day. But in many towns, and especially in America, the supply
  is considerably greater, but also in many cases a good deal of the
  supply is lost by leakage of the mains. The supply through the branch
  mains of a distributing system is calculated from the population
  supplied. But in determining the capacity of the mains the fluctuation
  of the demand must be allowed for. It is usual to take the maximum
  demand at twice the average demand. Hence if the average demand is 25
  gallons per head per day, the mains should be calculated for 50
  gallons per head per day.

  [Illustration: FIG. 98.]

  § 86. _Determination of the Diameters of Different Parts of a Water
  Main._--When the plan of the arrangement of mains is determined upon,
  and the supply to each locality and the pressure required is
  ascertained, it remains to determine the diameters of the pipes. Let
  fig. 97 show an elevation of a main ABCD ..., R being the reservoir
  from which the supply is derived. Let NN be the datum line of the
  levelling operations, and H_a, H_b ... the heights of the main above
  the datum line, H_r being the height of the water surface in the
  reservoir from the same datum. Set up next heights AA1, BB1, ...
  representing the minimum pressure height necessary for the adequate
  supply of each locality. Then A1B1C1D1 ... is a line which should form
  a lower limit to the line of virtual slope. Then if heights [h]_a,
  [h]_b, [h]_c ... are taken representing the actual losses of head in
  each length l_a, l_b, l_c ... of the main, A0B0C0 will be the line of
  virtual slope, and it will be obvious at what points such as D0 and
  E0, the pressure is deficient, and a different choice of diameter of
  main is required. For any point z in the length of the main, we have

    Pressure height = H_r - H_z - ([h]_a + [h]_b + ... [h]_z).

  Where no other circumstance limits the loss of head to be assigned to
  a given length of main, a consideration of the safety of the main from
  fracture by hydraulic shock leads to a limitation of the velocity of
  flow. Generally the velocity in water mains lies between 1½ and 4½ ft.
  per second. Occasionally the velocity in pipes reaches 10 ft. per
  second, and in hydraulic machinery working under enormous pressures
  even 20 ft. per second. Usually the velocity diminishes along the main
  as the discharge diminishes, so as to reduce somewhat the total loss
  of head which is liable to render the pressure insufficient at the end
  of the main.

  J. T. Fanning gives the following velocities as suitable in pipes for
  towns' supply:--

    Diameter in inches          4     8    12    18    24    30    36
    Velocity in feet per sec.  2.5   3.0   3.5   4.5   5.3   6.2   7.0

  § 87. _Branched Pipe connecting Reservoirs at Different Levels._--Let
  A, B, C (fig. 98) be three reservoirs connected by the arrangement of
  pipes shown,--l1, d1, Q1, v1; l2, d2, Q2, v2; h3, d3, Q3, v3 being the
  length, diameter, discharge and velocity in the three portions of the
  main pipe. Suppose the dimensions and positions of the pipes known and
  the discharges required.

  If a pressure column is introduced at X, the water will rise to a
  height XR, measuring the pressure at X, and aR, Rb, Rc will be the
  lines of virtual slope. If the free surface level at R is above b, the
  reservoir A supplies B and C, and if R is below b, A and B supply C.
  Consequently there are three cases:--

      I. R above b; Q1 = Q2 + Q3.
     II. R level with b; Q1 = Q3; Q2 = 0
    III. R below b; Q1 + Q2 = Q3.

  To determine which case has to be dealt with in the given conditions,
  suppose the pipe from X to B closed by a sluice. Then there is a
  simple main, and the height of free surface h´ at X can be determined.
  For this condition

    h_a - h´ = [zeta](v1²/2g)(4l1/d1)
             = 32[zeta]Q´² l1/g[pi]²d1^5;

    h´ - h_c = [zeta](v3²/2g)(4l3/d3)
             = 32[zeta]Q´²l3/g[pi]²d3^5;

  where Q´ is the common discharge of the two portions of the pipe.
  Hence

    (h_a - h´)/(h´ - h_c) = l1d3^5/l3d1^5,

  from which h´ is easily obtained. If then h´ is greater than hb,
  opening the sluice between X and B will allow flow towards B, and the
  case in hand is case I. If h´ is less than h_b, opening the sluice
  will allow flow from B, and the case is case III. If h´ = h_b, the
  case is case II., and is already completely solved.

  The true value of h must lie between h´ and h_b. Choose a new value of
  h, and recalculate Q1, Q2, Q3. Then if

    Q1 > Q2 + Q3 in case I.,

  or

    Q1 + Q2 > Q3 in case III.,

  the value chosen for h is too small, and a new value must be chosen.

  If

    Q1 < Q2 + Q3 in case I.,

  or

    Q1 + Q2 < Q3 in case III.,

  the value of h is too great.

  Since the limits between which h can vary are in practical cases not
  very distant, it is easy to approximate to values sufficiently
  accurate.

  § 88. _Water Hammer._--If in a pipe through which water is flowing a
  sluice is suddenly closed so as to arrest the forward movement of the
  water, there is a rise of pressure which in some cases is serious
  enough to burst the pipe. This action is termed water hammer or water
  ram. The fluctuation of pressure is an oscillating one and gradually
  dies out. Care is usually taken that sluices should only be closed
  gradually and then the effect is inappreciable. Very careful
  experiments on water hammer were made by N. J. Joukowsky at Moscow in
  1898 (_Stoss in Wasserleitungen_, St Petersburg, 1900), and the
  results are generally confirmed by experiments made by E. B. Weston
  and R. C. Carpenter in America. Joukowsky used pipes, 2, 4 and 6 in.
  diameter, from 1000 to 2500 ft. in length. The sluice closed in 0.03
  second, and the fluctuations of pressure were automatically
  registered. The maximum excess pressure due to water-hammer action was
  as follows:--

    +---------------------------------+---------------------------------+
    |       Pipe 4-in. diameter.      |       Pipe 6-in. diameter.      |
    +--------------+------------------+--------------+------------------+
    |   Velocity   | Excess Pressure. |   Velocity   | Excess Pressure. |
    | ft. per sec. |  lb. per sq. in. | ft. per sec. |  lb. per sq. in. |
    +--------------+------------------+--------------+------------------+
    |     0.5      |        31        |     0.6      |        43        |
    |     2.9      |       168        |     3.0      |       173        |
    |     4.1      |       232        |     5.6      |       369        |
    |     9.2      |       519        |     7.5      |       426        |
    +--------------+------------------+--------------+------------------+

  In some cases, in fixing the thickness of water mains, 100 lb. per sq.
  in. excess pressure is allowed to cover the effect of water hammer.
  With the velocities usual in water mains, especially as no valves can
  be quite suddenly closed, this appears to be a reasonable allowance
  (see also Carpenter, _Am. Soc. Mech. Eng._, 1893).


  IX. FLOW OF COMPRESSIBLE FLUIDS IN PIPES

  § 89. _Flow of Air in Long Pipes._--When air flows through a long
  pipe, by far the greater part of the work expended is used in
  overcoming frictional resistances due to the surface of the pipe. The
  work expended in friction generates heat, which for the most part must
  be developed in and given back to the air. Some heat may be
  transmitted through the sides of the pipe to surrounding materials,
  but in experiments hitherto made the amount so conducted away appears
  to be very small, and if no heat is transmitted the air in the tube
  must remain sensibly at the same temperature during expansion. In
  other words, the expansion may be regarded as isothermal expansion,
  the heat generated by friction exactly neutralizing the cooling due to
  the work done. Experiments on the pneumatic tubes used for the
  transmission of messages, by R. S. Culley and R. Sabine (_Proc. Inst.
  Civ. Eng._ xliii.), show that the change of temperature of the air
  flowing along the tube is much less than it would be in adiabatic
  expansion.

  § 90. _Differential Equation of the Steady Motion of Air Flowing in a
  Long Pipe of Uniform Section._--When air expands at a constant
  absolute temperature [tau], the relation between the pressure p in
  pounds per square foot and the density or weight per cubic foot G is
  given by the equation

    p/G = c[tau],   (1)

  where c = 53.15. Taking [tau] = 521, corresponding to a temperature of
  60° Fahr.,

    c[tau] = 27690 foot-pounds.   (2)

  The equation of continuity, which expresses the condition that in
  steady motion the same weight of fluid, W, must pass through each
  cross section of the stream in the unit of time, is

    G[Omega]u = W = constant,   (3)

  where [Omega] is the section of the pipe and u the velocity of the
  air. Combining (1) and (3),

    [Omega]up/W = c[tau] = constant.   (3a)

  [Illustration: FIG. 99.]

  Since the work done by gravity on the air during its flow through a
  pipe due to variations of its level is generally small compared with
  the work done by changes of pressure, the former may in many cases be
  neglected.

  Consider a short length dl of the pipe limited by sections A0, A1 at a
  distance dl (fig. 99). Let p, u be the pressure and velocity at A0, p
  + dp and u + du those at A1. Further, suppose that in a very short
  time dt the mass of air between A0A1 comes to A´0A´1 so that A0A´0 =
  udt and A1A´1 = (u + du)dt1. Let [Omega] be the section, and m the
  hydraulic mean radius of the pipe, and W the weight of air flowing
  through the pipe per second.

  From the steadiness of the motion the weight of air between the
  sections A0A´0, and A1A´1 is the same. That is,

    W dt = G[Omega]u dt = G[Omega](u + du) dt.

  By analogy with liquids the head lost in friction is, for the length
  dl (see § 72, eq. 3), [zeta](u²/2g)(dl/m). Let H = u²/2g. Then the
  head lost is [zeta](H/m)dl; and, since Wdt lb. of air flow through the
  pipe in the time considered, the work expended in friction is
  -[zeta](H/m)Wdl dt. The change of kinetic energy in dt seconds is the
  difference of the kinetic energy of A0A´0 and A1A´1, that is,

    (W/g) dt {(u + du)² - u²}/2 = (W/g)u du dt = W dH dt.

  The work of expansion when [Omega]udt cub. ft. of air at a pressure p
  expand to [Omega](u + du)dt cub. ft. is [Omega]p du dt. But from (3a)
  u = c[tau]W/[Omega]p, and therefore

    du/dp = -c[tau]W/[Omega]p².

  And the work done by expansion is -(c[tau]W/p)dpdt.

  The work done by gravity on the mass between A0 and A1 is zero if the
  pipe is horizontal, and may in other cases be neglected without great
  error. The work of the pressures at the sections A0A1 is

    p[Omega]u dt - (p + dp)[Omega](u + du) dt
      = -(pdu + udp)[Omega] dt

  But from (3a)

    pu = constant,

    p du + u dp = 0,

  and the work of the pressures is zero. Adding together the quantities
  of work, and equating them to the change of kinetic energy,

    WDH dt = -(c[tau]W/p) dp dt - [zeta](H/m)W dl dt

    dH + (c[tau]/p) dp + [zeta](H/m) dl = 0,

    dH/H + (c[tau]/Hp) dp + [zeta]dl/m) = 0   (4)

  But

    u = c[tau]W/[Omega]p,

  and

    H = u²/2g = c²[tau]²W²/2g[Omega]²p²,

    .: dH/H + (2g[Omega]²p/c[tau]W²) dp + [zeta] dl/m = 0.   (4a)

  For tubes of uniform section m is constant; for steady motion W is
  constant; and for isothermal expansion [tau] is constant. Integrating,

    log H + g[Omega]²p²/W²c[tau] + [zeta]l/m = constant;   (5)

  for

    l = 0, let H = H0, and p = p0;

  and for

    l = l, let H = H1, and p = p1.

  log (H1/H0) + (g[Omega]²}/W²c[tau]) (p1² - p0²) + [zeta]l/m = 0. (5a)
  where p0 is the greater pressure and p1 the less, and the flow is from
  A0 towards A1.

  By replacing W and H,

    log (p0/p1) + (gc[tau]/u0²p0²)(p1² - p0² + [zeta]l/m = 0  (6)

  Hence the initial velocity in the pipe is

    u0 = [root][{gc[tau](p0² - p1²)} / {p0²([zeta]l/m + log (p0/p1)}].   (7)

  When l is great, log p0/p1 is comparatively small, and then

    u0 = [root][(gc[tau]m/[zeta]l) {(p0² - p1²)/p0²}],   (7a)

  a very simple and easily used expression. For pipes of circular
  section m = d/4, where d is the diameter:--

    u0 = [root][(gc[tau]d/4[zeta]l) {(p0² - p1²)/p0²}];   (7b)

  or approximately

    u0 = (1.1319 - 0.7264 p1/p0) [root](gc[tau]d/4[zeta]l).   (7c)

  § 91. _Coefficient of Friction for Air._--A discussion by Professor
  Unwin of the experiments by Culley and Sabine on the rate of
  transmission of light carriers through pneumatic tubes, in which there
  is steady flow of air not sensibly affected by any resistances other
  than surface friction, furnished the value [zeta] = .007. The pipes
  were lead pipes, slightly moist, 2¼ in. (0.187 ft.) in diameter, and
  in lengths of 2000 to nearly 6000 ft.

  In some experiments on the flow of air through cast-iron pipes A.
  Arson found the coefficient of friction to vary with the velocity and
  diameter of the pipe. Putting

    [zeta] = [alpha]/v + [beta],   (8)

  he obtained the following values--

    +------------------+--------+-------+--------------------+
    | Diameter of Pipe |        |       | [zeta] for 100 ft. |
    |     in feet      | [alpha]| [beta]|   per second.      |
    +------------------+--------+-------+--------------------+
    |      1.64        | .00129 | .00483|      .00484        |
    |      1.07        | .00972 | .00640|      .00650        |
    |       .83        | .01525 | .00704|      .00719        |
    |       .338       | .03604 | .00941|      .00977        |
    |       .266       | .03790 | .00959|      .00997        |
    |       .164       | .04518 | .01167|      .01212        |
    +------------------+--------+-------+--------------------+

  It is worth while to try if these numbers can be expressed in the form
  proposed by Darcy for water. For a velocity of 100 ft. per second, and
  without much error for higher velocities, these numbers agree fairly
  with the formula

    [zeta] = 0.005(1 + (3/10)d),   (9)

  which only differs from Darcy's value for water in that the second
  term, which is always small except for very small pipes, is larger.

  Some later experiments on a very large scale, by E. Stockalper at the
  St Gotthard Tunnel, agree better with the value

    [zeta] = 0.0028(1 + (3/10)d).

  These pipes were probably less rough than Arson's.

  When the variation of pressure is very small, it is no longer safe to
  neglect the variation of level of the pipe. For that case we may
  neglect the work done by expansion, and then

    z0 - z1 - p0/G0 - p1/G1 - [zeta](v²/2g)(l/m) = 0,   (10)

  precisely equivalent to the equation for the flow of water, z0 and z1
  being the elevations of the two ends of the pipe above any datum, p0
  and p1 the pressures, G0 and G1 the densities, and v the mean velocity
  in the pipe. This equation may be used for the flow of coal gas.

  § 92. _Distribution of Pressure in a Pipe in which Air is
  Flowing._--From equation (7a) it results that the pressure p, at l ft.
  from that end of the pipe where the pressure is p0, is

    p = p0 [root](1 - [zeta]lu0²/mgc[tau]);   (11)

  which is of the form

    p = [root](al + b)

  for any given pipe with given end pressures. The curve of free surface
  level for the pipe is, therefore, a parabola with horizontal axis.
  Fig. 100 shows calculated curves of pressure for two of Sabine's
  experiments, in one of which the pressure was greater than atmospheric
  pressure, and in the other less than atmospheric pressure. The
  observed pressures are given in brackets and the calculated pressures
  without brackets. The pipe was the pneumatic tube between Fenchurch
  Street and the Central Station, 2818 yds. in length. The pressures are
  given in inches of mercury.

  [Illustration: FIG. 100.]

  _Variation of Velocity in the Pipe._--Let p0, u0 be the pressure and
  velocity at a given section of the pipe; p, u, the pressure and
  velocity at any other section. From equation (3a)

    up = c[tau]W/[Omega] = constant;

  so that, for any given uniform pipe,

    up = u0p0,
     u = u0p0/p;   (12)

  which gives the velocity at any section in terms of the pressure,
  which has already been determined. Fig. 101 gives the velocity curves
  for the two experiments of Culley and Sabine, for which the pressure
  curves have already been drawn. It will be seen that the velocity
  increases considerably towards that end of the pipe where the pressure
  is least.

  [Illustration: FIG. 101.]

  § 93. _Weight of Air Flowing per Second._--The weight of air
  discharged per second is (equation 3a)--

    W = [Omega]u0p0/c[tau].

  From equation (7b), for a pipe of circular section and diameter d,

    W = ¼[pi] [root](gd^5(p0² - p1²)/[zeta]lc[tau]),
      = .611[root](d^5(p0² - p1²)/[zeta]l[tau]).   (13)

  Approximately

    W = (.6916 p0 - .4438 p1)(d^5/[zeta]l[tau])^½.   (13a)

  § 94. _Application to the Case of Pneumatic Tubes for the Transmission
  of Messages._--In Paris, Berlin, London, and other towns, it has been
  found cheaper to transmit messages in pneumatic tubes than to
  telegraph by electricity. The tubes are laid underground with easy
  curves; the messages are made into a roll and placed in a light felt
  carrier, the resistance of which in the tubes in London is only ¾ oz.
  A current of air forced into the tube or drawn through it propels the
  carrier. In most systems the current of air is steady and continuous,
  and the carriers are introduced or removed without materially altering
  the flow of air.

  _Time of Transit through the Tube._--Putting t for the time of transit
  from 0 to l,
         _
        /l
    t = |   dl/u,
       _/0

  From (4a) neglecting dH/H, and putting m = d/4,

    dl = g d[Omega]²p dp/2[zeta]W²cr.

  From (1) and (3)

    u = Wc[tau]/p[Omega];

    dl/u = g d[Omega]³p² dp/2[zeta]W³c²[tau]²;
       _
      /p0
  t = |   g d[Omega]³p² dp/2[zeta]W³c²[tau]²,
     _/p1

    = gd[Omega]³(p0³ - p1³)/6[zeta]W³c²[tau]².   (14)

  But

    W = p0u0[Omega]/c[tau];

    .: t = gdc[tau](p0³ - p1³)/6[zeta]p0³u0³,

         = [zeta]^(½)l^(3/2)(p0³ - p1³)/6(gc[tau]d)^(½)(p0² - p1²)^(3/2);   (15)

  If [tau] = 521°, corresponding to 60° F.,

    t = .001412 [zeta]^(½)l^(3/2)(p0³ - p1³)/d^(½)(p0² - p1²)^(3/2);   (15a)

  which gives the time of transmission in terms of the initial and final
  pressures and the dimensions of the tube.

  _Mean Velocity of Transmission._--The mean velocity is l/t; or, for
  [tau] = 521°,

    u_mean = 0.708 [root]{d(p0² - p1²)^(3/2)/[zeta]l(p0³ - p1³)}.   (16)

  The following table gives some results:--

    +-----------+-----------------+----------------------------------+
    |           |    Absolute     |                                  |
    |           |  Pressures in   |    Mean Velocities for Tubes     |
    |           | lb. per sq. in. |       of a length in feet.       |
    +-----------+--------+--------+------+------+------+------+------+
    |           |   p0   |   p1   | 1000 | 2000 | 3000 | 4000 | 5000 |
    +-----------+--------+--------+------+------+------+------+------+
    | Vacuum    |   15   |    5   | 99.4 | 70.3 | 57.4 | 49.7 | 44.5 |
    |   Working |   15   |   10   | 67.2 | 47.5 | 38.8 | 34.4 | 30.1 |
    |           |        |        |      |      |      |      |      |
    | Pressure  |   20   |   15   | 57.2 | 40.5 | 33.0 | 28.6 | 25.6 |
    |   Working |   25   |   15   | 74.6 | 52.7 | 43.1 | 37.3 | 33.3 |
    |           |   30   |   15   | 84.7 | 60.0 | 49.0 | 42.4 | 37.9 |
    +-----------+-----------------+------+------+------+------+------+

  _Limiting Velocity in the Pipe when the Pressure at one End is
  diminished indefinitely._--If in the last equation there be put p1 =
  0, then

   u´_mean = 0.708 [root](d/[zeta]l);

  where the velocity is independent of the pressure p0 at the other end,
  a result which apparently must be absurd. Probably for long pipes, as
  for orifices, there is a limit to the ratio of the initial and
  terminal pressures for which the formula is applicable.


  X. FLOW IN RIVERS AND CANALS

  § 95. _Flow of Water in Open Canals and Rivers._--When water flows in
  a pipe the section at any point is determined by the form of the
  boundary. When it flows in an open channel with free upper surface,
  the section depends on the velocity due to the dynamical conditions.

  Suppose water admitted to an unfilled canal. The channel will
  gradually fill, the section and velocity at each point gradually
  changing. But if the inflow to the canal at its head is constant, the
  increase of cross section and diminution of velocity at each point
  attain after a time a limit. Thenceforward the section and velocity at
  each point are constant, and the motion is steady, or permanent regime
  is established.

  If when the motion is steady the sections of the stream are all equal,
  the motion is uniform. By hypothesis, the inflow [Omega]v is constant
  for all sections, and [Omega] is constant; therefore v must be
  constant also from section to section. The case is then one of uniform
  steady motion. In most artificial channels the form of section is
  constant, and the bed has a uniform slope. In that case the motion is
  uniform, the depth is constant, and the stream surface is parallel to
  the bed. If when steady motion is established the sections are
  unequal, the motion is steady motion with varying velocity from
  section to section. Ordinary rivers are in this condition, especially
  where the flow is modified by weirs or obstructions. Short
  unobstructed lengths of a river may be treated as of uniform section
  without great error, the mean section in the length being put for the
  actual sections.

  In all actual streams the different fluid filaments have different
  velocities, those near the surface and centre moving faster than those
  near the bottom and sides. The ordinary formulae for the flow of
  streams rest on a hypothesis that this variation of velocity may be
  neglected, and that all the filaments may be treated as having a
  common velocity equal to the mean velocity of the stream. On this
  hypothesis, a plane layer abab (fig. 102) between sections normal to
  the direction of motion is treated as sliding down the channel to
  a´a´b´b´ without deformation. The component of the weight parallel to
  the channel bed balances the friction against the channel, and in
  estimating the friction the velocity of rubbing is taken to be the
  mean velocity of the stream. In actual streams, however, the velocity
  of rubbing on which the friction depends is not the mean velocity of
  the stream, and is not in any simple relation with it, for channels of
  different forms. The theory is therefore obviously based on an
  imperfect hypothesis. However, by taking variable values for the
  coefficient of friction, the errors of the ordinary formulae are to a
  great extent neutralized, and they may be used without leading to
  practical errors. Formulae have been obtained based on less restricted
  hypotheses, but at present they are not practically so reliable, and
  are more complicated than the formulae obtained in the manner
  described above.

  [Illustration: FIG. 102.]

  § 96. _Steady Flow of Water with Uniform Velocity in Channels of
  Constant Section._--Let aa´, bb´ (fig. 103) be two cross sections
  normal to the direction of motion at a distance dl. Since the mass
  aa´bb´ moves uniformly, the external forces acting on it are in
  equilibrium. Let [Omega] be the area of the cross sections, [chi] the
  wetted perimeter, pq + qr + rs, of a section. Then the quantity m =
  [Omega]/[chi] is termed the hydraulic mean depth of the section. Let v
  be the mean velocity of the stream, which is taken as the common
  velocity of all the particles, i, the slope or fall of the stream in
  feet, per foot, being the ratio bc/ab.

  [Illustration: FIG. 103.]

  The external forces acting on aa´bb´ parallel to the direction of
  motion are three:--(a) The pressures on aa´ and bb´, which are equal
  and opposite since the sections are equal and similar, and the mean
  pressures on each are the same. (b) The component of the weight W of
  the mass in the direction of motion, acting at its centre of gravity
  g. The weight of the mass aa´bb´ is G[Omega]dl, and the component of
  the weight in the direction of motion is G[Omega]dl × the cosine of
  the angle between Wg and ab, that is, G[Omega]dl cos abc = G[Omega]dl
  bc/ab = G[Omega]idl. (c) There is the friction of the stream on the
  sides and bottom of the channel. This is proportional to the area
  [chi]dl of rubbing surface and to a function of the velocity which may
  be written f(v); f(v) being the friction per sq. ft. at a velocity v.
  Hence the friction is -[chi]dl f(v). Equating the sum of the forces to
  zero,

    G[Omega]i dl - [chi]dl f(v) = 0,

    f(v)/G = [Omega]i/[chi] = mi.   (1)

  But it has been already shown (§ 66) that f(v) = [zeta]Gv²/2g,

    .: [zeta]v²/2g = mi.   (2)

  This may be put in the form

    v = [root](2g/[zeta]) [root](mi) = c [root](mi);   (2a)

  where c is a coefficient depending on the roughness and form of the
  channel.

  The coefficient of friction [zeta] varies greatly with the degree of
  roughness of the channel sides, and somewhat also with the velocity.
  It must also be made to depend on the absolute dimensions of the
  section, to eliminate the error of neglecting the variations of
  velocity in the cross section. A common mean value assumed for [zeta]
  is 0.00757. The range of values will be discussed presently.

  It is often convenient to estimate the fall of the stream in feet per
  mile, instead of in feet per foot. If f is the fall in feet per mile,

    f = 5280 i.

  Putting this and the above value of [zeta] in (2a), we get the very
  simple and long-known approximate formula for the mean velocity of a
  stream--

    v = ¼ ½ [root](2mf).   (3)

  The flow down the stream per second, or discharge of the stream, is

    Q = [Omega]v = [Omega]c [root](mi).   (4)

  § 97. _Coefficient of Friction for Open Channels._--Various
  expressions have been proposed for the coefficient of friction for
  channels as for pipes. Weisbach, giving attention chiefly to the
  variation of the coefficient of friction with the velocity, proposed
  an expression of the form

    [zeta] = [alpha](1 + [beta]/v),   (5)

  and from 255 experiments obtained for the constants the values

    [alpha] = 0.007409; [beta] = 0.1920.

  This gives the following values at different velocities:--

    +----------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
    | v =      |  0.3  |  0.5  |  0.7  |  1    |  1½   |   2   |   3   |   5   |   7   |  10   |  15   |
    |          |       |       |       |       |       |       |       |       |       |       |       |
    | [zeta] = |0.01215|0.01025|0.00944|0.00883|0.00836|0.00812|0.90788|0.00769|0.00761|0.00755|0.00750|
    +----------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+

  In using this value of [zeta] when v is not known, it is best to
  proceed by approximation.

  § 98. _Darcy and Bazin's Expression for the Coefficient of
  Friction._--Darcy and Bazin's researches have shown that [zeta] varies
  very greatly for different degrees of roughness of the channel bed,
  and that it also varies with the dimensions of the channel. They give
  for [zeta] an empirical expression (similar to that for pipes) of the
  form

    [zeta] = a(1 + [beta]/m);   (6)

  where m is the hydraulic mean depth. For different kinds of channels
  they give the following values of the coefficient of friction:--

    +-------------------------------------------------+--------+------+
    |                 Kind of Channel.                | [alpha]|[beta]|
    +-------------------------------------------------+--------+------+
    |  I. Very smooth channels, sides of smooth       |        |      |
    |       cement or planed timber                   | .00294 | 0.10 |
    | II. Smooth channels, sides of ashlar, brickwork,|        |      |
    |       planks                                    | .00373 | 0.23 |
    |III. Rough channels, sides of rubble masonry or  |        |      |
    |       pitched with stone                        | .00471 | 0.82 |
    | IV. Very rough canals in earth                  | .00549 | 4.10 |
    |  V. Torrential streams encumbered with detritus | .00785 | 5.74 |
    +-------------------------------------------------+--------+------+

  The last values (Class V.) are not Darcy and Bazin's, but are taken
  from experiments by Ganguillet and Kutter on Swiss streams.

  The following table very much facilitates the calculation of the mean
  velocity and discharge of channels, when Darcy and Bazin's value of
  the coefficient of friction is used. Taking the general formula for
  the mean velocity already given in equation (2a) above,

    v = c [root](mi),

  where c = [root](2g/[zeta]), the following table gives values of c for
  channels of different degrees of roughness, and for such values of the
  hydraulic mean depths as are likely to occur in practical
  calculations:--

  Values of c in v = c[root](mi), deduced from Darcy and Bazin's Values.

    +----------+-----------+----------+---------+----------+--------------+
    |          |Very Smooth|  Smooth  |  Rough  |Very Rough| Excessively  |
    |   Mean   | Channels, | Channels,|Channels,| Channels,|Rough Channels|
    |Depth = m.|  Cement.  |Ashlar or | Rubble  |Canals in | encumbered   |
    |          |           |Brickwork.| Masonry.|  Earth.  |with Detritus.|
    +----------+-----------+----------+---------+----------+--------------+
    |     .25  |    125    |    95    |    57   |    26    |     18.5     |
    |     .5   |    135    |   110    |    72   |    36    |     25.6     |
    |     .75  |    139    |   116    |    81   |    42    |     30.8     |
    |    1.0   |    141    |   119    |    87   |    48    |     34.9     |
    |    1.5   |    143    |   122    |    94   |    56    |     41.2     |
    |    2.0   |    144    |   124    |    98   |    62    |     46.0     |
    |    2.5   |    145    |   126    |   101   |    67    |     ..       |
    |    3.0   |    145    |   126    |   104   |    70    |     53       |
    |    3.5   |    146    |   127    |   105   |    73    |     ..       |
    |    4.0   |    146    |   128    |   106   |    76    |     58       |
    |    4.5   |    146    |   128    |   107   |    78    |     ..       |
    |    5.0   |    146    |   128    |   108   |    80    |     62       |
    |    5.5   |    146    |   129    |   109   |    82    |     ..       |
    |    6.0   |    147    |   129    |   110   |    84    |     65       |
    |    6.5   |    147    |   129    |   110   |    85    |     ..       |
    |    7.0   |    147    |   129    |   110   |    86    |     67       |
    |    7.5   |    147    |   129    |   111   |    87    |     ..       |
    |    8.0   |    147    |   130    |   111   |    88    |     69       |
    |    8.5   |    147    |   130    |   112   |    89    |     ..       |
    |    9.0   |    147    |   130    |   112   |    90    |     71       |
    |    9.5   |    147    |   130    |   112   |    90    |     ..       |
    |   10.0   |    147    |   130    |   112   |    91    |     72       |
    |   11     |    147    |   130    |   113   |    92    |     ..       |
    |   12     |    147    |   130    |   113   |    93    |     74       |
    |   13     |    147    |   130    |   113   |    94    |     ..       |
    |   14     |    147    |   130    |   113   |    95    |     ..       |
    |   15     |    147    |   130    |   114   |    96    |     77       |
    |   16     |    147    |   130    |   114   |    97    |     ..       |
    |   17     |    147    |   130    |   114   |    97    |     ..       |
    |   18     |    147    |   130    |   114   |    98    |     ..       |
    |   20     |    147    |   131    |   114   |    98    |     80       |
    |   25     |    148    |   131    |   115   |   100    |     ..       |
    |   30     |    148    |   131    |   115   |   102    |     83       |
    |   40     |    148    |   131    |   116   |   103    |     85       |
    |   50     |    148    |   131    |   116   |   104    |     86       |
    |   [oo]   |    148    |   131    |   117   |   108    |     91       |
    +----------+-----------+----------+---------+----------+--------------+

  § 99. _Ganguillet and Kutter's Modified Darcy Formula._--Starting from
  the general expression v = c[root]mi, Ganguillet and Kutter examined
  the variations of c for a wider variety of cases than those discussed
  by Darcy and Bazin. Darcy and Bazin's experiments were confined to
  channels of moderate section, and to a limited variation of slope.
  Ganguillet and Kutter brought into the discussion two very distinct
  and important additional series of results. The gaugings of the
  Mississippi by A. A. Humphreys and H. L. Abbot afford data of
  discharge for the case of a stream of exceptionally large section and
  or very low slope. On the other hand, their own measurements of the
  flow in the regulated channels of some Swiss torrents gave data for
  cases in which the inclination and roughness of the channels were
  exceptionally great. Darcy and Bazin's experiments alone were
  conclusive as to the dependence of the coefficient c on the dimensions
  of the channel and on its roughness of surface. Plotting values of c
  for channels of different inclination appeared to indicate that it
  also depended on the slope of the stream. Taking the Mississippi data
  only, they found

    c = 256 for an inclination of 0.0034 per thousand,
      = 154      "       "        0.02         "

  so that for very low inclinations no constant value of c independent
  of the slope would furnish good values of the discharge. In small
  rivers, on the other hand, the values of c vary little with the slope.
  As regards the influence of roughness of the sides of the channel a
  different law holds. For very small channels differences of roughness
  have a great influence on the discharge, but for very large channels
  different degrees of roughness have but little influence, and for
  indefinitely large channels the influence of different degrees of
  roughness must be assumed to vanish. The coefficients given by Darcy
  and Bazin are different for each of the classes of channels of
  different roughness, even when the dimensions of the channel are
  infinite. But, as it is much more probable that the influence of the
  nature of the sides diminishes indefinitely as the channel is larger,
  this must be regarded as a defect in their formula.

  Comparing their own measurements in torrential streams in Switzerland
  with those of Darcy and Bazin, Ganguillet and Kutter found that the
  four classes of coefficients proposed by Darcy and Bazin were
  insufficient to cover all cases. Some of the Swiss streams gave
  results which showed that the roughness of the bed was markedly
  greater than in any of the channels tried by the French engineers. It
  was necessary therefore in adopting the plan of arranging the
  different channels in classes of approximately similar roughness to
  increase the number of classes. Especially an additional class was
  required for channels obstructed by detritus.

  To obtain a new expression for the coefficient in the formula

    v = [root](2g/[zeta]) [root](mi) = c [root](mi),

  Ganguillet and Kutter proceeded in a purely empirical way. They found
  that an expression of the form

    c = [alpha]/(1 + [beta]/[root]m)

  could be made to fit the experiments somewhat better than Darcy's
  expression. Inverting this, we get

    1/c = 1/[alpha] + [beta]/[alpha] [root]m,

  an equation to a straight line having 1/[root]m for abscissa, 1/c for
  ordinate, and inclined to the axis of abscissae at an angle the
  tangent of which is [beta]/[alpha].

  Plotting the experimental values of 1/c and 1/[root]m, the points so
  found indicated a curved rather than a straight line, so that [beta]
  must depend on [alpha]. After much comparison the following form was
  arrived at--

    c = (A + l/n)/(1 + An/[root]m),

  where n is a coefficient depending only on the roughness of the sides
  of the channel, and A and l are new coefficients, the value of which
  remains to be determined. From what has been already stated, the
  coefficient c depends on the inclination of the stream, decreasing as
  the slope i increases.

  Let

    A = a + p/i.

  Then

    c = (a + l/n + p/i)/{1 + (a + p/i)n/[root]m},

  the form of the expression for c ultimately adopted by Ganguillet and
  Kutter.

  For the constants a, l, p Ganguillet and Kutter obtain the values 23,
  1 and 0.00155 for metrical measures, or 41.6, 1.811 and 0.00281 for
  English feet. The coefficient of roughness n is found to vary from
  0.008 to 0.050 for either metrical or English measures.

  The most practically useful values of the coefficient of roughness n
  are given in the following table:--

              Nature of Sides of Channel.               Coefficient of
                                                         Roughness n.
    Well-planed timber                                     0.009
    Cement plaster                                         0.010
    Plaster of cement with one-third sand                  0.011
    Unplaned planks                                        0.012
    Ashlar and brickwork                                   0.013
    Canvas on frames                                       0.015
    Rubble masonry                                         0.017
    Canals in very firm gravel                             0.020
    Rivers and canals in perfect order, free from stones \
      or weeds                                           / 0.025
    Rivers and canals in moderately good order, not      \
      quite free from stones and weeds                   / 0.030
    Rivers and canals in bad order, with weeds and       \
      detritus                                           / 0.035
    Torrential streams encumbered with detritus            0.050

  Ganguillet and Kutter's formula is so cumbrous that it is difficult to
  use without the aid of tables.

  Lowis D'A. Jackson published complete and extensive tables for
  facilitating the use of the Ganguillet and Kutter formula (_Canal and
  Culvert Tables_, London, 1878). To lessen calculation he puts the
  formula in this form:--

    M = n(41.6 + 0.00281/i);

    v = ([root]m/n) {(M + 1.811)/(M + [root]m)} [root](mi).

  The following table gives a selection of values of M, taken from
  Jackson's tables:--

    +--------+--------------------------------------------------------------+
    |        |                     Values of M for n =                      |
    |  i =   +--------+--------+--------+--------+--------+--------+--------+
    |        |  0.010 |  0.012 |  0.015 |  0.017 |  0.020 |  0.025 |  0.030 |
    +--------+--------+--------+--------+--------+--------+--------+--------+
    | .00001 | 3.2260 | 3.8712 | 4.8390 | 5.4842 | 6.4520 | 8.0650 | 9.6780 |
    | .00002 | 1.8210 | 2.1852 | 2.7315 | 3.0957 | 3.6420 | 4.5525 | 5.4630 |
    | .00004 | 1.1185 | 1.3422 | 1.6777 | 1.9014 | 2.2370 | 2.7962 | 3.3555 |
    | .00006 | 0.8843 | 1.0612 | 1.3264 | 1.5033 | 1.7686 | 2.2107 | 2.6529 |
    | .00008 | 0.7672 | 0.9206 | 1.1508 | 1.3042 | 1.5344 | 1.9180 | 2.3016 |
    | .00010 | 0.6970 | 0.8364 | 1.0455 | 1.1849 | 1.3940 | 1.7425 | 2.0910 |
    | .00025 | 0.5284 | 0.6341 | 0.7926 | 0.8983 | 1.0568 | 1.3210 | 1.5852 |
    | .00050 | 0.4722 | 0.5666 | 0.7083 | 0.8027 | 0.9444 | 1.1805 | 1.4166 |
    | .00075 | 0.4535 | 0.5442 | 0.6802 | 0.7709 | 0.9070 | 1.1337 | 1.3605 |
    | .00100 | 0.4441 | 0.5329 | 0.6661 | 0.7550 | 0.8882 | 1.1102 | 1.3323 |
    | .00200 | 0.4300 | 0.5160 | 0.6450 | 0.7310 | 0.8600 | 1.0750 | 1.2900 |
    | .00300 | 0.4254 | 0.5105 | 0.6381 | 0.7232 | 0.8508 | 1.0635 | 1.2762 |
    +--------+--------+--------+--------+--------+--------+--------+--------+

  A difficulty in the use of this formula is the selection of the
  coefficient of roughness. The difficulty is one which no theory will
  overcome, because no absolute measure of the roughness of stream beds
  is possible. For channels lined with timber or masonry the difficulty
  is not so great. The constants in that case are few and sufficiently
  defined. But in the case of ordinary canals and rivers the case is
  different, the coefficients having a much greater range. For
  artificial canals in rammed earth or gravel n varies from 0.0163 to
  0.0301. For natural channels or rivers n varies from 0.020 to 0.035.

  In Jackson's opinion even Kutter's numerous classes of channels seem
  inadequately graduated, and he proposes for artificial canals the
  following classification:--

      I. Canals in very firm gravel, in perfect order      n = 0.02
     II. Canals in earth, above the average in order       n = 0.0225
    III. Canals in earth, in fair order                    n = 0.025
     IV. Canals in earth, below the average in order       n = 0.0275
      V. Canals in earth, in rather bad order, partially\
           overgrown with weeds and obstructed by        > n = 0.03
           detritus.                                    /

  Ganguillet and Kutter's formula has been considerably used partly from
  its adoption in calculating tables for irrigation work in India. But
  it is an empirical formula of an unsatisfactory form. Some engineers
  apparently have assumed that because it is complicated it must be more
  accurate than simpler formulae. Comparison with the results of
  gaugings shows that this is not the case. The term involving the slope
  was introduced to secure agreement with some early experiments on the
  Mississippi, and there is strong reason for doubting the accuracy of
  these results.

  § 100. _Bazin's New Formula._--Bazin subsequently re-examined all the
  trustworthy gaugings of flow in channels and proposed a modification
  of the original Darcy formula which appears to be more satisfactory
  than any hitherto suggested (_Étude d'une nouvelle formule_, Paris,
  1898). He points out that Darcy's original formula, which is of the
  form mi/v² = [alpha] + [beta]/m, does not agree with experiments on
  channels as well as with experiments on pipes. It is an objection to
  it that if m increases indefinitely the limit towards which mi/v²
  tends is different for different values of the roughness. It would
  seem that if the dimensions of a canal are indefinitely increased the
  variation of resistance due to differing roughness should vanish. This
  objection is met if it is assumed that [root](mi/v²) = [alpha] +
  [beta]/[root]m, so that if a is a constant mi/v² tends to the limit a
  when m increases. A very careful discussion of the results of gaugings
  shows that they can be expressed more satisfactorily by this new
  formula than by Ganguillet and Kutter's. Putting the equation in the
  form [zeta]v²/2g = mi, [zeta] = 0.002594(1 + [gamma]/[root]m), where
  [gamma] has the following values:--

      I. Very smooth sides, cement, planed plank, [gamma] = 0.109
     II. Smooth sides, planks, brickwork                    0.290
    III. Rubble masonry sides                               0.833
     IV. Sides of very smooth earth, or pitching            1.539
      V. Canals in earth in ordinary condition              2.353
     VI. Canals in earth exceptionally rough                3.168

  § 101. _The Vertical Velocity Curve._--If at each point along a
  vertical representing the depth of a stream, the velocity at that
  point is plotted horizontally, the curve obtained is the vertical
  velocity curve and it has been shown by many observations that it
  approximates to a parabola with horizontal axis. The vertex of the
  parabola is at the level of the greatest velocity. Thus in fig. 104 OA
  is the vertical at which velocities are observed; v0 is the surface;
  v_z the maximum and v_d the bottom velocity. B C D is the vertical
  velocity curve which corresponds with a parabola having its vertex at
  C. The mean velocity at the vertical is

    v_m = (1/3)[2v_z + v_d + (d_z/d)(v0 - v_d)].

  _The Horizontal Velocity Curve._--Similarly if at each point along a
  horizontal representing the width of the stream the velocities are
  plotted, a curve is obtained called the horizontal velocity curve. In
  streams of symmetrical section this is a curve symmetrical about the
  centre line of the stream. The velocity varies little near the centre
  of the stream, but very rapidly near the banks. In unsymmetrical
  sections the greatest velocity is at the point where the stream is
  deepest, and the general form of the horizontal velocity curve is
  roughly similar to the section of the stream.

  [Illustration: FIG. 104.]

  § 102. _Curves or Contours of Equal Velocity._--If velocities are
  observed at a number of points at different widths and depths in a
  stream, it is possible to draw curves on the cross section through
  points at which the velocity is the same. These represent contours of
  a solid, the volume of which is the discharge of the stream per
  second. Fig. 105 shows the vertical and horizontal velocity curves and
  the contours of equal velocity in a rectangular channel, from one of
  Bazin's gaugings.

  § 103. _Experimental Observations on the Vertical Velocity Curve._--A
  preliminary difficulty arises in observing the velocity at a given
  point in a stream because the velocity rapidly varies, the motion not
  being strictly steady. If an average of several velocities at the same
  point is taken, or the average velocity for a sensible period of time,
  this average is found to be constant. It may be inferred that though
  the velocity at a point fluctuates about a mean value, the
  fluctuations being due to eddying motions superposed on the general
  motion of the stream, yet these fluctuations produce effects which
  disappear in the mean of a series of observations and, in calculating
  the volume of flow, may be disregarded.

  [Illustration: FIG. 105.]

  In the next place it is found that in most of the best observations on
  the velocity in streams, the greatest velocity at any vertical is
  found not at the surface but at some distance below it. In various
  river gaugings the depth d_z at the centre of the stream has been
  found to vary from 0 to 0.3d.

  § 104. _Influence of the Wind._--In the experiments on the Mississippi
  the vertical velocity curve in calm weather was found to agree fairly
  with a parabola, the greatest velocity being at (3/10)ths of the depth
  of the stream from the surface. With a wind blowing down stream the
  surface velocity is increased, and the axis of the parabola approaches
  the surface. On the contrary, with a wind blowing up stream the
  surface velocity is diminished, and the axis of the parabola is
  lowered, sometimes to half the depth of the stream. The American
  observers drew from their observations the conclusion that there was
  an energetic retarding action at the surface of a stream like that due
  to the bottom and sides. If there were such a retarding action the
  position of the filament of maximum velocity below the surface would
  be explained.

  It is not difficult to understand that a wind acting on surface
  ripples or waves should accelerate or retard the surface motion of the
  stream, and the Mississippi results may be accepted so far as showing
  that the surface velocity of a stream is variable when the mean
  velocity of the stream is constant. Hence observations of surface
  velocity by floats or otherwise should only be made in very calm
  weather. But it is very difficult to suppose that, in still air, there
  is a resistance at the free surface of the stream at all analogous to
  that at the sides and bottom. Further, in very careful experiments, P.
  P. Boileau found the maximum velocity, though raised a little above
  its position for calm weather, still at a considerable distance below
  the surface, even when the wind was blowing down stream with a
  velocity greater than that of the stream, and when the action of the
  air must have been an accelerating and not a retarding action. A much
  more probable explanation of the diminution of the velocity at and
  near the free surface is that portions of water, with a diminished
  velocity from retardation by the sides or bottom, are thrown off in
  eddying masses and mingle with the rest of the stream. These eddying
  masses modify the velocity in all parts of the stream, but have their
  greatest influence at the free surface. Reaching the free surface they
  spread out and remain there, mingling with the water at that level and
  diminishing the velocity which would otherwise be found there.

  _Influence of the Wind on the Depth at which the Maximum Velocity is
  found._--In the gaugings of the Mississippi the vertical velocity
  curve was found to agree well with a parabola having a horizontal axis
  at some distance below the water surface, the ordinate of the parabola
  at the axis being the maximum velocity of the section. During the
  gaugings the force of the wind was registered on a scale ranging from
  0 for a calm to 10 for a hurricane. Arranging the velocity curves in
  three sets--(1) with the wind blowing up stream, (2) with the wind
  blowing down stream, (3) calm or wind blowing across stream--it was
  found that an upstream wind lowered, and a down-stream wind raised,
  the axis of the parabolic velocity curve. In calm weather the axis was
  at (3/10)ths of the total depth from the surface for all conditions of
  the stream.

  Let h´ be the depth of the axis of the parabola, m the hydraulic mean
  depth, f the number expressing the force of the wind, which may range
  from +10 to -10, positive if the wind is up stream, negative if it is
  down stream. Then Humphreys and Abbot find their results agree with
  the expression

    h´/m = 0.317 ± 0.06f.

  Fig. 106 shows the parabolic velocity curves according to the American
  observers for calm weather, and for an up- or down-stream wind of a
  force represented by 4.

  [Illustration: FIG. 106.]

  It is impossible at present to give a theoretical rule for the
  vertical velocity curve, but in very many gaugings it has been found
  that a parabola with horizontal axis fits the observed results fairly
  well. The mean velocity on any vertical in a stream varies from 0.85
  to 0.92 of the surface velocity at that vertical, and on the average
  if v0 is the surface and v_m the mean velocity at a vertical v_m =
  6/7 v0, a result useful in float gauging. On any vertical there is a
  point at which the velocity is equal to the mean velocity, and if this
  point were known it would be useful in gauging. Humphreys and Abbot in
  the Mississippi found the mean velocity at 0.66 of the depth; G. H. L.
  Hagen and H. Heinemann at 0.56 to 0.58 of the depth. The mean of
  observations by various observers gave the mean velocity at from 0.587
  to 0.62 of the depth, the average of all being almost exactly 0.6 of
  the depth. The mid-depth velocity is therefore nearly equal to, but a
  little greater than, the mean velocity on a vertical. If v_(md) is the
  mid-depth velocity, then on the average v_m = 0.98v_(md).

  § 105. _Mean Velocity on a Vertical from Two Velocity
  Observations._--A. J. C. Cunningham, in gaugings on the Ganges canal,
  found the following useful results. Let v0 be the surface, v_m the
  mean, and v_(xd) the velocity at the depth xd; then

    v_m = ¼[v0 + 3v_(2/3d)]
        = ½[v_(.211)^d + v_(.789)^d].

  § 106. _Ratio of Mean to Greatest Surface Velocity, for the whole
  Cross Section in Trapezoidal Channels._--It is often very important to
  be able to deduce the mean velocity, and thence the discharge, from
  observation of the greatest surface velocity. The simplest method of
  gauging small streams and channels is to observe the greatest surface
  velocity by floats, and thence to deduce the mean velocity. In general
  in streams of fairly regular section the mean velocity for the whole
  section varies from 0.7 to 0.85 of the greatest surface velocity. For
  channels not widely differing from those experimented on by Bazin, the
  expression obtained by him for the ratio of surface to mean velocity
  may be relied on as at least a good approximation to the truth. Let v0
  be the greatest surface velocity, v_m the mean velocity of the stream.
  Then, according to Bazin,

    v_m = v0 - 25.4 [root](mi).

  But

    v_m = c [root](mi),

  where c is a coefficient, the values of which have been already given
  in the table in § 98. Hence

    v_m = cv0/(c + 25.4).

    _Values of Coefficient c/(c + 25.4) in the Formula v_m = cv0/(c +
    25.4)._

    +----------+---------+----------+---------+----------+----------+
    |Hydraulic |  Very   |  Smooth  |  Rough  |Very Rough| Channels |
    |Mean Depth| Smooth  |Channels. |Channels.| Channels.|encumbered|
    |   = m.   |Channels.|Ashlar or | Rubble  | Canals in|   with   |
    |          | Cement. |Brickwork.| Masonry.|  Earth.  | Detritus.|
    +----------+---------+----------+---------+----------+----------+
    |          |         |          |         |          |          |
    |    0.25  |   .83   |   .79    |   .69   |   .51    |   .42    |
    |    0.5   |   .84   |   .81    |   .74   |   .58    |   .50    |
    |    0.75  |   .84   |   .82    |   .76   |   .63    |   .55    |
    |    1.0   |   .85   |    ..    |   .77   |   .65    |   .58    |
    |    2.0   |    ..   |   .83    |   .79   |   .71    |   .64    |
    |    3.0   |    ..   |    ..    |   .80   |   .73    |   .67    |
    |    4.0   |    ..   |    ..    |   .81   |   .75    |   .70    |
    |    5.0   |    ..   |    ..    |    ..   |   .76    |   .71    |
    |    6.0   |    ..   |   .84    |    ..   |   .77    |   .72    |
    |    7.0   |    ..   |    ..    |    ..   |   .78    |   .73    |
    |    8.0   |    ..   |    ..    |    ..   |    ..    |    ..    |
    |    9.0   |    ..   |    ..    |   .82   |    ..    |   .74    |
    |   10.0   |    ..   |    ..    |    ..   |    ..    |    ..    |
    |   15.0   |    ..   |    ..    |    ..   |   .79    |   .75    |
    |   20.0   |    ..   |    ..    |    ..   |   .80    |   .76    |
    |   30.0   |    ..   |    ..    |   .82   |    ..    |   .77    |
    |   40.0   |    ..   |    ..    |    ..   |    ..    |    ..    |
    |   50.0   |    ..   |    ..    |    ..   |    ..    |    ..    |
    |   [oo]   |    ..   |    ..    |    ..   |    ..    |   .79    |
    +----------+---------+----------+---------+----------+----------+

  [Illustration: FIG. 107.]

  § 107. _River Bends._--In rivers flowing in alluvial plains, the
  windings which already exist tend to increase in curvature by the
  scouring away of material from the outer bank and the deposition of
  detritus along the inner bank. The sinuosities sometimes increase till
  a loop is formed with only a narrow strip of land between the two
  encroaching branches of the river. Finally a "cut off" may occur, a
  waterway being opened through the strip of land and the loop left
  separated from the stream, forming a horseshoe shaped lagoon or marsh.
  Professor James Thomson pointed out (_Proc. Roy. Soc._, 1877, p. 356;
  _Proc. Inst. of Mech. Eng._, 1879, p. 456) that the usual supposition
  is that the water tending to go forwards in a straight line rushes
  against the outer bank and scours it, at the same time creating
  deposits at the inner bank. That view is very far from a complete
  account of the matter, and Professor Thomson gave a much more
  ingenious account of the action at the bend, which he completely
  confirmed by experiment.

  [Illustration: FIG. 108.]

  When water moves round a circular curve under the action of gravity
  only, it takes a motion like that in a free vortex. Its velocity is
  greater parallel to the axis of the stream at the inner than at the
  outer side of the bend. Hence the scouring at the outer side and the
  deposit at the inner side of the bend are not due to mere difference
  of velocity of flow in the general direction of the stream; but, in
  virtue of the centrifugal force, the water passing round the bend
  presses outwards, and the free surface in a radial cross section has a
  slope from the inner side upwards to the outer side (fig. 108). For
  the greater part of the water flowing in curved paths, this difference
  of pressure produces no tendency to transverse motion. But the water
  immediately in contact with the rough bottom and sides of the channel
  is retarded, and its centrifugal force is insufficient to balance the
  pressure due to the greater depth at the outside of the bend. It
  therefore flows inwards towards the inner side of the bend, carrying
  with it detritus which is deposited at the inner bank. Conjointly with
  this flow inwards along the bottom and sides, the general mass of
  water must flow outwards to take its place. Fig. 107 shows the
  directions of flow as observed in a small artificial stream, by means
  of light seeds and specks of aniline dye. The lines CC show the
  directions of flow immediately in contact with the sides and bottom.
  The dotted line AB shows the direction of motion of floating particles
  on the surface of the stream.

  § 108. _Discharge of a River when flowing at different Depths._--When
  frequent observations must be made on the flow of a river or canal,
  the depth of which varies at different times, it is very convenient to
  have to observe the depth only. A formula can be established giving
  the flow in terms of the depth. Let Q be the discharge in cubic feet
  per second; H the depth of the river in some straight and uniform
  part. Then Q = aH + bH², where the constants a and b must be found by
  preliminary gaugings in different conditions of the river. M. C.
  Moquerey found for part of the upper Saône, Q = 64.7H + 8.2H² in
  metric measures, or Q = 696H + 26.8H² in English measures.

  § 109. _Forms of Section of Channels._--The simplest form of section
  for channels is the semicircular or nearly semicircular channel (fig.
  109), a form now often adopted from the facility with which it can be
  executed in concrete. It has the advantage that the rubbing surface is
  less in proportion to the area than in any other form.

  [Illustration: FIG. 109.]

  Wooden channels or flumes, of which there are examples on a large
  scale in America, are rectangular in section, and the same form is
  adopted for wrought and cast-iron aqueducts. Channels built with
  brickwork or masonry may be also rectangular, but they are often
  trapezoidal, and are always so if the sides are pitched with masonry
  laid dry. In a trapezoidal channel, let b (fig. 110) be the bottom
  breadth, b0 the top breadth, d the depth, and let the slope of the
  sides be n horizontal to 1 vertical. Then the area of section is
  [Omega] = (b + nd)d = (b0 - nd)d, and the wetted perimeter [chi] = b +
  2d[root](n² + 1).

  [Illustration: FIG. 110.]

  When a channel is simply excavated in earth it is always originally
  trapezoidal, though it becomes more or less rounded in course of time.
  The slope of the sides then depends on the stability of the earth, a
  slope of 2 to 1 being the one most commonly adopted.

  Figs. 111, 112 show the form of canals excavated in earth, the former
  being the section of a navigation canal and the latter the section of
  an irrigation canal.

  § 110. _Channels of Circular Section._--The following short table
  facilitates calculations of the discharge with different depths of
  water in the channel. Let r be the radius of the channel section; then
  for a depth of water = [kappa]r, the hydraulic mean radius is [mu]r
  and the area of section of the waterway [nu]r², where [kappa], [mu],
  and [nu] have the following values:--

    +---------------------------------+------+-----+-----+-----+-----+-----+-----+-----+----+----+----+----+----+----+----+-----+-----+-----+-----+-----+-----+
    | Depth of water in   \ [kappa] = |.01   |.05  |.10  |.15  |.20  |.25  |.30  |.35  |.40 |.45 |.50 |.55 |.60 |.65 |.70 |.75  |.80  |.85  |.90  |.95  |1.0  |
    |   terms of radius   /           |      |     |     |     |     |     |     |     |    |    |    |    |    |    |    |     |     |     |     |     |     |
    | Hydraulic mean depth\ [mu]    = |.00668|.0321|.0523|.0963|.1278|.1574|.1852|.2142|.242|.269|.293|.320|.343|.365|.387|.408 |.429 |.449 |.466 |.484 |.500 |
    |   in terms of radius/           |      |     |     |     |     |     |     |     |    |    |    |    |    |    |    |     |     |     |     |     |     |
    | Waterway in terms of\ [nu]    = |.00189|.0211|.0598|.1067|.1651|.228 |.294 |.370 |.450|.532|.614|.709|.795|.885|.979|1.075|1.175|1.276|1.371|1.470|1.571|
    |   square of radius  /           |      |     |     |     |     |     |     |     |    |    |    |    |    |    |    |     |     |     |     |     |     |
    +---------------------------------+------+-----+-----+-----+-----+-----+-----+-----+----+----+----+----+----+----+----+-----+-----+-----+-----+-----+-----+

  [Illustration: FIG. 111.--Scale 20 ft. = 1 in.]

  [Illustration: FIG. 112.--Scale 80 ft. = 1 in.]

  § 111. _Egg-Shaped Channels or Sewers._--In sewers for discharging
  storm water and house drainage the volume of flow is extremely
  variable; and there is a great liability for deposits to be left when
  the flow is small, which are not removed during the short periods when
  the flow is large. The sewer in consequence becomes choked. To obtain
  uniform scouring action, the velocity of flow should be constant or
  nearly so; a complete uniformity of velocity cannot be obtained with
  any form of section suitable for sewers, but an approximation to
  uniform velocity is obtained by making the sewers of oval section.
  Various forms of oval have been suggested, the simplest being one in
  which the radius of the crown is double the radius of the invert, and
  the greatest width is two-thirds the height. The section of such a
  sewer is shown in fig. 113, the numbers marked on the figure being
  proportional numbers.

  [Illustration: FIG. 113.]

  § 112. _Problems on Channels in which the Flow is Steady and at
  Uniform Velocity._--The general equations given in §§ 96, 98 are

    [zeta] = [alpha](1 + [beta]/m);   (1)

    [zeta]v²/2g = mi;   (2)

    Q = [Omega]v.   (3)

  _Problem I._--Given the transverse section of stream and discharge, to
  find the slope. From the dimensions of the section find [Omega] and m;
  from (1) find [zeta], from (3) find v, and lastly from (2) find i.

  _Problem II._--Given the transverse section and slope, to find the
  discharge. Find v from (2), then Q from (3).

  _Problem III._--Given the discharge and slope, and either the breadth,
  depth, or general form of the section of the channel, to determine its
  remaining dimensions. This must generally be solved by approximations.
  A breadth or depth or both are chosen, and the discharge calculated.
  If this is greater than the given discharge, the dimensions are
  reduced and the discharge recalculated.

  [Illustration: FIG. 114.]

  Since m lies generally between the limits m = d and m = ½d, where d is
  the depth of the stream, and since, moreover, the velocity varies as
  [root](m) so that an error in the value of m leads only to a much less
  error in the value of the velocity calculated from it, we may proceed
  thus. Assume a value for m, and calculate v from it. Let v1 be this
  first approximation to v. Then Q/v1 is a first approximation to
  [Omega], say [Omega]1. With this value of [Omega] design the section
  of the channel; calculate a second value for m; calculate from it a
  second value of v, and from that a second value for [Omega]. Repeat
  the process till the successive values of m approximately coincide.

  § 113. _Problem IV. Most Economical Form of Channel for given Side
  Slopes._--Suppose the channel is to be trapezoidal in section (fig.
  114), and that the sides are to have a given slope. Let the
  longitudinal slope of the stream be given, and also the mean velocity.
  An infinite number of channels could be found satisfying the
  foregoing conditions. To render the problem determinate, let it be
  remembered that, since for a given discharge [Omega][oo] [cube
  root][chi], other things being the same, the amount of excavation will
  be least for that channel which has the least wetted perimeter. Let d
  be the depth and b the bottom width of the channel, and let the sides
  slope n horizontal to 1 vertical (fig. 114), then

    [Omega] = (b + nd)d;

    [chi] = b + 2d [root](n² + 1).

  Both [Omega] and [chi] are to be minima. Differentiating, and equating
  to zero.

    (db/dd + n)d + b + nd = 0,

    db/dd + 2[root](n² + 1) = 0;

  eliminating db/dd,

    {n - 2[root](n² + 1)}d + b + nd = 0;

    b = 2 {[root](n² + 1) - n}d.

  But

    [Omega]/[chi] = (b + nd)d/{b + 2d [root](n² + 1)}.

  Inserting the value of b,

    m = [Omega]/[chi] = {2d[root](n² + 1) - nd}/
      {4d [root](n² + 1) - 2nd} = ½d.

  That is, with given side slopes, the section is least for a given
  discharge when the hydraulic mean depth is half the actual depth.

  A simple construction gives the form of the channel which fulfils this
  condition, for it can be shown that when m = ½d the sides of the
  channel are tangential to a semicircle drawn on the water line.

  Since

    [Omega]/[chi] = ½d,

  therefore

    [Omega] = ½[chi]d.   (1)

  Let ABCD be the channel (fig. 115); from E the centre of AD drop
  perpendiculars EF, EG, EH on the sides.

  Let

    AB = CD = a; BC = b; EF = EH = c; and EG = d.

    [Omega] = area AEB + BEC + CED,
            = ac + ½bd.

    [chi] = 2a + b.

  Putting these values in (1),

    ac + ½bd = (a + ½b)d; and hence c = d.

  [Illustration: FIG. 115.]

  That is, EF, EG, EH are all equal, hence a semicircle struck from E
  with radius equal to the depth of the stream will pass through F and H
  and be tangential to the sides of the channel.

  [Illustration: FIG. 116.]

  To draw the channel, describe a semicircle on a horizontal line with
  radius = depth of channel. The bottom will be a horizontal tangent of
  that semicircle, and the sides tangents drawn at the required side
  slopes.

  The above result may be obtained thus (fig. 116):--

    [chi] = b + 2d/sin [beta].   (1)

    [Omega] = d(b + d cot [beta]);

    [Omega]/d = b + d cot [beta];   (2)

    [Omega]/d² = b/d + cot [beta].   (3)

  From (1) and (2),

    [chi] = [Omega]/d - d cot [beta] + 2d/sin [beta].

  This will be a minimum for

    d[chi]/dd = [Omega]/d² + cot[beta] - 2/sin [beta] = 0,

  or

    [Omega]/d² = 2 cosec. [beta] - cot [beta].   (4)

  or

    d = [root]{[Omega] sin [beta]/(2 - cos [beta])}.

  From (3) and (4),

    b/d = 2(1 - cos [beta])/sin [beta] = 2 tan ½[beta].

  _Proportions of Channels of Maximum Discharge for given Area and Side
  Slopes. Depth of channel = d; Hydraulic mean depth = ½d; Area of
  section =_ [Omega].

    +-------------+-----------+--------+----------+---------+------------+
    |             |Inclination|Ratio of| Area of  |         |Top width = |
    |             |of Sides to|  Side  | Section  |  Bottom |twice length|
    |             |  Horizon. | Slopes.| [Omega]. |  Width. |of each Side|
    |             |           |        |          |         |   Slope.   |
    +-------------+-----------+--------+----------+---------+------------+
    | Semicircle  |    ..     |   ..   | 1.571 d² |    0    |    2 d     |
    | Semi-hexagon|  60°  0´  | 3  : 5 | 1.732 d² | 1.155 d |  2.310 d   |
    | Semi-square |  90°  0´  | 0  : 1 |   2 d²   |  2 d    |    2 d     |
    |             |  75° 58´  | 1  : 4 | 1.812 d² | 1.562 d |  2.062 d   |
    |             |  63° 26´  | 1  : 2 | 1.736 d² | 1.236 d |  2.236 d   |
    |             |  53°  8´  | 3  : 4 | 1.750 d² |    d    |  2.500 d   |
    |             |  45°  0´  | 1  : 1 | 1.828 d² | 0.828 d |  2.828 d   |
    |             |  38° 40´  | 1¼ : 1 | 1.952 d² | 0.702 d |  3.202 d   |
    |             |  33° 42´  | 1½ : 1 | 2.106 d² | 0.606 d |  3.606 d   |
    |             |  29° 44´  | 1¾ : 1 | 2.282 d² | 0.532 d |  4.032 d   |
    |             |  26° 34´  | 2  : 1 | 2.472 d² | 0.472 d |  4.472 d   |
    |             |  23° 58´  | 2¼ : 1 | 2.674 d² | 0.424 d |  4.924 d   |
    |             |  21° 48´  | 2½ : 1 | 2.885 d² | 0.385 d |  5.385 d   |
    |             |  19° 58´  | 2¾ : 1 | 3.104 d² | 0.354 d |  5.854 d   |
    |             |  18° 26´  | 3  : 1 | 3.325 d² | 0.325 d |  6.325 d   |
    +-------------+-----------+--------+----------+---------+------------+

    Half the top width is the length of each side slope. The wetted
    perimeter is the sum of the top and bottom widths.

  § 114. _Form of Cross Section of Channel in which the Mean Velocity is
  Constant with Varying Discharge._--In designing waste channels from
  canals, and in some other cases, it is desirable that the mean
  velocity should be restricted within narrow limits with very different
  volumes of discharge. In channels of trapezoidal form the velocity
  increases and diminishes with the discharge. Hence when the discharge
  is large there is danger of erosion, and when it is small of silting
  or obstruction by weeds. A theoretical form of section for which the
  mean velocity would be constant can be found, and, although this is
  not very suitable for practical purposes, it can be more or less
  approximated to in actual channels.

  Let fig. 117 represent the cross section of the channel. From the
  symmetry of the section, only half the channel need be considered. Let
  obac be any section suitable for the minimum flow, and let it be
  required to find the curve beg for the upper part of the channel so
  that the mean velocity shall be constant. Take o as origin of
  coordinates, and let de, fg be two levels of the water above ob.

  [Illustration: FIG. 117.]

  Let

    ob = b/2; de = y, fg = y + dy, od = x, of = x + dx; eg = ds.

  The condition to be satisfied is that

    v = c [root](mi)

  should be constant, whether the water-level is at ob, de, or fg.
  Consequently

    m = constant = k

  for all three sections, and can be found from the section obac. Hence
  also

     Increment of section    y dx
    ---------------------- = ---- = k
    Increment of perimeter    ds

    y²dx² = k²ds² = k²(dx² + dy²) and dx = k dy/[root](y² - k²).

  Integrating,

    x = k log_[epsilon] {y + [root](y² - k²)} + constant;

  and, since y = b/2 when x = 0,

    x = k log_[epsilon] [{y + [root](y² - k²)}/{½b + [root](¼b² - k²)}].

  Assuming values for y, the values of x can be found and the curve
  drawn.

  The figure has been drawn for a channel the minimum section of which
  is a half hexagon of 4 ft. depth. Hence k = 2; b = 9.2; the rapid
  flattening of the side slopes is remarkable.


  STEADY MOTION OF WATER IN OPEN CHANNELS OF VARYING CROSS SECTION AND
  SLOPE

  § 115. In every stream the discharge of which is constant, or may be
  regarded as constant for the time considered, the velocity at
  different places depends on the slope of the bed. Except at certain
  exceptional points the velocity will be greater as the slope of the
  bed is greater, and, as the velocity and cross section of the stream
  vary inversely, the section of the stream will be least where the
  velocity and slope are greatest. If in a stream of tolerably uniform
  slope an obstruction such as a weir is built, that will cause an
  alteration of flow similar to that of an alteration of the slope of
  the bed for a greater or less distance above the weir, and the
  originally uniform cross section of the stream will become a varied
  one. In such cases it is often of much practical importance to
  determine the longitudinal section of the stream.

  The cases now considered will be those in which the changes of
  velocity and cross section are gradual and not abrupt, and in which
  the only internal work which needs to be taken into account is that
  due to the friction of the stream bed, as in cases of uniform motion.
  Further, the motion will be supposed to be steady, the mean velocity
  at each given cross section remaining constant, though it varies from
  section to section along the course of the stream.

  [Illustration: FIG. 118.]

  Let fig. 118 represent a longitudinal section of the stream, A0A1
  being the water surface, B0B1 the stream bed. Let A0B0, A1B1 be cross
  sections normal to the direction of flow. Suppose the mass of water
  A0B0A1B1 comes in a short time [theta] to C0D0C1D1, and let the work
  done on the mass be equated to its change of kinetic energy during
  that period. Let l be the length A0A1 of the portion of the stream
  considered, and z the fall, of surface level in that distance. Let Q
  be the discharge of the stream per second.

  [Illustration: FIG. 119.]

  _Change of Kinetic Energy._--At the end of the time [theta] there are
  as many particles possessing the same velocities in the space C0D0A1B1
  as at the beginning. The change of kinetic energy is therefore the
  difference of the kinetic energies of A0B0C0D0 and A1B1C1D1.

  Let fig. 119 represent the cross section A0B0, and let [omega] be a
  small element of its area at a point where the velocity is v. Let
  [Omega]0 be the whole area of the cross section and u0 the mean
  velocity for the whole cross section. From the definition of mean
  velocity we have

    u0 = [Sigma][omega]v/[Omega]0.

  Let v = u0 + w, where w is the difference between the velocity at the
  small element [omega] and the mean velocity. For the whole cross
  section, [Sigma][omega]w = 0.

  The mass of fluid passing through the element of section [omega], in
  [theta] seconds, is (G/g)[omega]v[theta], and its kinetic energy is
  (G/2g)[omega]v³[theta]. For the whole section, the kinetic energy of
  the mass A0B0C0D0 passing in [theta] seconds is

    (G[theta]/2g)[Sigma][omega]v³
      = (G[theta]/2g)[Sigma][omega](u0³ + 3u0²w + 3u0² + w³),
      = (G[theta]/2g){u0³[Omega] + [Sigma][omega]w²(3u0 + w)}.

  The factor 3u0 + w is equal to 2u0 + v, a quantity necessarily
  positive. Consequently [Sigma][omega]v³ > [Omega]0u0³, and
  consequently the kinetic energy of A0B0C0D0 is greater than

    (G[theta]/2g)[Omega]0u0³ or (G[theta])/2g)Qu0²,

  which would be its value if all the particles passing the section had
  the same velocity u0. Let the kinetic energy be taken at

    [alpha](G[theta]/2g)[Omega]0u0³ = [alpha](G[theta]/2g)Qu0²,

  where [alpha] is a corrective factor, the value of which was estimated
  by J. B. C. J. Bélanger at 1.1.[6] Its precise value is not of great
  importance.

  In a similar way we should obtain for the kinetic energy of A1B1C1D1
  the expression

    [alpha](G[theta]/2g)[Omega]1u1³ = [alpha](G[theta]/2g)Qu1²,

  where [Omega]1, u1 are the section and mean velocity at A1B1, and
  where a may be taken to have the same value as before without any
  important error.

  Hence the change of kinetic energy in the whole mass A0B0A1B1 in
  [theta] seconds is

    [alpha](G[theta]/2g) Q (u1² - u0²).   (1)

  _Motive Work of the Weight and Pressures._--Consider a small filament
  a0a1 which comes in [theta] seconds to c0c1. The work done by gravity
  during that movement is the same as if the portion a0c0 were carried
  to a1c1. Let dQ[theta] be the volume of a0c0 or a1c1, and y0, y1 the
  depths of a0, a1 from the surface of the stream. Then the volume
  dQ[theta] or GdQ[theta] pounds falls through a vertical height z + y1
  - y0, and the work done by gravity is

    G dQ[theta](z + y1 - y0).

  Putting p_a for atmospheric pressure, the whole pressure per unit of
  area at a0 is Gy0 + p_a, and that at a1 is - (Gy1 + p_a). The work of
  these pressures is

    G(y0 + p_a/G - y1 - p_a/G) dQ[theta] = G(y0 - y1) dQ[theta].

  Adding this to the work of gravity, the whole work is GzdQ[theta]; or,
  for the whole cross section,

    GzQ[theta].   (2)

  _Work expended in Overcoming the Friction of the Stream Bed._--Let
  A´B´, A´´B´´ be two cross sections at distances s and s + ds from
  A0B0. Between these sections the velocity may be treated as uniform,
  because by hypothesis the changes of velocity from section to section
  are gradual. Hence, to this short length of stream the equation for
  uniform motion is applicable. But in that case the work in overcoming
  the friction of the stream bed between A´B´ and A´´B´´ is

    GQ[theta][zeta](u²/2g)([chi]/[Omega]) ds,

  where u, [chi], [Omega] are the mean velocity, wetted perimeter, and
  section at A´B´. Hence the whole work lost in friction from A0B0 to
  A1B1 will be
               _
              / l
    GQ[theta] |   [zeta](u²/2g)([chi]/[Omega]) ds.   (3)
             _/ 0

  Equating the work given in (2) and (3) to the change of kinetic energy
  given in (1),

    [alpha](GQ[theta]/2g)(u1² - u0²)
                                _
                               / l
      = GQz[theta] - GQ[theta] |   [zeta](u²/2g)([chi]/[Omega]) ds;
                              _/ 0
                                    _
                                   / l
    .: z = [alpha](u1² - u0²)/2g + |   [zeta](u²/2g)([chi]/[Omega]) ds.
                                  _/ 0

  [Illustration: FIG. 120.]

  § 116. _Fundamental Differential Equation of Steady Varied
  Motion._--Suppose the equation just found to be applied to an
  indefinitely short length ds of the stream, limited by the end
  sections ab, a1b1, taken for simplicity normal to the stream bed (fig.
  120). For that short length of stream the fall of surface level, or
  difference of level of a and a1, may be written dz. Also, if we write
  u for u0, and u + du for u1, the term (u0² - u1²)/2g becomes udu/g.
  Hence the equation applicable to an indefinitely short length of the
  stream is

    dz = udu/g + ([chi]/[Omega])[zeta](u²/2g) ds.   (1)

  From this equation some general conclusions may be arrived at as to
  the form of the longitudinal section of the stream, but, as the
  investigation is somewhat complicated, it is convenient to simplify it
  by restricting the conditions of the problem.

  _Modification of the Formula for the Restricted Case of a Stream
  flowing in a Prismatic Stream Bed of Constant Slope._--Let i be the
  constant slope of the bed. Draw ad parallel to the bed, and ac
  horizontal. Then dz is sensibly equal to a´c. The depths of the
  stream, h and h + dh, are sensibly equal to ab and a´b´, and therefore
  dh = a´d. Also cd is the fall of the bed in the distance ds, and is
  equal to ids. Hence

    dz = a´c = cd - a´d = i ds - dh.   (2)

  Since the motion is steady--

    Q = [Omega]u = constant.

  Differentiating,

    [Omega] du + u d[Omega] = 0;

    .:du = -u d[Omega]/[Omega].

  Let x be the width of the stream, then d[Omega] = xdh very nearly.
  Inserting this value,

    du = -(ux/[Omega]) dh.   (3)

  Putting the values of du and dz found in (2) and (3) in equation (1),

    i ds - dh = -(u²x/g[Omega]) dh + ([chi]/[Omega])[zeta](u²/2g) ds.

    dh/ds = {i - ([chi]/[Omega]) [zeta] (u²/2g)}/{1 - (u²/g)(x/[Omega])}.   (4)

  _Further Restriction to the Case of a Stream of Rectangular Section
  and of Indefinite Width._--The equation might be discussed in the form
  just given, but it becomes a little simpler if restricted in the way
  just stated. For, if the stream is rectangular, [chi]h = [Omega], and
  if [chi] is large compared with h, [Omega]/[chi] = xh/x = h nearly.
  Then equation (4) becomes

    dh/ds = i(1 - [zeta]u²/2gih)/(1 - u²/gh).   (5)

  § 117. _General Indications as to the Form of Water Surface furnished
  by Equation_ (5).--Let A0A1 (fig. 121) be the water surface, B0B1 the
  bed in a longitudinal section of the stream, and ab any section at a
  distance s from B0, the depth ab being h. Suppose B0B1, B0A0 taken as
  rectangular coordinate axes, then dh/ds is the trigonometric tangent
  of the angle which the surface of the stream at a makes with the axis
  B0B1. This tangent dh/ds will be positive, if the stream is increasing
  in depth in the direction B0B1; negative, if the stream is diminishing
  in depth from B0 towards B1. If dh/ds = 0, the surface of the stream
  is parallel to the bed, as in cases of uniform motion. But from
  equation (4)

    dh/ds = 0, if i - ([chi]/[Omega])[zeta](u²/2g) = 0;

    .: [zeta](u²/2g) = ([Omega]/[chi])i = mi,

  which is the well-known general equation for uniform motion, based on
  the same assumptions as the equation for varied steady motion now
  being considered. The case of uniform motion is therefore a limiting
  case between two different kinds of varied motion.

  [Illustration: FIG. 121.]

  Consider the possible changes of value of the fraction

    (1 - [zeta]u²/2gih)/(1 - u²/gh).

  As h tends towards the limit 0, and consequently u is large, the
  numerator tends to the limit -[oo]. On the other hand if h = [oo], in
  which case u is small, the numerator becomes equal to 1. For a value H
  of h given by the equation

    1 - [zeta]u²/2giH = 0,

    H = [zeta]u²/2gi,

  we fall upon the case of uniform motion. The results just stated may
  be tabulated thus:--

    For h = 0, H, > H, [oo],

  the numerator has the value -[oo], 0, > 0, 1.

  Next consider the denominator. If h becomes very small, in which case
  u must be very large, the denominator tends to the limit -[oo]. As h
  becomes very large and u consequently very small, the denominator
  tends to the limit 1. For h = u²/g, or u = [root](gh), the denominator
  becomes zero. Hence, tabulating these results as before:--

    For h = 0, u²/g, > u²/g, [oo],

  the denominator becomes

    -[oo], 0, > 0, 1.

  [Illustration: FIG. 122.]

  § 118. _Case_ 1.--Suppose h > u²/g, and also h > H, or the depth
  greater than that corresponding to uniform motion. In this case dh/ds
  is positive, and the stream increases in depth in the direction of
  flow. In fig. 122 let B0B1 be the bed, C0C1 a line parallel to the bed
  and at a height above it equal to H. By hypothesis, the surface A0A1
  of the stream is above C0C1, and it has just been shown that the depth
  of the stream increases from B0 towards B1. But going up stream h
  approaches more and more nearly the value H, and therefore dh/ds
  approaches the limit 0, or the surface of the stream is asymptotic to
  C0C1. Going down stream h increases and u diminishes, the numerator
  and denominator of the fraction (1 - [zeta]u²/2gih)/(1 -u²/gh) both
  tend towards the limit 1, and dh/ds to the limit i. That is, the
  surface of the stream tends to become asymptotic to a horizontal line
  D0D1.

  The form of water surface here discussed is produced when the flow of
  a stream originally uniform is altered by the construction of a weir.
  The raising of the water surface above the level C0C1 is termed the
  backwater due to the weir.

  § 119. _Case_ 2.--Suppose h > u²/g, and also h < H. Then dh/ds is
  negative, and the stream is diminishing in depth in the direction of
  flow. In fig. 123 let B0B1 be the stream bed as before; C0C1 a line
  drawn parallel to B0B1 at a height above it equal to H. By hypothesis
  the surface A0A1 of the stream is below C0C1, and the depth has just
  been shown to diminish from B0 towards B1. Going up stream h
  approaches the limit H, and dh/ds tends to the limit zero. That is, up
  stream A0A1 is asymptotic to C0C1. Going down stream h diminishes and
  u increases; the inequality h>u²/g diminishes; the denominator of the
  fraction (1 - [zeta]u²/2gih)/(1 - u²/gh) tends to the limit zero, and
  consequently dh/ds tends to [infinity]. That is, down stream A0A1
  tends to a direction perpendicular to the bed. Before, however, this
  limit was reached the assumptions on which the general equation is
  based would cease to be even approximately true, and the equation
  would cease to be applicable. The filaments would have a relative
  motion, which would make the influence of internal friction in the
  fluid too important to be neglected. A stream surface of this form may
  be produced if there is an abrupt fall in the bed of the stream (fig.
  124).

  [Illustration: FIG. 123.]

  [Illustration: FIG. 124.]

  [Illustration: FIG. 125.]

  On the Ganges canal, as originally constructed, there were abrupt
  falls precisely of this kind, and it appears that the lowering of the
  water surface and increase of velocity which such falls occasion, for
  a distance of some miles up stream, was not foreseen. The result was
  that, the velocity above the falls being greater than was intended,
  the bed was scoured and considerable damage was done to the works.
  "When the canal was first opened the water was allowed to pass freely
  over the crests of the overfalls, which were laid on the level of the
  bed of the earthen channel; erosion of bed and sides for some miles up
  rapidly followed, and it soon became apparent that means must be
  adopted for raising the surface of the stream at those points (that
  is, the crests of the falls). Planks were accordingly fixed in the
  grooves above the bridge arches, or temporary weirs were formed over
  which the water was allowed to fall; in some cases the surface of the
  water was thus raised above its normal height, causing a backwater in
  the channel above" (Crofton's _Report on the Ganges Canal_, p. 14).
  Fig. 125 represents in an exaggerated form what probably occurred, the
  diagram being intended to represent some miles' length of the canal
  bed above the fall. AA parallel to the canal bed is the level
  corresponding to uniform motion with the intended velocity of the
  canal. In consequence of the presence of the ogee fall, however, the
  water surface would take some such form as BB, corresponding to Case 2
  above, and the velocity would be greater than the intended velocity,
  nearly in the inverse ratio of the actual to the intended depth. By
  constructing a weir on the crest of the fall, as shown by dotted
  lines, a new water surface CC corresponding to Case 1 would be
  produced, and by suitably choosing the height of the weir this might
  be made to agree approximately with the intended level AA.

  § 120. _Case_ 3.--Suppose a stream flowing uniformly with a depth
  h<u²/g. For a stream in uniform motion [zeta]u²/2g = mi, or if the
  stream is of indefinitely great width, so that m = H, then
  [zeta]u²/2g = iH, and H = [zeta]u²/2gi. Consequently the condition
  stated above involves that

    [zeta]u²/2gi < u²/g, or that i > [zeta]/2.

  If such a stream is interfered with by the construction of a weir
  which raises its level, so that its depth at the weir becomes h1 >
  u²/g, then for a portion of the stream the depth h will satisfy the
  conditions h < u²/g and h > H, which are not the same as those assumed in the two
  previous cases. At some point of the stream above the weir the depth h
  becomes equal to u²/g, and at that point dh/ds becomes infinite, or
  the surface of the stream is normal to the bed. It is obvious that at
  that point the influence of internal friction will be too great to be
  neglected, and the general equation will cease to represent the true
  conditions of the motion of the water. It is known that, in cases such
  as this, there occurs an abrupt rise of the free surface of the
  stream, or a standing wave is formed, the conditions of motion in
  which will be examined presently.

  It appears that the condition necessary to give rise to a standing
  wave is that i > [zeta]/2. Now [zeta] depends for different channels
  on the roughness of the channel and its hydraulic mean depth. Bazin
  calculated the values of [zeta] for channels of different degrees of
  roughness and different depths given in the following table, and the
  corresponding minimum values of i for which the exceptional case of
  the production of a standing wave may occur.

    +-----------------------------+----------------+-------------------------+
    |                             |  Slope below   |  Standing Wave Formed.  |
    |                             |which a Standing|                         |
    |   Nature of Bed of Stream.  |    Wave is     +-------------+-----------+
    |                             | impossible in  |Slope in feet|Least Depth|
    |                             | feet peer foot.|  per foot.  |  in feet. |
    +-----------------------------+----------------+-------------+-----------+
    |                             |                |  / 0.002    |   0.262   |
    | Very smooth cemented surface|    0.00147     | <  0.003    |    .098   |
    |                             |                |  \ 0.004    |    .065   |
    |                             |                |             |           |
    |                             |                |  / 0.003    |    .394   |
    | Ashlar or brickwork         |    0.00186     | <  0.004    |    .197   |
    |                             |                |  \ 0.006    |    .098   |
    |                             |                |             |           |
    |                             |                |  / 0.004    |   1.181   |
    | Rubble masonry              |    0.00235     | <  0.006    |    .525   |
    |                             |                |  \ 0.010    |    .262   |
    |                             |                |             |           |
    |                             |                |  / 0.006    |   3.478   |
    | Earth                       |    0.00275     | <  0.010    |   1.542   |
    |                             |                |  \ 0.015    |    .919   |
    +-----------------------------+----------------+-------------+-----------+


  STANDING WAVES

  § 121. The formation of a standing wave was first observed by Bidone.
  Into a small rectangular masonry channel, having a slope of 0.023 ft.
  per foot, he admitted water till it flowed uniformly with a depth of
  0.2 ft. He then placed a plank across the stream which raised the
  level just above the obstruction to 0.95 ft. He found that the stream
  above the obstruction was sensibly unaffected up to a point 15 ft.
  from it. At that point the depth suddenly increased from 0.2 ft. to
  0.56 ft. The velocity of the stream in the part unaffected by the
  obstruction was 5.54 ft. per second. Above the point where the abrupt
  change of depth occurred u² = 5.54² = 30.7, and gh = 32.2 × 0.2 =
  6.44; hence u² was > gh. Just below the abrupt change of depth u =
  5.54 × 0.2/0.56 = 1.97; u² = 3.88; and gh = 32.2 × 0.56 = 18.03; hence
  at this point u² < gh. Between these two points, therefore, u² = gh;
  and the condition for the production of a standing wave occurred.

  [Illustration: FIG. 126.]

  The change of level at a standing wave may be found thus. Let fig. 126
  represent the longitudinal section of a stream and ab, cd cross
  sections normal to the bed, which for the short distance considered
  may be assumed horizontal. Suppose the mass of water abcd to come to
  a´b´c´d´ in a short time t; and let u0, u1 be the velocities at ab and
  cd, [Omega]0, [Omega]1 the areas of the cross sections. The force
  causing change of momentum in the mass abcd estimated horizontally is
  simply the difference of the pressures on ab and cd. Putting h0, h1
  for the depths of the centres of gravity of ab and cd measured down
  from the free water surface, the force is G(h0[Omega]0 - h1[Omega]1)
  pounds, and the impulse in t seconds is G (h0[Omega]0 - h1[Omega]1) t
  second pounds. The horizontal change of momentum is the difference of
  the momenta of cdc´d´ and aba´b´; that is,

    (G/g)([Omega]1u1² - [Omega]0u0²)t.

  Hence, equating impulse and change of momentum,

    G(h0[Omega]0 - h1[Omega]1)t = (G/g)([Omega]1u1² - [Omega]0u0²)t;

    .: h0[Omega]0 - h1[Omega]1 = ([Omega]1u1² - [Omega]0u0²)/g.   (1)

  For simplicity let the section be rectangular, of breadth B and depths
  H0 and H1, at the two cross sections considered; then h0 = ½H0, and h1
  = ½H1. Hence

    H0² - H1² = (2/g)(H1u1² - H0u0²).

  But, since [Omega]0u0 = [Omega]1u1, we have

    u1² = u0²H0²/H1²,

    H0² - H1² = (2u0²/g)(H0²/H1 - H0).   (2)

  This equation is satisfied if H0 = H1, which corresponds to the case
  of uniform motion. Dividing by H0 - H1, the equation becomes

    (H1/H0)(H0 + H1) = 2u0²/g;   (3)

    .: H1 = [root](2u0²H0/g + ¼H0²) - ½H0.   (4)

  In Bidone's experiment u0 = 5.54, and H0 = 0.2. Hence H1 = 0.52, which
  agrees very well with the observed height.

  [Illustration: FIG. 127.]

  § 122. A standing wave is frequently produced at the foot of a weir.
  Thus in the ogee falls originally constructed on the Ganges canal a
  standing wave was observed as shown in fig. 127. The water falling
  over the weir crest A acquired a very high velocity on the steep slope
  AB, and the section of the stream at B became very small. It easily
  happened, therefore, that at B the depth h < u²/g. In flowing along
  the rough apron of the weir the velocity u diminished and the depth h
  increased. At a point C, where h became equal to u²/g, the conditions
  for producing the standing wave occurred. Beyond C the free surface
  abruptly rose to the level corresponding to uniform motion with the
  assigned slope of the lower reach of the canal.

  [Illustration: FIG. 128.]

  A standing wave is sometimes formed on the down stream side of bridges
  the piers of which obstruct the flow of the water. Some interesting
  cases of this kind are described in a paper on the "Floods in the
  Nerbudda Valley" in the _Proc. Inst. Civ. Eng._ vol. xxvii. p. 222, by
  A. C. Howden. Fig. 128 is compiled from the data given in that paper.
  It represents the section of the stream at pier 8 of the Towah
  Viaduct, during the flood of 1865. The ground level is not exactly
  given by Howden, but has been inferred from data given on another
  drawing. The velocity of the stream was not observed, but the author
  states it was probably the same as at the Gunjal river during a
  similar flood, that is 16.58 ft. per second. Now, taking the depth on
  the down stream face of the pier at 26 ft., the velocity necessary for
  the production of a standing wave would be u = [root](gh) =
  [root](32.2 × 26) = 29 ft. per second nearly. But the velocity at this
  point was probably from Howden's statements 16.58 × {40/26} = 25.5 ft.
  per second, an agreement as close as the approximate character of the
  data would lead us to expect.


  XI. ON STREAMS AND RIVERS

  § 123. _Catchment Basin._--A stream or river is the channel for the
  discharge of the available rainfall of a district, termed its
  catchment basin. The catchment basin is surrounded by a ridge or
  watershed line, continuous except at the point where the river finds
  an outlet. The area of the catchment basin may be determined from a
  suitable contoured map on a scale of at least 1 in 100,000. Of the
  whole rainfall on the catchment basin, a part only finds its way to
  the stream. Part is directly re-evaporated, part is absorbed by
  vegetation, part may escape by percolation into neighbouring
  districts. The following table gives the relation of the average
  stream discharge to the average rainfall on the catchment basin
  (Tiefenbacher).

    +-----------------------------+-----------------+--------------------+
    |                             |Ratio of average |Loss by Evaporation,|
    |                             |   Discharge to  | &c., in per cent of|
    |                             |average Rainfall.|   total Rainfall.  |
    +-----------------------------+-----------------+--------------------+
    | Cultivated land and spring- |                 |                    |
    |   forming declivities.      |    .3 to .33    |      67 to 70      |
    | Wooded hilly slopes.        |   .35 to .45    |      55 to 65      |
    | Naked unfissured mountains  |   .55 to .60    |      40 to 45      |
    +-----------------------------+-----------------+--------------------+

  § 124. _Flood Discharge._--The flood discharge can generally only be
  determined by examining the greatest height to which floods have been
  known to rise. To produce a flood the rainfall must be heavy and
  widely distributed, and to produce a flood of exceptional height the
  duration of the rainfall must be so great that the flood waters of the
  most distant affluents reach the point considered, simultaneously with
  those from nearer points. The larger the catchment basin the less
  probable is it that all the conditions tending to produce a maximum
  discharge should simultaneously occur. Further, lakes and the river
  bed itself act as storage reservoirs during the rise of water level
  and diminish the rate of discharge, or serve as flood moderators. The
  influence of these is often important, because very heavy rain storms
  are in most countries of comparatively short duration. Tiefenbacher
  gives the following estimate of the flood discharge of streams in
  Europe:--

                                         Flood discharge of Streams
                                         per Second per Square Mile
                                             of Catchment Basin.

    In flat country                           8.7 to 12.5 cub. ft.
    In hilly districts                       17.5 to 22.5    "
    In moderately mountainous districts      36.2 to 45.0    "
    In very mountainous districts            50.0 to 75.0    "

  It has been attempted to express the decrease of the rate of flood
  discharge with the increase of extent of the catchment basin by
  empirical formulae. Thus Colonel P. P. L. O'Connell proposed the
  formula y = M [root]x, where M is a constant called the modulus of the
  river, the value of which depends on the amount of rainfall, the
  physical characters of the basin, and the extent to which the floods
  are moderated by storage of the water. If M is small for any given
  river, it shows that the rainfall is small, or that the permeability
  or slope of the sides of the valley is such that the water does not
  drain rapidly to the river, or that lakes and river bed moderate the
  rise of the floods. If values of M are known for a number of rivers,
  they may be used in inferring the probable discharge of other similar
  rivers. For British rivers M varies from 0.43 for a small stream
  draining meadow land to 37 for the Tyne. Generally it is about 15 or
  20. For large European rivers M varies from 16 for the Seine to 67.5
  for the Danube. For the Nile M = 11, a low value which results from
  the immense length of the Nile throughout which it receives no
  affluent, and probably also from the influence of lakes. For different
  tributaries of the Mississippi M varies from 13 to 56. For various
  Indian rivers it varies from 40 to 303, this variation being due to
  the great variations of rainfall, slope and character of Indian
  rivers.

  In some of the tank projects in India, the flood discharge has been
  calculated from the formula D = C[3root]n², where D is the discharge
  in cubic yards per hour from n square miles of basin. The constant C
  was taken = 61,523 in the designs for the Ekrooka tank, = 75,000 on
  Ganges and Godavery works, and = 10,000 on Madras works.

  [Illustration: FIG. 129.]

  [Illustration: FIG. 130.]

  § 125. _Action of a Stream on its Bed._--If the velocity of a stream
  exceeds a certain limit, depending on its size, and on the size,
  heaviness, form and coherence of the material of which its bed is
  composed, it scours its bed and carries forward the materials. The
  quantity of material which a given stream can carry in suspension
  depends on the size and density of the particles in suspension, and is
  greater as the velocity of the stream is greater. If in one part of
  its course the velocity of a stream is great enough to scour the bed
  and the water becomes loaded with silt, and in a subsequent part of
  the river's course the velocity is diminished, then part of the
  transported material must be deposited. Probably deposit and scour go
  on simultaneously over the whole river bed, but in some parts the rate
  of scour is in excess of the rate of deposit, and in other parts the
  rate of deposit is in excess of the rate of scour. Deep streams appear
  to have the greatest scouring power at any given velocity. It is
  possible that the difference is strictly a difference of transporting,
  not of scouring action. Let fig. 129 represent a section of a stream.
  The material lifted at a will be diffused through the mass of the
  stream and deposited at different distances down stream. The average
  path of a particle lifted at a will be some such curve as abc, and the
  average distance of transport each time a particle is lifted will be
  represented by ac. In a deeper stream such as that in fig. 130, the
  average height to which particles are lifted, and, since the rate of
  vertical fall through the water may be assumed the same as before, the
  average distance a´c´ of transport will be greater. Consequently,
  although the scouring action may be identical in the two streams, the
  velocity of transport of material down stream is greater as the depth
  of the stream is greater. The effect is that the deep stream excavates
  its bed more rapidly than the shallow stream.

  § 126. _Bottom Velocity at which Scour commences._--The following
  bottom velocities were determined by P. L. G. Dubuat to be the maximum
  velocities consistent with stability of the stream bed for different
  materials.

  Darcy and Bazin give, for the relation of the mean velocity v_m and
  bottom velocity v_b.

    v_m = v_b + 10.87 [root](mi).

  But

    [root]mi = v_m [root]([zeta]/2g);

    .: v_m = v_b/(1 - 10.87 [root]([zeta]/2g)).

  Taking a mean value for [zeta], we get

    v_m = 1.312 v_b,

  and from this the following values of the mean velocity are
  obtained:--

    +-----------------------+---------------+-------------+
    |                       |Bottom Velocity|Mean Velocity|
    |                       |     = v_b.    |    = v_m.   |
    +-----------------------+---------------+-------------+
    | 1. Soft earth         |     0.25      |     .33     |
    | 2. Loam               |     0.50      |     .65     |
    | 3. Sand               |     1.00      |    1.30     |
    | 4. Gravel             |     2.00      |    2.62     |
    | 5. Pebbles            |     3.40      |    4.46     |
    | 6. Broken stone, flint|     4.00      |    5.25     |
    | 7. Chalk, soft shale  |     5.00      |    6.56     |
    | 8. Rock in beds       |     6.00      |    7.87     |
    | 9. Hard rock.         |    10.00      |   13.12     |
    +-----------------------+---------------+-------------+

  The following table of velocities which should not be exceeded in
  channels is given in the _Ingenieurs Taschenbuch_ of the Verein
  "Hütte":--

    +--------------------------------+---------+---------+---------+
    |                                | Surface |  Mean   | Bottom  |
    |                                |Velocity.|Velocity.|Velocity.|
    +--------------------------------+---------+---------+---------+
    | Slimy earth or brown clay      |    .49  |    .36  |    .26  |
    | Clay                           |    .98  |    .75  |    .52  |
    | Firm sand                      |   1.97  |   1.51  |   1.02  |
    | Pebbly bed                     |   4.00  |   3.15  |   2.30  |
    | Boulder bed                    |   5.00  |   4.03  |   3.08  |
    | Conglomerate of slaty fragments|   7.28  |   6.10  |   4.90  |
    | Stratified rocks               |   8.00  |   7.45  |   6.00  |
    | Hard rocks                     |  14.00  |  12.15  |  10.36  |
    +--------------------------------+---------+---------+---------+

  § 127. _Regime of a River Channel._--A river channel is said to be in
  a state of regime, or stability, when it changes little in draught or
  form in a series of years. In some rivers the deepest part of the
  channel changes its position perpetually, and is seldom found in the
  same place in two successive years. The sinuousness of the river also
  changes by the erosion of the banks, so that in time the position of
  the river is completely altered. In other rivers the change from year
  to year is very small, but probably the regime is never perfectly
  stable except where the rivers flow over a rocky bed.

  [Illustration: FIG. 131.]

  If a river had a constant discharge it would gradually modify its bed
  till a permanent regime was established. But as the volume discharged
  is constantly changing, and therefore the velocity, silt is deposited
  when the velocity decreases, and scour goes on when the velocity
  increases in the same place. When the scouring and silting are
  considerable, a perfect balance between the two is rarely established,
  and hence continual variations occur in the form of the river and the
  direction of its currents. In other cases, where the action is less
  violent, a tolerable balance may be established, and the deepening of
  the bed by scour at one time is compensated by the silting at another.
  In that case the general regime is permanent, though alteration is
  constantly going on. This is more likely to happen if by artificial
  means the erosion of the banks is prevented. If a river flows in soil
  incapable of resisting its tendency to scour it is necessarily sinuous
  (§ 107), for the slightest deflection of the current to either side
  begins an erosion which increases progressively till a considerable
  bend is formed. If such a river is straightened it becomes sinuous
  again unless its banks are protected from scour.

  § 128. _Longitudinal Section of River Bed._--The declivity of rivers
  decreases from source to mouth. In their higher parts rapid and
  torrential, flowing over beds of gravel or boulders, they enlarge in
  volume by receiving affluent streams, their slope diminishes, their
  bed consists of smaller materials, and finally they reach the sea.
  Fig. 131 shows the length in miles, and the surface fall in feet per
  mile, of the Tyne and its tributaries.

  The decrease of the slope is due to two causes. (1) The action of the
  transporting power of the water, carrying the smallest debris the
  greatest distance, causes the bed to be less stable near the mouth
  than in the higher parts of the river; and, as the river adjusts its
  slope to the stability of the bed by scouring or increasing its
  sinuousness when the slope is too great, and by silting or
  straightening its course if the slope is too small, the decreasing
  stability of the bed would coincide with a decreasing slope. (2) The
  increase of volume and section of the river leads to a decrease of
  slope; for the larger the section the less slope is necessary to
  ensure a given velocity.

  The following investigation, though it relates to a purely arbitrary
  case, is not without interest. Let it be assumed, to make the
  conditions definite--(1) that a river flows over a bed of uniform
  resistance to scour, and let it be further assumed that to maintain
  stability the velocity of the river in these circumstances is constant
  from source to mouth; (2) suppose the sections of the river at all
  points are similar, so that, b being the breadth of the river at any
  point, its hydraulic mean depth is ab and its section is cb², where a
  and c are constants applicable to all parts of the river; (3) let us
  further assume that the discharge increases uniformly in consequence
  of the supply from affluents, so that, if l is the length of the river
  from its source to any given point, the discharge there will be kl,
  where k is another constant applicable to all points in the course of
  the river.

  [Illustration: FIG. 132.]

  Let AB (fig. 132) be the longitudinal section of the river, whose
  source is at A; and take A for the origin of vertical and horizontal
  coordinates. Let C be a point whose ordinates are x and y, and let the
  river at C have the breadth b, the slope i, and the velocity v. Since
  velocity × area of section = discharge, vcb² = kl, or b =
  [root](kl/cv).

  Hydraulic mean depth = ab = a [root](kl/cv).

  But, by the ordinary formula for the flow of rivers, mi = [zeta]v²;

    .: i = [zeta]v²/m = ([zeta]v^(5/2)/a) [root](c/kl).

  But i is the tangent of the angle which the curve at C makes with the
  axis of X, and is therefore = dy/dx. Also, as the slope is small, l =
  AC = AD = x nearly.

    .: dy/dx = ([zeta]v^(5/2)/a) [root](c/kx);

  and, remembering that v is constant,

    y = (2[zeta]v^(5/2)/a) [root](cx/k);

  or

    y² = constant × x;

  so that the curve is a common parabola, of which the axis is
  horizontal and the vertex at the source. This may be considered an
  ideal longitudinal section, to which actual rivers approximate more or
  less, with exceptions due to the varying hardness of their beds, and
  the irregular manner in which their volume increases.

  § 129. _Surface Level of River._--The surface level of a river is a
  plane changing constantly in position from changes in the volume of
  water discharged, and more slowly from changes in the river bed, and
  the circumstances affecting the drainage into the river.

  For the purposes of the engineer, it is important to determine (1) the
  extreme low water level, (2) the extreme high water or flood level,
  and (3) the highest navigable level.

  1. _Low Water Level_ cannot be absolutely known, because a river
  reaches its lowest level only at rare intervals, and because
  alterations in the cultivation of the land, the drainage, the removal
  of forests, the removal or erection of obstructions in the river bed,
  &c., gradually alter the conditions of discharge. The lowest level of
  which records can be found is taken as the conventional or approximate
  low water level, and allowance is made for possible changes.

  2. _High Water or Flood Level._--The engineer assumes as the highest
  flood level the highest level of which records can be obtained. In
  forming a judgment of the data available, it must be remembered that
  the highest level at one point of a river is not always simultaneous
  with the attainment of the highest level at other points, and that
  the rise of a river in flood is very different in different parts of
  its course. In temperate regions, the floods of rivers seldom rise
  more than 20 ft. above low-water level, but in the tropics the rise of
  floods is greater.

  3. _Highest Navigable Level._--When the river rises above a certain
  level, navigation becomes difficult from the increase of the velocity
  of the current, or from submersion of the tow paths, or from the
  headway under bridges becoming insufficient. Ordinarily the highest
  navigable level may be taken to be that at which the river begins to
  overflow its banks.

  § 130. _Relative Value of Different Materials for Submerged
  Works._--That the power of water to remove and transport different
  materials depends on their density has an important bearing on the
  selection of materials for submerged works. In many cases, as in the
  aprons or floorings beneath bridges, or in front of locks or falls,
  and in the formation of training walls and breakwaters by _pierres
  perdus_, which have to resist a violent current, the materials of
  which the structures are composed should be of such a size and weight
  as to be able individually to resist the scouring action of the water.
  The heaviest materials will therefore be the best; and the different
  value of materials in this respect will appear much more striking, if
  it is remembered that all materials lose part of their weight in
  water. A block whose volume is V cubic feet, and whose density in air
  is w lb. per cubic foot, weighs in air wV lb., but in water only
  (w--62.4) V lb.

    +----------------------+-----------------------------+
    |                      | Weight of a Cub. Ft. in lb. |
    |                      +--------------+--------------+
    |                      |    In Air.   |   In Water.  |
    +----------------------+--------------+--------------+
    | Basalt               |    187.3     |    124.9     |
    | Brick                |    130.0     |     67.6     |
    | Brickwork            |    112.0     |     49.6     |
    | Granite and limestone|    170.0     |    107.6     |
    | Sandstone            |    144.0     |     81.6     |
    | Masonry              |   116-144    |  53.6-81.6   |
    +----------------------+--------------+--------------+

  § 131. _Inundation Deposits from a River._--When a river carrying silt
  periodically overflows its banks, it deposits silt over the area
  flooded, and gradually raises the surface of the country. The silt is
  deposited in greatest abundance where the water first leaves the
  river. It hence results that the section of the country assumes a
  peculiar form, the river flowing in a trough along the crest of a
  ridge, from which the land slopes downwards on both sides. The silt
  deposited from the water forms two wedges, having their thick ends
  towards the river (fig. 133).

  [Illustration: FIG. 133.]

  This is strikingly the case with the Mississippi, and that river is
  now kept from flooding immense areas by artificial embankments or
  levees. In India, the term _deltaic segment_ is sometimes applied to
  that portion of a river running through deposits formed by inundation,
  and having this characteristic section. The irrigation of the country
  in this case is very easy; a comparatively slight raising of the river
  surface by a weir or annicut gives a command of level which permits
  the water to be conveyed to any part of the district.

  § 132. _Deltas._--The name delta was originally given to the [Greek:
  Delta]-shaped portion of Lower Egypt, included between seven branches
  of the Nile. It is now given to the whole of the alluvial tracts round
  river mouths formed by deposition of sediment from the river, where
  its velocity is checked on its entrance to the sea. The characteristic
  feature of these alluvial deltas is that the river traverses them, not
  in a single channel, but in two or many bifurcating branches. Each
  branch has a tract of the delta under its influence, and gradually
  raises the surface of that tract, and extends it seaward. As the delta
  extends itself seaward, the conditions of discharge through the
  different branches change. The water finds the passage through one of
  the branches less obstructed than through the others; the velocity and
  scouring action in that branch are increased; in the others they
  diminish. The one channel gradually absorbs the whole of the water
  supply, while the other branches silt up. But as the mouth of the new
  main channel extends seaward the resistance increases both from the
  greater length of the channel and the formation of shoals at its
  mouth, and the river tends to form new bifurcations AC or AD (fig.
  134), and one of these may in time become the main channel of the
  river.

  § 133. _Field Operations preliminary to a Study of River
  Improvement._--There are required (1) a plan of the river, on which
  the positions of lines of levelling and cross sections are marked; (2)
  a longitudinal section and numerous cross sections of the river; (3) a
  series of gaugings of the discharge at different points and in
  different conditions of the river.

  _Longitudinal Section._--This requires to be carried out with great
  accuracy. A line of stakes is planted, following the sinuosities of
  the river, and chained and levelled. The cross sections are referred
  to the line of stakes, both as to position and direction. The
  determination of the surface slope is very difficult, partly from its
  extreme smallness, partly from oscillation of the water. Cunningham
  recommends that the slope be taken in a length of 2000 ft. by four
  simultaneous observations, two on each side of the river.

  [Illustration: FIG. 134.]

  § 134. _Cross Sections_--A stake is planted flush with the water, and
  its level relatively to some point on the line of levels is
  determined. Then the depth of the water is determined at a series of
  points (if possible at uniform distances) in a line starting from the
  stake and perpendicular to the thread of the stream. To obtain these,
  a wire may be stretched across with equal distances marked on it by
  hanging tags. The depth at each of these tags may be obtained by a
  light wooden staff, with a disk-shaped shoe 4 to 6 in. in diameter. If
  the depth is great, soundings may be taken by a chain and weight. To
  ensure the wire being perpendicular to the thread of the stream, it is
  desirable to stretch two other wires similarly graduated, one above
  and the other below, at a distance of 20 to 40 yds. A number of floats
  being then thrown in, it is observed whether they pass the same
  graduation on each wire.

  [Illustration: FIG. 135.]

  For large and rapid rivers the cross section is obtained by sounding
  in the following way. Let AC (fig. 135) be the line on which soundings
  are required. A base line AB is measured out at right angles to AC,
  and ranging staves are set up at AB and at D in line with AC. A boat
  is allowed to drop down stream, and, at the moment it comes in line
  with AD, the lead is dropped, and an observer in the boat takes, with
  a box sextant, the angle AEB subtended by AB. The sounding line may
  have a weight of 14 lb. of lead, and, if the boat drops down stream
  slowly, it may hang near the bottom, so that the observation is made
  instantly. In extensive surveys of the Mississippi observers with
  theodolites were stationed at A and B. The theodolite at A was
  directed towards C, that at B was kept on the boat. When the boat came
  on the line AC, the observer at A signalled, the sounding line was
  dropped, and the observer at B read off the angle ABE. By repeating
  observations a number of soundings are obtained, which can be plotted
  in their proper position, and the form of the river bed drawn by
  connecting the extremities of the lines. From the section can be
  measured the sectional area of the stream [Omega] and its wetted
  perimeter [chi]; and from these the hydraulic mean depth m can be
  calculated.

  § 135. _Measurement of the Discharge of Rivers._--The area of cross
  section multiplied by the mean velocity gives the discharge of the
  stream. The height of the river with reference to some fixed mark
  should be noted whenever the velocity is observed, as the velocity and
  area of cross section are different in different states of the river.
  To determine the mean velocity various methods may be adopted; and,
  since no method is free from liability to error, either from the
  difficulty of the observations or from uncertainty as to the ratio of
  the mean velocity to the velocity observed, it is desirable that more
  than one method should be used.


  INSTRUMENTS FOR MEASURING THE VELOCITY OF WATER

  § 136. _Surface Floats_ are convenient for determining the surface
  velocities of a stream, though their use is difficult near the banks.
  The floats may be small balls of wood, of wax or of hollow metal, so
  loaded as to float nearly flush with the water surface. To render
  them visible they may have a vertical painted stem. In experiments on
  the Seine, cork balls 1(3/4) in. diameter were used, loaded to float
  flush with the water, and provided with a stem. In A. J. C.
  Cunningham's observations at Roorkee, the floats were thin circular
  disks of English deal, 3 in. diameter and ¼ in. thick. For
  observations near the banks, floats 1 in. diameter and 1/8 in. thick
  were used. To render them visible a tuft of cotton wool was used
  loosely fixed in a hole at the centre.

  The velocity is obtained by allowing the float to be carried down, and
  noting the time of passage over a measured length of the stream. If v
  is the velocity of any float, t the time of passing over a length l,
  then v = l/t. To mark out distinctly the length of stream over which
  the floats pass, two ropes may be stretched across the stream at a
  distance apart, which varies usually from 50 to 250 ft., according to
  the size and rapidity of the river. In the Roorkee experiments a
  length of run of 50 ft. was found best for the central two-fifths of
  the width, and 25 ft. for the remainder, except very close to the
  banks, where the run was made 12½ ft. only. The longer the run the
  less is the proportionate error of the time observations, but on the
  other hand the greater the deviation of the floats from a straight
  course parallel to the axis of the stream. To mark the precise
  position at which the floats cross the ropes, Cunningham used short
  white rope pendants, hanging so as nearly to touch the surface of the
  water. In this case the streams were 80 to 180 ft. in width. In wider
  streams the use of ropes to mark the length of run is impossible, and
  recourse must be had to box sextants or theodolites to mark the path
  of the floats.

  [Illustration: FIG. 136.]

  Let AB (fig. 136) be a measured base line strictly parallel to the
  thread of the stream, and AA1, BB1 lines at right angles to AB marked
  out by ranging rods at A1 and B1. Suppose observers stationed at A and
  B with sextants or theodolites, and let CD be the path of any float
  down stream. As the float approaches AA1, the observer at B keeps it
  on the cross wire of his instrument. The observer at A observes the
  instant of the float reaching the line AA1, and signals to B who then
  reads off the angle ABC. Similarly, as the float approaches BB1, the
  observer at A keeps it in sight, and when signalled to by B reads the
  angle BAD. The data so obtained are sufficient for plotting the path
  of the float and determining the distances AC, BD.

  The time taken by the float in passing over the measured distance may
  be observed by a chronograph, started as the float passes the upper
  rope or line, and stopped when it passes the lower. In Cunningham's
  observations two chronometers were sometimes used, the time of passing
  one end of the run being noted on one, and that of passing the other
  end of the run being noted on the other. The chronometers were
  compared immediately before the observations. In other cases a single
  chronometer was used placed midway of the run. The moment of the
  floats passing the ends of the run was signalled to a time-keeper at
  the chronometer by shouting. It was found quite possible to count the
  chronometer beats to the nearest half second, and in some cases to the
  nearest quarter second.

  [Illustration: FIG. 137.]

  § 137. _Sub-surface Floats._--The velocity at different depths below
  the surface of a stream may be obtained by sub-surface floats, used
  precisely in the same way as surface floats. The most usual
  arrangement is to have a large float, of slightly greater density than
  water, connected with a small and very light surface float. The motion
  of the combined arrangement is not sensibly different from that of the
  large float, and the small surface float enables an observer to note
  the path and velocity of the sub-surface float. The instrument is,
  however, not free from objection. If the large submerged float is made
  of very nearly the same density as water, then it is liable to be
  thrown upwards by very slight eddies in the water, and it does not
  maintain its position at the depth at which it is intended to float.
  On the other hand, if the large float is made sensibly heavier than
  water, the indicating or surface float must be made rather large, and
  then it to some extent influences the motion of the submerged float.
  Fig. 137 shows one form of sub-surface float. It consists of a couple
  of tin plates bent at a right angle and soldered together at the
  angle. This is connected with a wooden ball at the surface by a very
  thin wire or cord. As the tin alone makes a heavy submerged float, it
  is better to attach to the tin float some pieces of wood to diminish
  its weight in water. Fig. 138 shows the form of submerged float used
  by Cunningham. It consists of a hollow metal ball connected to a
  slice of cork, which serves as the surface float.

  [Illustration: FIG. 138.]

  [Illustration: FIG. 139.]

  § 138. _Twin Floats._--Suppose two equal and similar floats (fig. 139)
  connected by a wire. Let one float be a little lighter and the other a
  little heavier than water. Then the velocity of the combined floats
  will be the mean of the surface velocity and the velocity at the depth
  at which the heavier float swims, which is determined by the length of
  the connecting wire. Thus if v_s is the surface velocity and v_d the
  velocity at the depth to which the lower float is sunk, the velocity
  of the combined floats will be

    v = ½(v_s + v_d).

  Consequently, if v is observed, and v_s determined by an experiment
  with a single float,

    v_d = 2v - v_s

  According to Cunningham, the twin float gives better results than the
  sub-surface float.

  [Illustration: FIG. 140.]

  § 139. _Velocity Rods._--Another form of float is shown in fig. 140.
  This consists of a cylindrical rod loaded at the lower end so as to
  float nearly vertical in water. A wooden rod, with a metal cap at the
  bottom in which shot can be placed, answers better than anything else,
  and sometimes the wooden rod is made in lengths, which can be screwed
  together so as to suit streams of different depths. A tuft of cotton
  wool at the top serves to make the float more easily visible. Such a
  rod, so adjusted in length that it sinks nearly to the bed of the
  stream, gives directly the mean velocity of the whole vertical section
  in which it floats.

  § 140. _Revy's Current Meter._--No instrument has been so much used in
  directly determining the velocity of a stream at a given point as the
  screw current meter. Of this there are a dozen varieties at least. As
  an example of the instrument in its simplest form, Revy's meter may be
  selected. This is an ordinary screw meter of a larger size than usual,
  more carefully made, and with its details carefully studied (figs.
  141, 142). It was designed after experience in gauging the great South
  American rivers. The screw, which is actuated by the water, is 6 in.
  in diameter, and is of the type of the Griffiths screw used in ships.
  The hollow spherical boss serves to make the weight of the screw
  sensibly equal to its displacement, so that friction is much reduced.
  On the axis aa of the screw is a worm which drives the counter. This
  consists of two worm wheels g and h fixed on a common axis. The worm
  wheels are carried on a frame attached to the pin l. By means of a
  string attached to l they can be pulled into gear with the worm, or
  dropped out of gear and stopped at any instant. A nut m can be screwed
  up, if necessary, to keep the counter permanently in gear. The worm is
  two-threaded, and the worm wheel g has 200 teeth. Consequently it
  makes one rotation for 100 rotations of the screw, and the number of
  rotations up to 100 is marked by the passage of the graduations on its
  edge in front of a fixed index. The second worm wheel has 196 teeth,
  and its edge is divided into 49 divisions. Hence it falls behind the
  first wheel one division for a complete rotation of the latter. The
  number of hundreds of rotations of the screw are therefore shown by
  the number of divisions on h passed over by an index fixed to g. One
  difficulty in the use of the ordinary screw meter is that particles of
  grit, getting into the working parts, very sensibly alter the
  friction, and therefore the speed of the meter. Revy obviates this by
  enclosing the counter in a brass box with a glass face. This box is
  filled with pure water, which ensures a constant coefficient of
  friction for the rubbing parts, and prevents any mud or grit finding
  its way in. In order that the meter may place itself with the axis
  parallel to the current, it is pivoted on a vertical axis and directed
  by a large vane shown in fig. 142. To give the vane more
  directing power the vertical axis is nearer the screw than in ordinary
  meters, and the vane is larger. A second horizontal vane is attached
  by the screws x, x, the object of which is to allow the meter to rest
  on the ground without the motion of the screw being interfered with.
  The string or wire for starting and stopping the meter is carried
  through the centre of the vertical axis, so that the strain on it may
  not tend to pull the meter oblique to the current. The pitch of the
  screw is about 9 in. The screws at x serve for filling the meter with
  water. The whole apparatus is fixed to a rod (fig. 142), of a length
  proportionate to the depth, or for very great depths it is fixed to a
  weighted bar lowered by ropes, a plan invented by Revy. The instrument
  is generally used thus. The reading of the counter is noted, and it is
  put out of gear. The meter is then lowered into the water to the
  required position from a platform between two boats, or better from a
  temporary bridge. Then the counter is put into gear for one, two or
  five minutes. Lastly, the instrument is raised and the counter again
  read. The velocity is deduced from the number of rotations in unit
  time by the formulae given below. For surface velocities the counter
  may be kept permanently in gear, the screw being started and stopped
  by hand.

  [Illustration: FIG. 141.]

  [Illustration: FIG. 142.]

  § 141. _The Harlacher Current Meter._--In this the ordinary counting
  apparatus is abandoned. A worm drives a worm wheel, which makes an
  electrical contact once for each 100 rotations of the worm. This
  contact gives a signal above water. With this arrangement, a series of
  velocity observations can be made, without removing the instrument
  from the water, and a number of practical difficulties attending the
  accurate starting and stopping of the ordinary counter are entirely
  got rid of. Fig. 143 shows the meter. The worm wheel z makes one
  rotation for 100 of the screw. A pin moving the lever x makes the
  electrical contact. The wires b, c are led through a gas pipe B; this
  also serves to adjust the meter to any required position on the wooden
  rod dd. The rudder or vane is shown at WH. The galvanic current acts
  on the electromagnet m, which is fixed in a small metal box containing
  also the battery. The magnet exposes and withdraws a coloured disk at
  an opening in the cover of the box.

  § 142. _Amsler Laffon Current Meter._--A very convenient and accurate
  current meter is constructed by Amsler Laffon of Schaffhausen. This
  can be used on a rod, and put into and out of gear by a ratchet. The
  peculiarity in this case is that there is a double ratchet, so that
  one pull on the string puts the counter into gear and a second puts it
  out of gear. The string may be slack during the action of the meter,
  and there is less uncertainty than when the counter has to be held in
  gear. For deep streams the meter A is suspended by a wire with a heavy
  lenticular weight below (fig. 144). The wire is payed out from a small
  winch D, with an index showing the depth of the meter, and passes over
  a pulley B. The meter is in gimbals and is directed by a conical
  rudder which keeps it facing the stream with its axis horizontal.
  There is an electric circuit from a battery C through the meter, and a
  contact is made closing the circuit every 100 revolutions. The moment
  the circuit closes a bell rings. By a subsidiary arrangement, when the
  foot of the instrument, 0.3 metres below the axis of the meter,
  touches the ground the circuit is also closed and the bell rings. It
  is easy to distinguish the continuous ring when the ground is reached
  from the short ring when the counter signals. A convenient winch for
  the wire is so graduated that if set when the axis of the meter is at
  the water surface it indicates at any moment the depth of the meter
  below the surface. Fig. 144 shows the meter as used on a boat. It is a
  very convenient instrument for obtaining the velocity at different
  depths and can also be used as a sounding instrument.

  [Illustration: FIG. 143.]

  § 143. _Determination of the Coefficients of the Current
  Meter._--Suppose a series of observations has been made by towing the
  meter in still water at different speeds, and that it is required to
  ascertain from these the constants of the meter. If v is the velocity
  of the water and n the observed number of rotations per second, let

    v = [alpha] + [beta]n   (1)

  where [alpha] and [beta] are constants. Now let the meter be towed
  over a measured distance L, and let N be the revolutions of the meter
  and t the time of transit. Then the speed of the meter relatively to
  the water is L/t = v feet per second, and the number of revolutions
  per second is N/t = n. Suppose m observations have been made in this
  way, furnishing corresponding values of v and n, the speed in each
  trial being as uniform as possible,

    [Sigma]n = n1 + n2 + ...

    [Sigma]v = v1 + v2 + ...

    [Sigma]nv = n1v1 + n2v2 + ...

    [Sigma]n² = n1² + n2² + ...

    [[Sigma]n]² = [n1 + n2 + ...]²

  Then for the determination of the constants [alpha] and [beta] in (1),
  by the method of least squares--

              [Sigma]n²[Sigma]v - [Sigma]n[Sigma]nv
    [alpha] = -------------------------------------,
                     m[Sigma]n² - [[Sigma]n]²

             m[Sigma]nv - [Sigma]v[Sigma]n
    [beta] = -----------------------------.
                m[Sigma]n² - [[Sigma]n]²

  [Illustration: FIG. 144.]

  In a few cases the constants for screw current meters have been
  determined by towing them in R. E. Froude's experimental tank in which
  the resistance of ship models is ascertained. In that case the data
  are found with exceptional accuracy.

  § 144. Darcy Gauge or modified Pitot Tube.--A very old instrument for
  measuring velocities, invented by Henri Pitot in 1730 (_Histoire de
  l'Académie des Sciences_, 1732, p. 376), consisted simply of a
  vertical glass tube with a right-angled bend, placed so that its mouth
  was normal to the direction of flow (fig. 145).

  [Illustration: FIG. 145.]

  The impact of the stream on the mouth of the tube balances a column in
  the tube, the height of which is approximately h = v²/2g, where v is
  the velocity at the depth x. Placed with its mouth parallel to the
  stream the water inside the tube is nearly at the same level as the
  surface of the stream, and turned with the mouth down stream, the
  fluid sinks a depth h´ = v²/2g nearly, though the tube in that case
  interferes with the free flow of the liquid and somewhat modifies the
  result. Pitot expanded the mouth of the tube so as to form a funnel or
  bell mouth. In that case he found by experiment

    h = 1.5v²/2g.

  But there is more disturbance of the stream. Darcy preferred to make
  the mouth of the tube very small to avoid interference with the
  stream and to check oscillations of the water column. Let the
  difference of level of a pair of tubes A and B (fig. 145) be taken to
  be h = kv²/2g, then k may be taken to be a corrective coefficient
  whose value in well-shaped instruments is very nearly unity. By
  placing his instrument in front of a boat towed through water Darcy
  found k = 1.034; by placing the instrument in a stream the velocity of
  which had been ascertained by floats, he found k = 1.006; by readings
  taken in different parts of the section of a canal in which a known
  volume of water was flowing, he found k = 0.993. He believed the first
  value to be too high in consequence of the disturbance caused by the
  boat. The mean of the other two values is almost exactly unity
  (_Recherches hydrauliques_, Darcy and Bazin, 1865, p. 63). W. B.
  Gregory used somewhat differently formed Pitot tubes for which the k =
  1 (_Am. Soc. Mech. Eng._, 1903, 25). T. E. Stanton used a Pitot tube
  in determining the velocity of an air current, and for his instrument
  he found k = 1.030 to k = 1.032 ("On the Resistance of Plane Surfaces
  in a Current of Air," _Proc. Inst. Civ. Eng._, 1904, 156).

  One objection to the Pitot tube in its original form was the great
  difficulty and inconvenience of reading the height h in the immediate
  neighbourhood of the stream surface. This is obviated in the Darcy
  gauge, which can be removed from the stream to be read.

  Fig. 146 shows a Darcy gauge. It consists of two Pitot tubes having
  their mouths at right angles. In the instrument shown, the two tubes,
  formed of copper in the lower part, are united into one for strength,
  and the mouths of the tubes open vertically and horizontally. The
  upper part of the tubes is of glass, and they are provided with a
  brass scale and two verniers b, b. The whole instrument is supported
  on a vertical rod or small pile AA, the fixing at B permitting the
  instrument to be adjusted to any height on the rod, and at the same
  time allowing free rotation, so that it can be held parallel to the
  current. At c is a two-way cock, which can be opened or closed by
  cords. If this is shut, the instrument can be lifted out of the stream
  for reading. The glass tubes are connected at top by a brass fixing,
  with a stop cock a, and a flexible tube and mouthpiece m. The use of
  this is as follows. If the velocity is required at a point near the
  surface of the stream, one at least of the water columns would be
  below the level at which it could be read. It would be in the copper
  part of the instrument. Suppose then a little air is sucked out by the
  tube m, and the cock a closed, the two columns will be forced up an
  amount corresponding to the difference between atmospheric pressure
  and that in the tubes. But the difference of level will remain
  unaltered.

  When the velocities to be measured are not very small, this instrument
  is an admirable one. It requires observation only of a single linear
  quantity, and does not require any time observation. The law
  connecting the velocity and the observed height is a rational one, and
  it is not absolutely necessary to make any experiments on the
  coefficient of the instrument. If we take v = k[root](2gh), then it
  appears from Darcy's experiments that for a well-formed instrument k
  does not sensibly differ from unity. It gives the velocity at a
  definite point in the stream. The chief difficulty arises from the
  fact that at any given point in a stream the velocity is not
  absolutely constant, but varies a little from moment to moment. Darcy
  in some of his experiments took several readings, and deduced the
  velocity from the mean of the highest and lowest.

  § 145. _Perrodil Hydrodynamometer._--This consists of a frame abcd
  (fig. 147) placed vertically in the stream, and of a height not less
  than the stream's depth. The two vertical members of this frame are
  connected by cross bars, and united above water by a circular bar,
  situated in the vertical plane and carrying a horizontal graduated
  circle ef. This whole system is movable round its axis, being
  suspended on a pivot at g connected with the fixed support mn. Other
  horizontal arms serve as guides. The central vertical rod gr forms a
  torsion rod, being fixed at r to the frame abcd, and, passing freely
  upwards through the guides, it carries a horizontal needle moving
  over the graduated circle ef. The support g, which carries the
  apparatus, also receives in a tubular guide the end of the torsion rod
  gr and a set screw for fixing the upper end of the torsion rod when
  necessary. The impulse of the stream of water is received on a
  circular disk x, in the plane of the torsion rod and the frame abcd.
  To raise and lower the apparatus easily, it is not fixed directly to
  the rod mn, but to a tube kl sliding on mn.

  [Illustration: FIG. 146.]

  Suppose the apparatus arranged so that the disk x is at that level in
  the stream where the velocity is to be determined. The plane abcd is
  placed parallel to the direction of motion of the water. Then the disk
  x (acting as a rudder) will place itself parallel to the stream on the
  down stream side of the frame. The torsion rod will be unstrained, and
  the needle will be at zero on the graduated circle. If, then, the
  instrument is turned by pressing the needle, till the plane abcd of
  the disk and the zero of the graduated circle is at right angles to
  the stream, the torsion rod will be twisted through an angle which
  measures the normal impulse of the stream on the disk x. That angle
  will be given by the distance of the needle from zero. Observation
  shows that the velocity of the water at a given point is not constant.
  It varies between limits more or less wide. When the apparatus is
  nearly in its right position, the set screw at g is made to clamp the
  torsion spring. Then the needle is fixed, and the apparatus carrying
  the graduated circle oscillates. It is not, then, difficult to note
  the mean angle marked by the needle.

  [Illustration: FIG. 147.]

  Let r be the radius of the torsion rod, l its length from the needle
  over ef to r, and [alpha] the observed torsion angle. Then the moment
  of the couple due to the molecular forces in the torsion rod is

    M = E_t I[alpha]/l;

  where E_t is the modulus of elasticity for torsion, and I the polar
  moment of inertia of the section of the rod. If the rod is of circular
  section, I = ½[pi]r^4. Let R be the radius of the disk, and b its
  leverage, or the distance of its centre from the axis of the torsion
  rod. The moment of the pressure of the water on the disk is

    Fb = kb(G/2g)[pi]R²v²,

  where G is the heaviness of water and k an experimental coefficient.
  Then

    E_t I[alpha]/l = kb(G/2g)[pi]R²v².

  For any given instrument,

    v = c [root][alpha],

  where c is a constant coefficient for the instrument.

  The instrument as constructed had three disks which could be used at
  will. Their radii and leverages were in feet

                  R =      b =

    1st disk     0.052     0.16
    2nd   "      0.105     0.32
    3rd   "      0.210     0.66

  For a thin circular plate, the coefficient k = 1.12. In the actual
  instrument the torsion rod was a brass wire 0.06 in. diameter and 6½
  ft. long. Supposing [alpha] measured in degrees, we get by calculation

    v = 0.335 [root][alpha]; 0.115 [root][alpha]; 0.042 [root][alpha].

  Very careful experiments were made with the instrument. It was fixed
  to a wooden turning bridge, revolving over a circular channel of 2 ft.
  width, and about 76 ft. circumferential length. An allowance was made
  for the slight current produced in the channel. These experiments gave
  for the coefficient c, in the formula v = c [root][alpha],

    1st disk, c =  0.3126 for velocities of 3 to 16  ft.
    2nd   "        0.1177  "       "        1¼ to 3¼  "
    3rd   "        0.0349  "       "    less than 1¼  "

  The instrument is preferable to the current meter in giving the
  velocity in terms of a single observed quantity, the angle of torsion,
  while the current meter involves the observation of two quantities,
  the number of rotations and the time. The current meter, except in
  some improved forms, must be withdrawn from the water to read the
  result of each experiment, and the law connecting the velocity and
  number of rotations of a current meter is less well-determined than
  that connecting the pressure on a disk and the torsion of the wire of
  a hydrodynamometer.

  The Pitot tube, like the hydrodynamometer, does not require a time
  observation. But, where the velocity is a varying one, and
  consequently the columns of water in the Pitot tube are oscillating,
  there is room for doubt as to whether, at any given moment of closing
  the cock, the difference of level exactly measures the impulse of the
  stream at the moment. The Pitot tube also fails to give measurable
  indications of very low velocities.


  PROCESSES FOR GAUGING STREAMS

  § 146. _Gauging by Observation of the Maximum Surface Velocity._--The
  method of gauging which involves the least trouble is to determine the
  surface velocity at the thread of the stream, and to deduce from it
  the mean velocity of the whole cross section. The maximum surface
  velocity may be determined by floats or by a current meter.
  Unfortunately the ratio of the maximum surface to the mean velocity is
  extremely variable. Thus putting v_o for the surface velocity at the
  thread of the stream, and v_m for the mean velocity of the whole cross
  section, v_m/v_o has been found to have the following values:--

                                                     v_m/v_o

    De Prony, experiments on small wooden channels   0.8164
    Experiments on the Seine                         0.62
    Destrem and De Prony, experiments on the Neva    0.78
    Boileau, experiments on canals                   0.82
    Baumgartner, experiments on the Garonne          0.80
    Brünings (mean)                                  0.85
    Cunningham, Solani aqueduct                      0.823

  Various formulae, either empirical or based on some theory of the
  vertical and horizontal velocity curves, have been proposed for
  determining the ratio v_m/v_o. Bazin found from his experiments the
  empirical expression

    v_m = v_o - 25.4 [root](mi);

  where m is the hydraulic mean depth and i the slope of the stream.

  In the case of irrigation canals and rivers, it is often important to
  determine the discharge either daily or at other intervals of time,
  while the depth and consequently the mean velocity is varying.
  Cunningham (_Roorkee Prof. Papers_, iv. 47), has shown that, for a
  given part of such a stream, where the bed is regular and of permanent
  section, a simple formula may be found for the variation of the
  central surface velocity with the depth. When once the constants of
  this formula have been determined by measuring the central surface
  velocity and depth, in different conditions of the stream, the surface
  velocity can be obtained by simply observing the depth of the stream,
  and from this the mean velocity and discharge can be calculated. Let z
  be the depth of the stream, and v_o the surface velocity, both measured
  at the thread of the stream. Then v_o² = cz; where c is a constant
  which for the Solani aqueduct had the values 1.9 to 2, the depths
  being 6 to 10 ft., and the velocities 3½ to 4½ ft. Without any
  assumption of a formula, however, the surface velocities, or still
  better the mean velocities, for different conditions of the stream may
  be plotted on a diagram in which the abscissae are depths and the
  ordinates velocities. The continuous curve through points so found
  would then always give the velocity for any observed depth of the
  stream, without the need of making any new float or current meter
  observations.

  § 147. _Mean Velocity determined by observing a Series of Surface
  Velocities._--The ratio of the mean velocity to the surface velocity
  in one longitudinal section is better ascertained than the ratio of
  the central surface velocity to the mean velocity of the whole cross
  section. Suppose the river divided into a number of compartments by
  equidistant longitudinal planes, and the surface velocity observed in
  each compartment. From this the mean velocity in each compartment and
  the discharge can be calculated. The sum of the partial discharges
  will be the total discharge of the stream. When wires or ropes can be
  stretched across the stream, the compartments can be marked out by
  tags attached to them. Suppose two such ropes stretched across the
  stream, and floats dropped in above the upper rope. By observing
  within which compartment the path of the float lies, and noting the
  time of transit between the ropes, the surface velocity in each
  compartment can be ascertained. The mean velocity in each compartment
  is 0.85 to 0.91 of the surface velocity in that compartment. Putting k
  for this ratio, and v1, v2 ... for the observed velocities, in
  compartments of area [Omega]1, [Omega]2 ... then the total discharge
  is

    Q = k([Omega]1v1 + [Omega]2v2 + ... ).

  If several floats are allowed to pass over each compartment, the mean
  of all those corresponding to one compartment is to be taken as the
  surface velocity of that compartment.

  [Illustration: FIG. 148.]

  This method is very applicable in the case of large streams or rivers
  too wide to stretch a rope across. The paths of the floats are then
  ascertained in this way. Let fig. 148 represent a portion of the
  river, which should be straight and free from obstructions. Suppose a
  base line AB measured parallel to the thread of the stream, and let
  the mean cross section of the stream be ascertained either by sounding
  the terminal cross sections AE, BF, or by sounding a series of
  equidistant cross sections. The cross sections are taken at right
  angles to the base line. Observers are placed at A and B with
  theodolites or box sextants. The floats are dropped in from a boat
  above AE, and picked up by another boat below BF. An observer with a
  chronograph or watch notes the time in which each float passes from AE
  to BF. The method of proceeding is this. The observer A sets his
  theodolite in the direction AE, and gives a signal to drop a float. B
  keeps his instrument on the float as it comes down. At the moment the
  float arrives at C in the line AE, the observer at A calls out. B
  clamps his instrument and reads off the angle ABC, and the time
  observer begins to note the time of transit. B now points his
  instrument in the direction BF, and A keeps the float on the cross
  wire of his instrument. At the moment the float arrives at D in the
  line BF, the observer B calls out, A clamps his instrument and reads
  off the angle BAD, and the time observer notes the time of transit
  from C to D. Thus all the data are determined for plotting the path CD
  of the float and determining its velocity. By dropping in a series of
  floats, a number of surface velocities can be determined. When all
  these have been plotted, the river can be divided into convenient
  compartments. The observations belonging to each compartment are then
  averaged, and the mean velocity and discharge calculated. It is
  obvious that, as the surface velocity is greatly altered by wind,
  experiments of this kind should be made in very calm weather.

  The ratio of the surface velocity to the mean velocity in the same
  vertical can be ascertained from the formulae for the vertical
  velocity curve already given (§ 101). Exner, in _Erbkam's Zeitschrift_
  for 1875, gave the following convenient formula. Let v be the mean and
  V the surface velocity in any given vertical longitudinal section, the
  depth of which is h

    v/V = (1 + 0.1478 [root]h)/(1 + 0.2216 [root]h).

  If vertical velocity rods are used instead of common floats, the mean
  velocity is directly determined for the vertical section in which the
  rod floats. No formula of reduction is then necessary. The observed
  velocity has simply to be multiplied by the area of the compartment to
  which it belongs.

  § 148. _Mean Velocity of the Stream from a Series of Mid Depth
  Velocities._--In the gaugings of the Mississippi it was found that the
  mid depth velocity differed by only a very small quantity from the
  mean velocity in the vertical section, and it was uninfluenced by
  wind. If therefore a series of mid depth velocities are determined by
  double floats or by a current meter, they may be taken to be the mean
  velocities of the compartments in which they occur, and no formula of
  reduction is necessary. If floats are used, the method is precisely
  the same as that described in the last paragraph for surface floats.
  The paths of the double floats are observed and plotted, and the mean
  taken of those corresponding to each of the compartments into which
  the river is divided. The discharge is the sum of the products of the
  observed mean mid depth velocities and the areas of the compartments.

  § 149. _P. P. Boileau's Process for Gauging Streams._--Let U be the
  mean velocity at a given section of a stream, V the maximum velocity,
  or that of the principal filament, which is generally a little below
  the surface, W and w the greatest and least velocities at the surface.
  The distance of the principal filament from the surface is generally
  less than one-fourth of the depth of the stream; W is a little less
  than V; and U lies between W and w. As the surface velocities change
  continuously from the centre towards the sides there are at the
  surface two filaments having a velocity equal to U. The determination
  of the position of these filaments, which Boileau terms the gauging
  filaments, cannot be effected entirely by theory. But, for sections of
  a stream in which there are no abrupt changes of depth, their position
  can be very approximately assigned. Let [Delta] and l be the
  horizontal distances of the surface filament, having the velocity W,
  from the gauging filament, which has the velocity U, and from the bank
  on one side. Then

    [Delta]/l = c^4 [root]{(W + 2w)/7(W - w)},

  c being a numerical constant. From gaugings by Humphreys and Abbot,
  Bazin and Baumgarten, the values c = 0.919, 0.922 and 0.925 are
  obtained. Boileau adopts as a mean value 0.922. Hence, if W and w are
  determined by float gauging or otherwise, [Delta] can be found, and
  then a single velocity observation at [Delta] ft. from the filament of
  maximum velocity gives, without need of any reduction, the mean
  velocity of the stream. More conveniently W, w, and U can be measured
  from a horizontal surface velocity curve, obtained from a series of
  float observations.

  § 150. _Direct Determination of the Mean Velocity by a Current Meter
  or Darcy Gauge._--The only method of determining the mean velocity at
  a cross section of a stream which involves no assumption of the ratio
  of the mean velocity to other quantities is this--a plank bridge is
  fixed across the stream near its surface. From this, velocities are
  observed at a sufficient number of points in the cross section of the
  stream, evenly distributed over its area. The mean of these is the
  true mean velocity of the stream. In Darcy and Bazin's experiments on
  small streams, the velocity was thus observed at 36 points in the
  cross section.

  When the stream is too large to fix a bridge across it, the
  observations may be taken from a boat, or from a couple of boats with
  a gangway between them, anchored successively at a series of points
  across the width of the stream. The position of the boat for each
  series of observations is fixed by angular observations to a base line
  on shore.

  [Illustration: FIG. 149.]

  § 151. _A. R. Harlacher's Graphic Method of determining the Discharge
  from a Series of Current Meter Observations._--Let ABC (fig. 149) be
  the cross section of a river at which a complete series of current
  meter observations have been taken. Let I., II., III ... be the
  verticals at different points of which the velocities were measured.
  Suppose the depths at I., II., III., ... (fig. 149), set off as
  vertical ordinates in fig. 150, and on these vertical ordinates
  suppose the velocities set off horizontally at their proper depths.
  Thus, if v is the measured velocity at the depth h from the surface in
  fig. 149, on vertical marked III., then at III. in fig. 150 take cd =
  h and ac = v. Then d is a point in the vertical velocity curve for the
  vertical III., and, all the velocities for that ordinate being
  similarly set off, the curve can be drawn. Suppose all the vertical
  velocity curves I.... V. (fig. 150), thus drawn. On each of these
  figures draw verticals corresponding to velocities of x, 2x, 3x ...
  ft. per second. Then for instance cd at III. (fig. 150) is the depth
  at which a velocity of 2x ft. per second existed on the vertical III.
  in fig. 149 and if cd is set off at III. in fig. 149 it gives a point
  in a curve passing through points of the section where the velocity
  was 2x ft. per second. Set off on each of the verticals in fig. 149
  all the depths thus found in the corresponding diagram in fig. 150.
  Curves drawn through the corresponding points on the verticals are
  curves of equal velocity.

  [Illustration: FIG. 150.]

  The discharge of the stream per second may be regarded as a solid
  having the cross section of the river (fig. 149) as a base, and cross
  sections normal to the plane of fig. 149 given by the diagrams in fig.
  150. The curves of equal velocity may therefore be considered as
  contour lines of the solid whose volume is the discharge of the stream
  per second. Let [Omega]0 be the area of the cross section of the
  river, [Omega]1, [Omega]2 ... the areas contained by the successive
  curves of equal velocity, or, if these cut the surface of the stream,
  by the curves and that surface. Let x be the difference of velocity
  for which the successive curves are drawn, assumed above for
  simplicity at 1 ft. per second. Then the volume of the successive
  layers of the solid body whose volume represents the discharge,
  limited by successive planes passing through the contour curves, will
  be

    ½x([Omega]0 + [Omega]1), ½x([Omega]1 + [Omega]2), and so on.

  Consequently the discharge is

    Q = x{½([Omega]0 + [Omega]_n) + [Omega]1 = [Omega]2 + ... + [Omega](n-1)}.

  The areas [Omega]0, [Omega]1 ... are easily ascertained by means of
  the polar planimeter. A slight difficulty arises in the part of the
  solid lying above the last contour curve. This will have generally a
  height which is not exactly x, and a form more rounded than the other
  layers and less like a conical frustum. The volume of this may be
  estimated separately, and taken to be the area of its base (the area
  [Omega]_n) multiplied by 1/3 to ½ its height.

  [Illustration: FIG. 151.]

  Fig. 151 shows the results of one of Harlacher's gaugings worked out
  in this way. The upper figure shows the section of the river and the
  positions of the verticals at which the soundings and gaugings were
  taken. The lower gives the curves of equal velocity, worked out from
  the current meter observations, by the aid of vertical velocity
  curves. The vertical scale in this figure is ten times as great as in
  the other. The discharge calculated from the contour curves is 14.1087
  cubic metres per second. In the lower figure some other interesting
  curves are drawn. Thus, the uppermost dotted curve is the curve
  through points at which the maximum velocity was found; it shows that
  the maximum velocity was always a little below the surface, and at a
  greater depth at the centre than at the sides. The next curve shows
  the depth at which the mean velocity for each vertical was found. The
  next is the curve of equal velocity corresponding to the mean velocity
  of the stream; that is, it passes through points in the cross section
  where the velocity was identical with the mean velocity of the stream.


HYDRAULIC MACHINES

§ 152. Hydraulic machines may be broadly divided into two classes: (1)
_Motors_, in which water descending from a higher to a lower level, or
from a higher to a lower pressure, gives up energy which is available
for mechanical operations; (2) _Pumps_, in which the energy of a steam
engine or other motor is expended in raising water from a lower to a
higher level. A few machines such as the ram and jet pump combine the
functions of motor and pump. It may be noted that constructively pumps
are essentially reversed motors. The reciprocating pump is a reversed
pressure engine, and the centrifugal pump a reversed turbine. Hydraulic
machine tools are in principle motors combined with tools, and they now
form an important special class.

Water under pressure conveyed in pipes is a convenient and economical
means of transmitting energy and distributing it to many scattered
working points. Hence large and important hydraulic systems are adopted
in which at a central station water is pumped at high pressure into
distributing mains, which convey it to various points where it actuates
hydraulic motors operating cranes, lifts, dock gates, and in some cases
riveting and shearing machines. In this case the head driving the
hydraulic machinery is artificially created, and it is the convenience of
distributing power in an easily applied form to distant points which
makes the system advantageous. As there is some unavoidable loss in
creating an artificial head this system is most suitable for driving
machines which work intermittently (see POWER TRANSMISSION). The
development of electrical methods of transmitting and distributing energy
has led to the utilization of many natural waterfalls so situated as to
be useless without such a means of transferring the power to points where
it can be conveniently applied. In some cases, as at Niagara, the
hydraulic power can only be economically developed in very large units,
and it can be most conveniently subdivided and distributed by
transformation into electrical energy. Partly from the development of new
industries such as paper-making from wood pulp and electro-metallurgical
processes, which require large amounts of cheap power, partly from the
facility with which energy can now be transmitted to great distances
electrically, there has been a great increase in the utilization of
water-power in countries having natural waterfalls. According to the
twelfth census of the United States the total amount of water-power
reported as used in manufacturing establishments in that country was
1,130,431 h.p. in 1870; 1,263,343 h.p. in 1890; and 1,727,258 h.p. in
1900. The increase was 8.4% in the decade 1870-1880, 3.1% in 1880-1890,
and no less than 36.7% in 1890-1900. The increase is the more striking
because in this census the large amounts of hydraulic power which are
transmitted electrically are not included.


  XII. IMPACT AND REACTION OF WATER

  § 153. When a stream of fluid in steady motion impinges on a solid
  surface, it presses on the surface with a force equal and opposite to
  that by which the velocity and direction of motion of the fluid are
  changed. Generally, in problems on the impact of fluids, it is
  necessary to neglect the effect of friction between the fluid and the
  surface on which it moves.

  _During Impact the Velocity of the Fluid relatively to the Surface on
  which it impinges remains unchanged in Magnitude._--Consider a mass of
  fluid flowing in contact with a solid surface also in motion, the
  motion of both fluid and solid being estimated relatively to the
  earth. Then the motion of the fluid may be resolved into two parts,
  one a motion equal to that of the solid, and in the same direction,
  the other a motion relatively to the solid. The motion which the fluid
  has in common with the solid cannot at all be influenced by the
  contact. The relative component of the motion of the fluid can only be
  altered in direction, but not in magnitude. The fluid moving in
  contact with the surface can only have a relative motion parallel to
  the surface, while the pressure between the fluid and solid, if
  friction is neglected, is normal to the surface. The pressure
  therefore can only deviate the fluid, without altering the magnitude
  of the relative velocity. The unchanged common component and, combined
  with it, the deviated relative component give the resultant final
  velocity, which may differ greatly in magnitude and direction from the
  initial velocity.

  From the principle of momentum, the impulse of any mass of fluid
  reaching the surface in any given time is equal to the change of
  momentum estimated in the same direction. The pressure between the
  fluid and surface, in any direction, is equal to the change of
  momentum in that direction of so much fluid as reaches the surface in
  one second. If P_a is the pressure in any direction, m the mass of
  fluid impinging per second, v_a the change of velocity in the
  direction of P_a due to impact, then

    P_a = mv_a.

  If v1 (fig. 152) is the velocity and direction of motion before
  impact, v2 that after impact, then v is the total change of motion due
  to impact. The resultant pressure of the fluid on the surface is in
  the direction of v, and is equal to v multiplied by the mass impinging
  per second. That is, putting P for the resultant pressure,

    P = mv.

  Let P be resolved into two components, N and T, normal and tangential
  to the direction of motion of the solid on which the fluid impinges.
  Then N is a lateral force producing a pressure on the supports of the
  solid, T is an effort which does work on the solid. If u is the
  velocity of the solid, Tu is the work done per second by the fluid in
  moving the solid surface.

  [Illustration: FIG. 152.]

  Let Q be the volume, and GQ the weight of the fluid impinging per
  second, and let v1 be the initial velocity of the fluid before
  striking the surface. Then GQv1²/2g is the original kinetic energy of
  Q cub. ft. of fluid, and the efficiency of the stream considered as an
  arrangement for moving the solid surface is

    [eta] = Tu/(GQv1²/2g).

  § 154. _Jet deviated entirely in one Direction.--Geometrical Solution_
  (fig. 153).--Suppose a jet of water impinges on a surface ac with a
  velocity ab, and let it be wholly deviated in planes parallel to the
  figure. Also let ae be the velocity and direction of motion of the
  surface. Join eb; then the water moves with respect to the surface in
  the direction and with the velocity eb. As this relative velocity is
  unaltered by contact with the surface, take cd = eb, tangent to the
  surface at c, then cd is the relative motion of the water with respect
  to the surface at c. Take df equal and parallel to ae. Then fc
  (obtained by compounding the relative motion of water to surface and
  common velocity of water and surface) is the absolute velocity and
  direction of the water leaving the surface. Take ag equal and parallel
  to fc. Then, since ab is the initial and ag the final velocity and
  direction of motion, gb is the total change of motion of the water.
  The resultant pressure on the plane is in the direction gb. Join eg.
  In the triangle gae, ae is equal and parallel to df, and ag to fc.
  Hence eg is equal and parallel to cd. But cd = eb = relative motion of
  water and surface. Hence the change of motion of the water is
  represented in magnitude and direction by the third side of an
  isosceles triangle, of which the other sides are equal to the relative
  velocity of the water and surface, and parallel to the initial and
  final directions of relative motion.

  [Illustration: FIG. 153.]


  SPECIAL CASES

  § 155. (1) _A Jet impinges on a plane surface at rest, in a direction
  normal to the plane_ (fig. 154).--Let a jet whose section is [omega]
  impinge with a velocity v on a plane surface at rest, in a direction
  normal to the plane. The particles approach the plane, are gradually
  deviated, and finally flow away parallel to the plane, having then no
  velocity in the original direction of the jet. The quantity of water
  impinging per second is [omega]v. The pressure on the plane, which is
  equal to the change of momentum per second, is P = (G/g)[omega]v².

  [Illustration: FIG. 154.]

  (2) _If the plane is moving in the direction of the jet with the
  velocity_ ±u, the quantity impinging per second is [omega](v ± u).
  The momentum of this quantity before impact is (G/g)[omega](v ± u)v.
  After impact, the water still possesses the velocity ±u in the
  direction of the jet; and the momentum, in that direction, of so much
  water as impinges in one second, after impact, is
  ±(G/g)[omega](v ± u)u. The pressure on the plane, which is the change
  of momentum per second, is the difference of these quantities or P =
  (G/g)[omega](v ± u)². This differs from the expression obtained in
  the previous case, in that the relative velocity of the water and
  plane v ± u is substituted for v. The expression may be written P = 2
  × G × [omega](v ± u)²/2g, where the last two terms are the volume of
  a prism of water whose section is the area of the jet and whose length
  is the head due to the relative velocity. The pressure on the plane is
  twice the weight of that prism of water. The work done when the plane
  is moving in the same direction as the jet is Pu = (G/g)[omega](v -
  u)²u foot-pounds per second. There issue from the jet [omega]v cub.
  ft. per second, and the energy of this quantity before impact is
  (G/2g)[omega]v³. The efficiency of the jet is therefore [eta] = 2(v -
  u)²u/v³. The value of u which makes this a maximum is found by
  differentiating and equating the differential coefficient to zero:--

    d[eta]/du = 2(v² - 4vu + 3u²)/v³ = 0;

    .: u = v or (1/3)v.

  The former gives a minimum, the latter a maximum efficiency.

  Putting u = (1/3)v in the expression above,

    [eta] max. = 8/27.

  (3) If, instead of one plane moving before the jet, a series of planes
  are introduced at short intervals at the same point, the quantity of
  water impinging on the series will be [omega]v instead of [omega](v -
  u), and the whole pressure = (G/g)[omega]v(v - u). The work done is
  (G/g)[omega]vu(v - u). The efficiency [eta] = (G/g)[omega]vu(v - u) ÷
  (G/2g)[omega]v³ = 2u(v - u)/v². This becomes a maximum for d[eta]/du =
  2(v - 2u) = 0, or u = ½v, and the [eta] = ½. This result is often used
  as an approximate expression for the velocity of greatest efficiency
  when a jet of water strikes the floats of a water wheel. The work
  wasted in this case is half the whole energy of the jet when the
  floats run at the best speed.

  § 156. (4) _Case of a Jet impinging on a Concave Cup Vane_, velocity
  of water v, velocity of vane in the same direction u (fig. 155),
  weight impinging per second = Gw(v - u).

  [Illustration: FIG. 155.]

  If the cup is hemispherical, the water leaves the cup in a direction
  parallel to the jet. Its relative velocity is v - u when approaching
  the cup, and -(v - u) when leaving it. Hence its absolute velocity
  when leaving the cup is u - (v - u) = 2u - v. The change of momentum
  per second = (G/g)[omega](v - u) {v - (2u - v)} = 2(G/g)[omega](v -
  u)². Comparing this with case 2, it is seen that the pressure on a
  hemispherical cup is double that on a flat plane. The work done on the
  cup = 2(G/g)[omega] (v - u)²u foot-pounds per second. The efficiency
  of the jet is greatest when v = 3u; in that case the efficiency =
  {16/27}.

  If a series of cup vanes are introduced in front of the jet, so that
  the quantity of water acted upon is [omega]v instead of [omega](v -
  u), then the whole pressure on the chain of cups is (G/g)[omega]v{v -
  (2u - v)} = 2(G/g)[omega]v(v - u). In this case the efficiency is
  greatest when v = 2u, and the maximum efficiency is unity, or all the
  energy of the water is expended on the cups.

  [Illustration: FIG. 156.]

  § 157. (5) _Case of a Flat Vane oblique to the Jet_ (fig. 156).--This
  case presents some difficulty. The water spreading on the plane in all
  directions from the point of impact, different particles leave the
  plane with different absolute velocities. Let AB = v = velocity of
  water, AC = u = velocity of plane. Then, completing the parallelogram,
  AD represents in magnitude and direction the relative velocity of
  water and plane. Draw AE normal to the plane and DE parallel to the
  plane. Then the relative velocity AD may be regarded as consisting of
  two components, one AE normal, the other DE parallel to the plane. On
  the assumption that friction is insensible, DE is unaffected by
  impact, but AE is destroyed. Hence AE represents the entire change of
  velocity due to impact and the direction of that change. The pressure
  on the plane is in the direction AE, and its amount is = mass of water
  impinging per second × AE.

  Let DAE = [theta], and let AD = v_r. Then AE = v_r cos [theta]; DE =
  v_r sin [theta]. If Q is the volume of water impinging on the plane
  per second, the change of momentum is (G/g)Qv_r cos [theta]. Let AC =
  u = velocity of the plane, and let AC make the angle CAE = [delta]
  with the normal to the plane. The velocity of the plane in the
  direction AE = u cos [delta]. The work of the jet on the plane =
  (G/g)Qv_r cos [theta] u cos [delta]. The same problem may be thus
  treated algebraically (fig. 157). Let BAF = [alpha], and CAF =
  [delta]. The velocity v of the water may be decomposed into AF = v cos
  [alpha] normal to the plane, and FB = v sin [alpha] parallel to the
  plane. Similarly the velocity of the plane = u = AC = BD can be
  decomposed into BG = FE = u cos [delta] normal to the plane, and DG =
  u sin [delta] parallel to the plane. As friction is neglected, the
  velocity of the water parallel to the plane is unaffected by the
  impact, but its component v cos [alpha] normal to the plane becomes
  after impact the same as that of the plane, that is, u cos [delta].
  Hence the change of velocity during impact = AE = v cos [alpha] - u
  cos [delta]. The change of momentum per second, and consequently the
  normal pressure on the plane is N = (G/g) Q(v cos [alpha] - u cos
  [delta]). The pressure in the direction in which the plane is moving
  is P = N cos [delta] = (G/g)Q (v cos [alpha] - u cos [delta]) cos
  [delta], and the work done on the plane is Pu = (G/g)Q(v cos [alpha] -
  u cos [delta]) u cos [delta], which is the same expression as before,
  since AE = v_r cos [theta] = v cos [alpha] - u cos [delta].

  [Illustration: FIG. 157.]

  [Illustration: FIG. 158.]

  In one second the plane moves so that the point A (fig. 158) comes to
  C, or from the position shown in full lines to the position shown in
  dotted lines. If the plane remained stationary, a length AB = v of the
  jet would impinge on the plane, but, since the plane moves in the same
  direction as the jet, only the length HB = AB - AH impinges on the
  plane.

  But AH = AC cos [delta]/ cos [alpha] = u cos [delta]/ cos [alpha], and
  therefore HB = v - u cos [delta]/ cos [alpha]. Let [omega] = sectional
  area of jet; volume impinging on plane per second = Q = [omega](v - u
  cos [delta]/cos [alpha]) = [omega](v cos [alpha] - u cos [delta])/ cos
  [alpha]. Inserting this in the formulae above, we get

         G    [omega]
    N = --- ----------- (v cos [alpha] - u cos [delta])²;   (1)
         g  cos [alpha]

         G  [omega] cos [delta]
    P = --- ------------------- (v cos [alpha] - u cos [delta])²;   (2)
         g      cos [alpha]

          G           cos [delta]
    Pu = --- [omega]u ----------- (v cos [alpha] - u cos [delta])².   (3)
          g           cos [alpha]

  Three cases may be distinguished:--

  (a) The plane is at rest. Then u = 0, N = (G/g)[omega]v² cos [alpha];
  and the work done on the plane and the efficiency of the jet are zero.

  (b) The plane moves parallel to the jet. Then [delta] = [alpha], and
  Pu = (G/g)[omega]u cos²[alpha](v - u)², which is a maximum when u =
  1/3 v.

  When u = 1/3 v then Pu max. = 4/27 (G/g)[omega]v³ cos² [alpha], and
  the efficiency = [eta] = 4/9 cos² [alpha].

  (c) The plane moves perpendicularly to the jet. Then [delta] = 90° -
  [alpha]; cos [delta] = sin [alpha]; and Pu = G/g [omega]u (sin
  [alpha]/cos [alpha]) (v cos [alpha] - u sin [alpha])². This is a
  maximum when u = 1/3 v cos [alpha].

  When u = 1/3 v cos [alpha], the maximum work and the efficiency are
  the same as in the last case.

  [Illustration: FIG. 159.]

  § 158. _Best Form of Vane to receive Water._--When water impinges
  normally or obliquely on a plane, it is scattered in all directions
  after impact, and the work carried away by the water is then generally
  lost, from the impossibility of dealing afterwards with streams of
  water deviated in so many directions. By suitably forming the vane,
  however, the water may be entirely deviated in one direction, and the
  loss of energy from agitation of the water is entirely avoided.

  Let AB (fig. 159) be a vane, on which a jet of water impinges at the
  point A and in the direction AC. Take AC = v = velocity of water, and
  let AD represent in magnitude and direction the velocity of the vane.
  Completing the parallelogram, DC or AE represents the direction in
  which the water is moving relatively to the vane. If the lip of the
  vane at A is tangential to AE, the water will not have its direction
  suddenly changed when it impinges on the vane, and will therefore have
  no tendency to spread laterally. On the contrary it will be so
  gradually deviated that it will glide up the vane in the direction AB.
  This is sometimes expressed by saying that the vane _receives the
  water without shock_.

  [Illustration: FIG. 160.]

  § 159. _Floats of Poncelet Water Wheels._--Let AC (fig. 160) represent
  the direction of a thin horizontal stream of water having the velocity
  v. Let AB be a curved float moving horizontally with velocity u. The
  relative motion of water and float is then initially horizontal, and
  equal to v - u.

  In order that the float may receive the water without shock, it is
  necessary and sufficient that the lip of the float at A should be
  tangential to the direction AC of relative motion. At the end of (v -
  u)/g seconds the float moving with the velocity u comes to the
  position A1B1, and during this time a particle of water received at A
  and gliding up the float with the relative velocity v - u, attains a
  height DE = (v - u)²/2g. At E the water comes to relative rest. It
  then descends along the float, and when after 2(v - u)/g seconds the
  float has come to A2B2 the water will again have reached the lip at A2
  and will quit it tangentially, that is, in the direction CA2, with a
  relative velocity -(v - u) = -[root](2gDE) acquired under the
  influence of gravity. The absolute velocity of the water leaving the
  float is therefore u - (v - u) = 2u - v. If u = ½v, the water will
  drop off the bucket deprived of all energy of motion. The whole of the
  work of the jet must therefore have been expended in driving the
  float. The water will have been received without shock and discharged
  without velocity. This is the principle of the Poncelet wheel, but in
  that case the floats move over an arc of a large circle; the stream of
  water has considerable thickness (about 8 in.); in order to get the
  water into and out of the wheel, it is then necessary that the lip of
  the float should make a small angle (about 15°) with the direction of
  its motion. The water quits the wheel with a little of its energy of
  motion remaining.

  § 160. _Pressure on a Curved Surface when the Water is deviated wholly
  in one Direction._--When a jet of water impinges on a curved surface
  in such a direction that it is received without shock, the pressure on
  the surface is due to its gradual deviation from its first direction.
  On any portion of the area the pressure is equal and opposite to the
  force required to cause the deviation of so much water as rests on
  that surface. In common language, it is equal to the centrifugal force
  of that quantity of water.

  [Illustration: FIG. 161.]

  _Case 1. Surface Cylindrical and Stationary._--Let AB (fig. 161) be
  the surface, having its axis at O and its radius = r. Let the water
  impinge at A tangentially, and quit the surface tangentially at B.
  Since the surface is at rest, v is both the absolute velocity of the
  water and the velocity relatively to the surface, and this remains
  unchanged during contact with the surface, because the deviating force
  is at each point perpendicular to the direction of motion. The water
  is deviated through an angle BCD = AOB = [phi]. Each particle of water
  of weight p exerts radially a centrifugal force pv²/rg. Let the
  thickness of the stream = t ft. Then the weight of water resting on
  unit of surface = Gt lb.; and the normal pressure per unit of surface
  = n = Gtv²/gr. The resultant of the radial pressures uniformly
  distributed from A to B will be a force acting in the direction OC
  bisecting AOB, and its magnitude will equal that of a force of
  intensity = n, acting on the projection of AB on a plane perpendicular
  to the direction OC. The length of the chord AB = 2r sin ½[phi]; let b
  = breadth of the surface perpendicular to the plane of the figure. The
  resultant pressure on surface

                  [phi]   Gt v²     G           [phi]
    = R = 2rb sin ----- × --.-- = 2--- btv² sin -----,
                    2     g  r      g             2

  which is independent of the radius of curvature. It may be inferred
  that the resultant pressure is the same for any curved surface of the
  same projected area, which deviates the water through the same angle.

  _Case 2. Cylindrical Surface moving in the Direction AC with
  Velocity u._--The relative velocity = v - u. The final velocity BF
  (fig. 162) is found by combining the relative velocity BD = v - u
  tangential to the surface with the velocity BE = u of the surface. The
  intensity of normal pressure, as in the last case, is (G/g)t(v -
  u)²/r. The resultant normal pressure R = 2(G/g)bt(v - u)² sin ½[phi].
  This resultant pressure may be resolved into two components P and L,
  one parallel and the other perpendicular to the direction of the
  vane's motion. The former is an effort doing work on the vane. The
  latter is a lateral force which does no work.

    P = R sin ½[phi] = (G/g) bt (v - u)² (1 - cos [phi]);

    L = R cos ½[phi] = (G/g) bt (v - u)² sin [phi].

  [Illustration: FIG. 162.]

  The work done by the jet on the vane is Pu = (G/g)btu(v - u)²(1 - cos
  [phi]), which is a maximum when u = 1/3 v. This result can also be
  obtained by considering that the work done on the plane must be equal
  to the energy lost by the water, when friction is neglected.

  If [phi] = 180°, cos [phi] = -1, 1 - cos [phi] = 2; then P =
  2(G/g)bt(v - u)², the same result as for a concave cup.

  [Illustration: FIG. 163.]

  § 161. _Position which a Movable Plane takes in Flowing Water._--When
  a rectangular plane, movable about an axis parallel to one of its
  sides, is placed in an indefinite current of fluid, it takes a
  position such that the resultant of the normal pressures on the two
  sides of the axis passes through the axis. If, therefore, planes
  pivoted so that the ratio a/b (fig. 163) is varied are placed in
  water, and the angle they make with the direction of the stream is
  observed, the position of the resultant of the pressures on the plane
  is determined for different angular positions. Experiments of this
  kind have been made by Hagen. Some of his results are given in the
  following table:--

    +-----------+-------------+--------------+
    |           |Larger plane.|Smaller Plane.|
    +-----------+-------------+--------------+
    | a/b = 1.0 |[phi] = ...  |[phi] = 90°   |
    |       0.9 |        75°  |        72½°  |
    |       0.8 |        60°  |        57°   |
    |       0.7 |        48°  |        43°   |
    |       0.6 |        25°  |        29°   |
    |       0.5 |        13°  |        13°   |
    |       0.4 |         8°  |         6½°  |
    |       0.3 |         6°  |        ..    |
    |       0.2 |         4°  |        ..    |
    +-----------+-------------+--------------+

  § 162. _Direct Action distinguished from Reaction_ (Rankine, _Steam
  Engine_, § 147).

  The pressure which a jet exerts on a vane can be distinguished into
  two parts, viz.:--

  (1) The pressure arising from changing the direct component of the
  velocity of the water into the velocity of the vane. In fig. 153, §
  154, ab cos bae is the direct component of the water's velocity, or
  component in the direction of motion of vane. This is changed into the
  velocity ae of the vane. The pressure due to direct impulse is then

    P1 = GQ(ab cos bae - ae)/g.

  For a flat vane moving normally, this direct action is the only action
  producing pressure on the vane.

  (2) The term reaction is applied to the additional action due to the
  direction and velocity with which the water glances off the vane. It
  is this which is diminished by the friction between the water and the
  vane. In Case 2, § 160, the direct pressure is

    P1 = Gbt(v - u)²/g.

  That due to reaction is

    P2 = -Gbt(v - u)² cos [phi]/g.

  If [phi] < 90°, the direct component of the water's motion is not
  wholly converted into the velocity of the vane, and the whole
  pressure due to direct impulse is not obtained. If [phi] > 90°, cos
  [phi] is negative and an additional pressure due to reaction is
  obtained.

  [Illustration: FIG. 164.]

  § 163. _Jet Propeller._--In the case of vessels propelled by a jet of
  water (fig. 164), driven sternwards from orifices at the side of the
  vessel, the water, originally at rest outside the vessel, is drawn
  into the ship and caused to move with the forward velocity V of the
  ship. Afterwards it is projected sternwards from the jets with a
  velocity v relatively to the ship, or v - V relatively to the earth.
  If [Omega] is the total sectional area of the jets, [Omega]v is the
  quantity of water discharged per second. The momentum generated per
  second in a sternward direction is (G/g)[Omega]v(v - V), and this is
  equal to the forward acting reaction P which propels the ship.

  The energy carried away by the water

    = ½(G/g)[Omega]v (v - V)².   (1)

  The useful work done on the ship

    PV = (G/g)[Omega]v (v - V)V.   (2)

  Adding (1) and (2), we get the whole work expended on the water,
  neglecting friction:--

    W = ½(G/g)[Omega]v (v² - V²).

  Hence the efficiency of the jet propeller is

    PV/W = 2V/(v + V).   (3)

  This increases towards unity as v approaches V. In other words, the
  less the velocity of the jets exceeds that of the ship, and therefore
  the greater the area of the orifice of discharge, the greater is the
  efficiency of the propeller.

  In the "Waterwitch" v was about twice V. Hence in this case the
  theoretical efficiency of the propeller, friction neglected, was about
  2/3.

  [Illustration: FIG. 165.]

  § 164. _Pressure of a Steady Stream in a Uniform Pipe on a Plane
  normal to the Direction of Motion._--Let CD (fig. 165) be a plane
  placed normally to the stream which, for simplicity, may be supposed
  to flow horizontally. The fluid filaments are deviated in front of the
  plane, form a contraction at A1A1, and converge again, leaving a mass
  of eddying water behind the plane. Suppose the section A0A0 taken at a
  point where the parallel motion has not begun to be disturbed, and
  A2A2 where the parallel motion is re-established. Then since the same
  quantity of water with the same velocity passes A0A0, A2A2 in any
  given time, the external forces produce no change of momentum on the
  mass A0A0A2A2, and must therefore be in equilibrium. If [Omega] is the
  section of the stream at A0A0 or A2A2, and [omega] the area of the
  plate CD, the area of the contracted section of the stream at A1A1
  will be c_c([Omega] - [omega]), where c_c is the coefficient of
  contraction. Hence, if v is the velocity at A0A0 or A2A2, and v1 the
  velocity at A1A1,

    v[Omega] = c_c v([Omega] - [omega]);

    .:v1 = v[Omega]/c_c ([Omega] - [omega]).   (1)

  Let p0, p1, p2 be the pressures at the three sections. Applying
  Bernoulli's theorem to the sections A0A0 and A1A1,

    p0   v²   p1   v1²
    -- + -- = -- + ---.
    G    2g   G    2g

  Also, for the sections A1A1 and A2A2, allowing that the head due to
  the relative velocity v1 - v is lost in shock:--

    p1   v1²    p2    v²   (v1 - v)²
    -- + --- =  --  + -- + ---------;
    G    2g     G     2g      2g

    .: p0 - p2 = G(v1 - v)²/2g;   (2)

  or, introducing the value in (1),

              G   /        [Omega]            \²
    p0 - p2 = -- ( ----------------------- - 1 ) v²   (3)
              2g  \c_c ([Omega] - [omega])    /

  Now the external forces in the direction of motion acting on the mass
  A0A0A2A2 are the pressures p0[Omega]1 - p2[Omega] at the ends, and the
  reaction -R of the plane on the water, which is equal and opposite to
  the pressure of the water on the plane. As these are in equilibrium,

    (p0 - p2)[Omega] - R = 0;

                     /         [Omega]           \² v²
    .: R = G[Omega] ( ----------------------- - 1 ) --;   (4)
                     \c_c ([Omega] - [omega])    /  2g

  an expression like that for the pressure of an isolated jet on an
  indefinitely extended plane, with the addition of the term in
  brackets, which depends only on the areas of the stream and the plane.
  For a given plane the expression in brackets diminishes as [Omega]
  increases. If [Omega]/[omega] = [rho], the equation (4) becomes
                     _                            _
                 v² |       /     [rho]         \² |
    R = G[omega] -- |[rho] ( --------------- - 1 ) |,   (4a)
                 2g |_      \c_c ([rho] - 1)    / _|

  which is of the form

    R = G[omega](v²/2g)K,

  where K depends only on the ratio of the sections of the stream and
  plane.

  For example, let c_c = 0.85, a value which is probable, if we allow
  that the sides of the pipe act as internal borders to an orifice. Then

               /        [rho]      \²
    K = [rho] ( 1.176 --------- - 1 ).
               \      [rho] - 1    /

    [rho] =        K =

       1        [infinity]
       2           3.66
       3           1.75
       4           1.29
       5           1.10
      10            .94
      50           2.00
     100           3.50

  The assumption that the coefficient of contraction c_c is constant for
  different values of [rho] is probably only true when [rho] is not very
  large. Further, the increase of K for large values of [rho] is
  contrary to experience, and hence it may be inferred that the
  assumption that all the filaments have a common velocity v1 at the
  section A1A1 and a common velocity v at the section A2A2 is not true
  when the stream is very much larger than the plane. Hence, in the
  expression

    R = KG[omega]v²/2g,

  K must be determined by experiment in each special case. For a
  cylindrical body putting [omega] for the section, c_c for the
  coefficient of contraction, c_c([Omega] - [omega]) for the area of the
  stream at A1A1,

    v1 = v[Omega]/c_c([Omega] - [omega]); v2 = v[Omega]/([Omega] - [omega]);

  or, putting [rho] = [Omega]/[omega],

    v1 = v[rho]/c_c ([rho] - 1), v2 = v[rho]/([rho] - 1).

  Then

    R = K1G[omega]v²/2g,

  where

                _                                           _
               |  /  [rho]  \²  / 1     \²  /  [rho]      \² |
    K1 = [rho] | ( --------- ) ( --- - 1 ) ( --------- - 1 ) |.
               |_ \[rho] - 1/   \c_c    /   \[rho] - 1    / _|

  Taking c_c = 0.85 and [rho] = 4, K1 = 0.467, a value less than before.
  Hence there is less pressure on the cylinder than on the thin plane.

  [Illustration: FIG. 166.]

  § 165. _Distribution of Pressure on a Surface on which a Jet impinges
  normally._--The principle of momentum gives readily enough the total
  or resultant pressure of a jet impinging on a plane surface, but in
  some cases it is useful to know the distribution of the pressure. The
  problem in the case in which the plane is struck normally, and the jet
  spreads in all directions, is one of great complexity, but even in
  that case the maximum intensity of the pressure is easily assigned.
  Each layer of water flowing from an orifice is gradually deviated
  (fig. 166) by contact with the surface, and during deviation exercises
  a centrifugal pressure towards the axis of the jet. The force exerted
  by each small mass of water is normal to its path and inversely as the
  radius of curvature of the path. Hence the greatest pressure on the
  plane must be at the axis of the jet, and the pressure must decrease
  from the axis outwards, in some such way as is shown by the curve of
  pressure in fig. 167, the branches of the curve being probably
  asymptotic to the plane.

  For simplicity suppose the jet is a vertical one. Let h1 (fig. 167) be
  the depth of the orifice from the free surface, and v1 the velocity of
  discharge. Then, if [omega] is the area of the orifice, the quantity
  of water impinging on the plane is obviously

    Q = [omega]v1 = [omega] [root](2gh1);

  that is, supposing the orifice rounded, and neglecting the coefficient
  of discharge.

  The velocity with which the fluid reaches the plane is, however,
  greater than this, and may reach the value

    v = [root](2gh);

  where h is the depth of the plane below the free surface. The external
  layers of fluid subjected throughout, after leaving the orifice, to
  the atmospheric pressure will attain the velocity v, and will flow
  away with this velocity unchanged except by friction. The layers
  towards the interior of the jet, being subjected to a pressure greater
  than atmospheric pressure, will attain a less velocity, and so much
  less as they are nearer the centre of the jet. But the pressure can
  in no case exceed the pressure v²/2g or h measured in feet of water,
  or the direction of motion of the water would be reversed, and there
  would be reflux. Hence the maximum intensity of the pressure of the
  jet on the plane is h ft. of water. If the pressure curve is drawn
  with pressures represented by feet of water, it will touch the free
  water surface at the centre of the jet.

  [Illustration: FIG. 167.]

  Suppose the pressure curve rotated so as to form a solid of
  revolution. The weight of water contained in that solid is the total
  pressure of the jet on the surface, which has already been determined.
  Let V = volume of this solid, then GV is its weight in pounds.
  Consequently

    GV = (G/g)[omega]v1v;

    V = 2[omega] [root](hh1).

  We have already, therefore, two conditions to be satisfied by the
  pressure curve.

  [Illustration: FIG. 168.--Curves of Pressure of Jets impinging
  normally on a Plane.]

  Some very interesting experiments on the distribution of pressure on a
  surface struck by a jet have been made by J. S. Beresford (_Prof.
  Papers on Indian Engineering_, No. cccxxii.), with a view to afford
  information as to the forces acting on the aprons of weirs.
  Cylindrical jets ½ in. to 2 in. diameter, issuing from a vessel in
  which the water level was constant, were allowed to fall vertically on
  a brass plate 9 in. in diameter. A small hole in the brass plate
  communicated by a flexible tube with a vertical pressure column.
  Arrangements were made by which this aperture could be moved 1/20
  in. at a time across the area struck by the jet. The height of the
  pressure column, for each position of the aperture, gave the pressure
  at that point of the area struck by the jet. When the aperture was
  exactly in the axis of the jet, the pressure column was very nearly
  level with the free surface in the reservoir supplying the jet; that
  is, the pressure was very nearly v²/2g. As the aperture moved away
  from the axis of the jet, the pressure diminished, and it became
  insensibly small at a distance from the axis of the jet about equal to
  the diameter of the jet. Hence, roughly, the pressure due to the jet
  extends over an area about four times the area of section of the jet.

  Fig. 168 shows the pressure curves obtained in three experiments with
  three jets of the sizes shown, and with the free surface level in the
  reservoir at the heights marked.

    +------------------------------------------------------+
    |          Experiment 1. Jet .475 in. diameter.        |
    +----------------+------------------+------------------+
    |  Height from   |   Distance from  |                  |
    |  Free Surface  |    Axis of Jet   |   Pressure in.   |
    | to Brass Plate |    in inches.    | inches of Water. |
    |   in inches.   |                  |                  |
    +----------------+------------------+------------------+
    |       43       |        0         |       40.5       |
    |        "       |        .05       |       39.40      |
    |        "       |        .1        |     37.5-39.5    |
    |        "       |        .15       |       35         |
    |        "       |        .2        |     33.5-37      |
    |        "       |        .25       |       31         |
    |        "       |        .3        |      21-27       |
    |        "       |        .35       |       21         |
    |        "       |        .4        |       14         |
    |        "       |        .45       |        8         |
    |        "       |        .5        |        3.5       |
    |        "       |        .55       |        1         |
    |        "       |        .6        |        0.5       |
    |        "       |        .65       |        0         |
    +----------------+------------------+------------------+
    |         Experiment 2. Jet .988 in. diameter.         |
    +----------------+------------------+------------------+
    |      42.15     |         0        |       42         |
    |        "       |        .05       |       41.9       |
    |        "       |        .1        |    41.5-41.8     |
    |        "       |        .15       |       41         |
    |        "       |        .2        |       40.3       |
    |        "       |        .25       |       39.2       |
    |        "       |        .3        |       37.5       |
    |        "       |        .35       |       34.8       |
    |        "       |        .45       |       27         |
    |      42.25     |        .5        |       23         |
    |        "       |        .55       |       18.5       |
    |        "       |        .6        |       13         |
    |        "       |        .65       |        8.3       |
    |        "       |        .7        |        5         |
    |        "       |        .75       |        3         |
    |        "       |        .8        |        2.2       |
    |      42.15     |        .85       |        1.6       |
    |        "       |        .95       |        1         |
    +----------------+------------------+------------------+
    |         Experiment 3. Jet 19.5 in. diameter.         |
    +----------------+------------------+------------------+
    |      27.15     |         0        |       26.9       |
    |        "       |        .08       |       26.9       |
    |        "       |        .13       |       26.8       |
    |        "       |        .18       |     26.5-26.6    |
    |        "       |        .23       |     26.4-26.5    |
    |        "       |        .28       |     26.3-26.6    |
    |       27       |        .33       |       26.2       |
    |        "       |        .38       |       25.9       |
    |        "       |        .43       |       25.5       |
    |        "       |        .48       |       25         |
    |        "       |        .53       |       24.5       |
    |        "       |        .58       |       24         |
    |        "       |        .63       |       23.3       |
    |        "       |        .68       |       22.5       |
    |        "       |        .73       |       21.8       |
    |        "       |        .78       |       21         |
    |        "       |        .83       |       20.3       |
    |        "       |        .88       |       19.3       |
    |        "       |        .93       |       18         |
    |        "       |        .98       |       17         |
    |      26.5      |       1.13       |       13.5       |
    |        "       |       1.18       |       12.5       |
    |        "       |       1.23       |       10.8       |
    |        "       |       1.28       |        9.5       |
    |        "       |       1.33       |        8         |
    |        "       |       1.38       |        7         |
    |        "       |       1.43       |        6.3       |
    |        "       |       1.48       |        5         |
    |        "       |       1.53       |        4.3       |
    |        "       |       1.58       |        3.5       |
    |        "       |       1.9        |        2         |
    +----------------+------------------+------------------+

  As the general form of the pressure curve has been already indicated,
  it may be assumed that its equation is of the form

    y = ab^(-x²).

  But it has already been shown that for x = 0, y = h, hence a = h. To
  determine the remaining constant, the other condition may be used,
  that the solid formed by rotating the pressure curve represents the
  total pressure on the plane. The volume of the solid is
          _
         /[oo]
    V =  |    2[pi]xy dx
        _/0
                _
               /[oo]
      = 2[pi]h |    b^(-x²)x dx
              _/0
                        _      _
                       |        |[oo]
      = ([pi]h/log_eb) |-b^(-x²)|
                       |_      _|0

      = [pi]h/log_e b.

  Using the condition already stated,

    2[omega] [root](hh1) = [pi]h/log_e b,

    log_e b = ([pi]/2[omega]) [root](h/h1).

  Putting the value of b in (2) in eq. (1), and also r for the radius of
  the jet at the orifice, so that [omega] = [pi]r², the equation to the
  pressure curve is

                              h  x²
    y = h[epsilon]^(-½) [root]-- --.
                              h1 r²

  § 166. _Resistance of a Plane moving through a Fluid, or Pressure of a
  Current on a Plane._--When a thin plate moves through the air, or
  through an indefinitely large mass of still water, in a direction
  normal to its surface, there is an excess of pressure on the anterior
  face and a diminution of pressure on the posterior face. Let v be the
  relative velocity of the plate and fluid, [Omega] the area of the
  plate, G the density of the fluid, h the height due to the velocity,
  then the total resistance is expressed by the equation

    R = fG[Omega]v²/2g pounds = fG[Omega]h;

  where f is a coefficient having about the value 1.3 for a plate moving
  in still fluid, and 1.8 for a current impinging on a fixed plane,
  whether the fluid is air or water. The difference in the value of the
  coefficient in the two cases is perhaps due to errors of experiment.
  There is a similar resistance to motion in the case of all bodies of "
  _unfair_ " form, that is, in which the surfaces over which the water
  slides are not of gradual and continuous curvature.

  The stress between the fluid and plate arises chiefly in this way.
  The streams of fluid deviated in front of the plate, supposed for
  definiteness to be moving through the fluid, receive from it forward
  momentum. Portions of this forward moving water are thrown off
  laterally at the edges of the plate, and diffused through the
  surrounding fluid, instead of falling to their original position
  behind the plate. Other portions of comparatively still water are
  dragged into motion to fill the space left behind the plate; and there
  is thus a pressure less than hydrostatic pressure at the back of the
  plate. The whole resistance to the motion of the plate is the sum of
  the excess of pressure in front and deficiency of pressure behind.
  This resistance is independent of any friction or viscosity in the
  fluid, and is due simply to its inertia resisting a sudden change of
  direction at the edge of the plate.

  Experiments made by a whirling machine, in which the plate is fixed on
  a long arm and moved circularly, gave the following values of the
  coefficient _f_. The method is not free from objection, as the
  centrifugal force causes a flow outwards across the plate.

    +---------------+------------------------+
    |  Approximate  |      Values of f.      |
    | Area of Plate +------+-------+---------+
    |   in sq. ft.  |Borda.|Hutton.|Thibault.|
    +---------------+------+-------+---------+
    |     0.13      | 1.39 |  1.24 |   ..    |
    |     0.25      | 1.49 |  1.43 |  1.525  |
    |     0.63      | 1.64 |   ..  |   ..    |
    |     1.11      |  ..  |   ..  |  1.784  |
    +---------------+------+-------+---------+

  There is a steady increase of resistance with the size of the plate,
  in part or wholly due to centrifugal action.

  P. L. G. Dubuat (1734-1809) made experiments on a plane 1 ft. square,
  moved in a straight line in water at 3 to 6½ ft. per second. Calling m
  the coefficient of excess of pressure in front, and n the coefficient
  of deficiency of pressure behind, so that f = m + n, he found the
  following values:--

    m = 1; n = 0.433; f = 1.433.

  The pressures were measured by pressure columns. Experiments by A. J.
  Morin (1795-1880), G. Piobert (1793-1871) and I. Didion (1798-1878) on
  plates of 0.3 to 2.7 sq. ft. area, drawn vertically through water,
  gave f = 2.18; but the experiments were made in a reservoir of
  comparatively small depth. For similar plates moved through air they
  found f = 1.36, a result more in accordance with those which precede.

  For a fixed plane in a moving current of water E. Mariotte found f =
  1.25. Dubuat, in experiments in a current of water like those
  mentioned above, obtained the values m = 1.186; n = 0.670; f = 1.856.
  Thibault exposed to wind pressure planes of 1.17 and 2.5 sq. ft. area,
  and found f to vary from 1.568 to 2.125, the mean value being f =
  1.834, a result agreeing well with Dubuat.

  [Illustration: FIG. 169.]

  § 167. _Stanton's Experiments on the Pressure of Air on Surfaces._--At
  the National Physical Laboratory, London, T. E. Stanton carried out a
  series of experiments on the distribution of pressure on surfaces in a
  current of air passing through an air trunk. These were on a small
  scale but with exceptionally accurate means of measurement. These
  experiments differ from those already given in that the plane is small
  relatively to the cross section of the current (_Proc. Inst. Civ.
  Eng._ clvi., 1904). Fig. 169 shows the distribution of pressure on a
  square plate. ab is the plate in vertical section. acb the
  distribution of pressure on the windward and adb that on the leeward
  side of the central section. Similarly aeb is the distribution of
  pressure on the windward and afb on the leeward side of a diagonal
  section. The intensity of pressure at the centre of the plate on the
  windward side was in all cases p = Gv²/2g lb. per sq. ft., where G is
  the weight of a cubic foot of air and v the velocity of the current in
  ft. per sec. On the leeward side the negative pressure is uniform
  except near the edges, and its value depends on the form of the plate.
  For a circular plate the pressure on the leeward side was 0.48 Gv²/2g
  and for a rectangular plate 0.66 Gv²/2g. For circular or square plates
  the resultant pressure on the plate was P = 0.00126 v² lb. per sq. ft.
  where v is the velocity of the current in ft. per sec. On a long
  narrow rectangular plate the resultant pressure was nearly 60% greater
  than on a circular plate. In later tests on larger planes in free air,
  Stanton found resistances 18% greater than those observed with small
  planes in the air trunk.

  § 168. _Case when the Direction of Motion is oblique to the
  Plane._--The determination of the pressure between a fluid and surface
  in this case is of importance in many practical questions, for
  instance, in assigning the load due to wind pressure on sloping and
  curved roofs, and experiments have been made by Hutton, Vince, and
  Thibault on planes moved circularly through air and water on a
  whirling machine.

  [Illustration: FIG. 170.]

  Let AB (fig. 170) be a plane moving in the direction R making an angle
  [phi] with the plane. The resultant pressure between the fluid and the
  plane will be a normal pressure N. The component R of this normal
  pressure is the resistance to the motion of the plane and the other
  component L is a lateral force resisted by the guides which support
  the plane. Obviously

    R = N sin [phi];

    L = N cos [phi].

  In the case of wind pressure on a sloping roof surface, R is the
  horizontal and L the vertical component of the normal pressure.

  In experiments with the whirling machine it is the resistance to
  motion, R, which is directly measured. Let P be the pressure on a
  plane moved normally through a fluid. Then, for the same plane
  inclined at an angle [phi] to its direction of motion, the resistance
  was found by Hutton to be

    R = P(sin [phi])^{1.842 cos [phi]}.

  A simpler and more convenient expression given by Colonel Duchemin is

    R = 2P sin² [phi]/(1 + sin² [phi]).

  Consequently, the total pressure between the fluid and plane is

    N = 2P sin [phi]/(1 + sin² [phi]) = 2P/(cosec [phi] + sin [phi]),

  and the lateral force is

    L = 2P sin [phi] cos [phi]/(1 + sin² [phi]).

  In 1872 some experiments were made for the Aeronautical Society on the
  pressure of air on oblique planes. These plates, of 1 to 2 ft. square,
  were balanced by ingenious mechanism designed by F. H. Wenham and
  Spencer Browning, in such a manner that both the pressure in the
  direction of the air current and the lateral force were separately
  measured. These planes were placed opposite a blast from a fan issuing
  from a wooden pipe 18 in. square. The pressure of the blast varied
  from 6/10 to 1 in. of water pressure. The following are the results
  given in pounds per square foot of the plane, and a comparison of the
  experimental results with the pressures given by Duchemin's rule.
  These last values are obtained by taking P = 3.31, the observed
  pressure on a normal surface:--

    +-----------------------------------+-------+-------+-------+------+
    | Angle between Plane and Direction |  15°  |  20°  |  60°  |  90° |
    |   of Blast                        |       |       |       |      |
    +-----------------------------------+-------+-------+-------+------+
    | Horizontal pressure R             | 0.4   | 0.61  | 2.73  | 3.31 |
    | Lateral pressure L                | 1.6   | 1.96  | 1.26  |  ..  |
    | Normal pressure [root](L² + R²)   | 1.65  | 2.05  | 3.01  | 3.31 |
    | Normal pressure by Duchemin's rule| 1.605 | 2.027 | 3.276 | 3.31 |
    +-----------------------------------+-------+-------+-------+------+


WATER MOTORS

In every system of machinery deriving energy from a natural waterfall
there exist the following parts:--

1. A supply channel or head race, leading the water from the highest
accessible level to the site of the machine. This may be an open channel
of earth, masonry or wood, laid at as small a slope as is consistent
with the delivery of the necessary supply of water, or it may be a
closed cast or wrought-iron pipe, laid at the natural slope of the
ground, and about 3 ft. below the surface. In some cases part of the
head race is an open channel, part a closed pipe. The channel often
starts from a small storage reservoir, constructed near the stream
supplying the water motor, in which the water accumulates when the motor
is not working. There are sluices or penstocks by which the supply can
be cut off when necessary.

2. Leading from the motor there is a tail race, culvert, or discharge
pipe delivering the water after it has done its work at the lowest
convenient level.

3. A waste channel, weir, or bye-wash is placed at the origin of the
head race, by which surplus water, in floods, escapes.

4. The motor itself, of one of the kinds to be described presently,
which either overcomes a useful resistance directly, as in the case of a
ram acting on a lift or crane chain, or indirectly by actuating
transmissive machinery, as when a turbine drives the shafting, belting
and gearing of a mill. With the motor is usually combined regulating
machinery for adjusting the power and speed to the work done. This may
be controlled in some cases by automatic governing machinery.

§ 169. _Water Motors with Artificial Sources of Energy._--The great
convenience and simplicity of water motors has led to their adoption in
certain cases, where no natural source of water power is available. In
these cases, an artificial source of water power is created by using a
steam-engine to pump water to a reservoir at a great elevation, or to
pump water into a closed reservoir in which there is great pressure. The
water flowing from the reservoir through hydraulic engines gives back
the energy expended, less so much as has been wasted by friction. Such
arrangements are most useful where a continuously acting steam engine
stores up energy by pumping the water, while the work done by the
hydraulic engines is done intermittently.

  § 170. _Energy of a Water-fall._--Let H_t be the total fall of level
  from the point where the water is taken from a natural stream to the
  point where it is discharged into it again. Of this total fall a
  portion, which can be estimated independently, is expended in
  overcoming the resistances of the head and tail races or the supply
  and discharge pipes. Let this portion of head wasted be [h]_r. Then
  the available head to work the motor is H = H_t - [h]_r. It is this
  available head which should be used in all calculations of the
  proportions of the motor. Let Q be the supply of water per second.
  Then GQH foot-pounds per second is the gross available work of the
  fall. The power of the fall may be utilized in three ways. (a) The GQ
  pounds of water may be placed on a machine at the highest level, and
  descending in contact with it a distance of H ft., the work done will
  be (neglecting losses from friction or leakage) GQH foot-pounds per
  second. (b) Or the water may descend in a closed pipe from the higher
  to the lower level, in which case, with the same reservation as
  before, the pressure at the foot of the pipe will be p = GH pounds per
  square foot. If the water with this pressure acts on a movable piston
  like that of a steam engine, it will drive the piston so that the
  volume described is Q cubic feet per second. Then the work done will
  be pQ = GHQ foot-pounds per second as before. (c) Or lastly, the water
  may be allowed to acquire the velocity v = [root](2gH) by its descent.
  The kinetic energy of Q cubic feet will then be ½GQv²/g = GQH, and if
  the water is allowed to impinge on surfaces suitably curved which
  bring it finally to rest, it will impart to these the same energy as
  in the previous cases. Motors which receive energy mainly in the three
  ways described in (a), (b), (c) may be termed gravity, pressure and
  inertia motors respectively. Generally, if Q ft. per second of water
  act by weight through a distance h1, at a pressure p due to h2 ft. of
  fall, and with a velocity v due to h3 ft. of fall, so that h1 + h2 +
  h3 = H, then, apart from energy wasted by friction or leakage or
  imperfection of the machine, the work done will be

    GQh1 + pQ + (G/g) Q (v²/2g) = GQH foot pounds,

  the same as if the water acted simply by its weight while descending H
  ft.

§ 171. _Site for Water Motor._--Wherever a stream flows from a higher to
a lower level it is possible to erect a water motor. The amount of power
obtainable depends on the available head and the supply of water. In
choosing a site the engineer will select a portion of the stream where
there is an abrupt natural fall, or at least a considerable slope of the
bed. He will have regard to the facility of constructing the channels
which are to convey the water, and will take advantage of any bend in
the river which enables him to shorten them. He will have accurate
measurements made of the quantity of water flowing in the stream, and he
will endeavour to ascertain the average quantity available throughout
the year, the minimum quantity in dry seasons, and the maximum for which
bye-wash channels must be provided. In many cases the natural fall can
be increased by a dam or weir thrown across the stream. The engineer
will also examine to what extent the head will vary in different
seasons, and whether it is necessary to sacrifice part of the fall and
give a steep slope to the tail race to prevent the motor being drowned
by backwater in floods. Streams fed from lakes which form natural
reservoirs or fed from glaciers are less variable than streams depending
directly on rainfall, and are therefore advantageous for water-power
purposes.

  § 172. _Water Power at Holyoke, U.S.A._--About 85 m. from the mouth of
  the Connecticut river there was a fall of about 60 ft. in a short
  distance, forming what were called the Grand Rapids, below which the
  river turned sharply, forming a kind of peninsula on which the city of
  Holyoke is built. In 1845 the magnitude of the water-power available
  attracted attention, and it was decided to build a dam across the
  river. The ordinary flow of the river is 6000 cub. ft. per sec.,
  giving a gross power of 30,000 h.p. In dry seasons the power is 20,000
  h.p., or occasionally less. From above the dam a system of canals
  takes the water to mills on three levels. The first canal starts with
  a width of 140 ft. and depth of 22 ft., and supplies the highest
  range of mills. A second canal takes the water which has driven
  turbines in the highest mills and supplies it to a second series of
  mills. There is a third canal on a still lower level supplying the
  lowest mills. The water then finds its way back to the river. With the
  grant of a mill site is also leased the right to use the water-power.
  A mill-power is defined as 38 cub. ft. of water per sec. during 16
  hours per day on a fall of 20 ft. This gives about 60 h.p. effective.
  The charge for the power water is at the rate of 20s. per h.p. per
  annum.

§ 173. _Action of Water in a Water Motor._--Water motors may be divided
into water-pressure engines, water-wheels and turbines.

Water-pressure engines are machines with a cylinder and piston or ram,
in principle identical with the corresponding part of a steam-engine.
The water is alternately admitted to and discharged from the cylinder,
causing a reciprocating action of the piston or plunger. It is admitted
at a high pressure and discharged at a low one, and consequently work is
done on the piston. The water in these machines never acquires a high
velocity, and for the most part the kinetic energy of the water is
wasted. The useful work is due to the difference of the pressure of
admission and discharge, whether that pressure is due to the weight of a
column of water of more or less considerable height, or is artificially
produced in ways to be described presently.

Water-wheels are large vertical wheels driven by water falling from a
higher to a lower level. In most water-wheels, the water acts directly
by its weight loading one side of the wheel and so causing rotation. But
in all water-wheels a portion, and in some a considerable portion, of
the work due to gravity is first employed to generate kinetic energy in
the water; during its action on the water-wheel the velocity of the
water diminishes, and the wheel is therefore in part driven by the
impulse due to the change of the water's momentum. Water-wheels are
therefore motors on which the water acts, partly by weight, partly by
impulse.

Turbines are wheels, generally of small size compared with water wheels,
driven chiefly by the impulse of the water. Before entering the moving
part of the turbine, the water is allowed to acquire a considerable
velocity; during its action on the turbine this velocity is diminished,
and the impulse due to the change of momentum drives the turbine.

In designing or selecting a water motor it is not sufficient to consider
only its efficiency in normal conditions of working. It is generally
quite as important to know how it will act with a scanty water supply or
a diminished head. The greatest difference in water motors is in their
adaptability to varying conditions of working.


_Water-pressure Engines._

§ 174. In these the water acts by pressure either due to the height of
the column in a supply pipe descending from a high-level reservoir, or
created by pumping. Pressure engines were first used in mine-pumping on
waterfalls of greater height than could at that time be utilized by
water wheels. Usually they were single acting, the water-pressure
lifting the heavy pump rods which then made the return or pumping stroke
by their own weight. To avoid losses by fluid friction and shock the
velocity of the water in the pipes and passages was restricted to from 3
to 10 ft. per second, and the mean speed of plunger to 1 ft. per second.
The stroke was long and the number of strokes 3 to 6 per minute. The
pumping lift being constant, such engines worked practically always at
full load, and the efficiency was high, about 84%. But they were
cumbrous machines. They are described in Weisbach's _Mechanics of
Engineering_.

The convenience of distributing energy from a central station to
scattered working-points by pressure water conveyed in pipes--a system
invented by Lord Armstrong--has already been mentioned. This system has
led to the development of a great variety of hydraulic pressure engines
of very various types. The cost of pumping the pressure water to some
extent restricts its use to intermittent operations, such as working
lifts and cranes, punching, shearing and riveting machines, forging and
flanging presses. To keep down the cost of the distributing mains
very high pressures are adopted, generally 700 lb. per sq. in. or 1600
ft. of head or more.

In a large number of hydraulic machines worked by water at high
pressure, especially lifting machines, the motor consists of a direct,
single acting ram and cylinder. In a few cases double-acting pistons and
cylinders are used; but they involve a water-tight packing of the piston
not easily accessible. In some cases pressure engines are used to obtain
rotative movement, and then two double-acting cylinders or three
single-acting cylinders are used, driving a crank shaft. Some
double-acting cylinders have a piston rod half the area of the piston.
The pressure water acts continuously on the annular area in front of the
piston. During the forward stroke the pressure on the front of the
piston balances half the pressure on the back. During the return stroke
the pressure on the front is unopposed. The water in front of the piston
is not exhausted, but returns to the supply pipe. As the frictional
losses in a fluid are independent of the pressure, and the work done
increases directly as the pressure, the percentage loss decreases for
given velocities of flow as the pressure increases. Hence for
high-pressure machines somewhat greater velocities are permitted in the
passages than for low-pressure machines. In supply mains the velocity is
from 3 to 6 ft. per second, in valve passages 5 to 10 ft. per second, or
in extreme cases 20 ft. per second, where there is less object in
economizing energy. As the water is incompressible, slide valves must
have neither lap nor lead, and piston valves are preferable to ordinary
slide valves. To prevent injurious compression from exhaust valves
closing too soon in rotative engines with a fixed stroke, small
self-acting relief valves are fitted to the cylinder ends, opening
outwards against the pressure into the valve chest. Imprisoned water can
then escape without over-straining the machines.

In direct single-acting lift machines, in which the stroke is fixed, and
in rotative machines at constant speed it is obvious that the cylinder
must be filled at each stroke irrespective of the amount of work to be
done. The same amount of water is used whether much or little work is
done, or whether great or small weights are lifted. Hence while pressure
engines are very efficient at full load, their efficiency decreases as
the load decreases. Various arrangements have been adopted to diminish
this defect in engines working with a variable load. In lifting
machinery there is sometimes a double ram, a hollow ram enclosing a
solid ram. By simple arrangements the solid ram only is used for small
loads, but for large loads the hollow ram is locked to the solid ram,
and the two act as a ram of larger area. In rotative engines the case is
more difficult. In Hastie's and Rigg's engines the stroke is
automatically varied with the load, increasing when the load is large
and decreasing when it is small. But such engines are complicated and
have not achieved much success. Where pressure engines are used
simplicity is generally a first consideration, and economy is of less
importance.

  § 175. _Efficiency of Pressure Engines._--It is hardly possible to
  form a theoretical expression for the efficiency of pressure engines,
  but some general considerations are useful. Consider the case of a
  long stroke hydraulic ram, which has a fairly constant velocity v
  during the stroke, and valves which are fairly wide open during most
  of the stroke. Let r be the ratio of area of ram to area of valve
  passage, a ratio which may vary in ordinary cases from 4 to 12. Then
  the loss in shock of the water entering the cylinder will be (r -
  1)²v²/2g in ft. of head. The friction in the supply pipe is also
  proportional to v². The energy carried away in exhaust will be
  proportional to v². Hence the total hydraulic losses may be taken to
  be approximately [zeta]v²/2g ft., where [zeta] is a coefficient
  depending on the proportions of the machine. Let f be the friction of
  the ram packing and mechanism reckoned in lb. per sq. ft. of ram area.
  Then if the supply-pipe pressure driving the machine is p lb. per sq.
  ft., the effective working pressure will be

    p - G[zeta]v²/2g - f lb. per sq. ft.

  Let A be the area of the ram in sq. ft., v its velocity in ft. per
  sec. The useful work done will be

    (p - G[zeta]v²/2g - f)Av ft. lb. per sec.,

  and the efficiency of the machine will be

    [eta] = (p - G[zeta]v²/2g - f)/p.

  This shows that the efficiency increases with the pressure p, and
  diminishes with the speed v, other things being the same. If in
  regulating the engine for varying load the pressure is throttled,
  part of the available head is destroyed at the throttle valve, and p
  in the bracket above is reduced. Direct-acting hydraulic lifts,
  without intermediate gearing, may have an efficiency of 95% during the
  working stroke. If a hydraulic jigger is used with ropes and sheaves
  to change the speed of the ram to the speed of the lift, the
  efficiency may be only 50%. E. B. Ellington has given the efficiency
  of lifts with hydraulic balance at 85% during the working stroke.
  Large pressure engines have an efficiency of 85%, but small rotative
  engines probably not more than 50% and that only when fully loaded.

[Illustration: FIG. 171.]

§ 176. _Direct-Acting Hydraulic Lift_ (fig. 171).--This is the simplest
of all kinds of hydraulic motor. A cage W is lifted directly by water
pressure acting in a cylinder C, the length of which is a little greater
than the lift. A ram or plunger R of the same length is attached to the
cage. The water-pressure admitted by a cock to the cylinder forces up
the ram, and when the supply valve is closed and the discharge valve
opened, the ram descends. In this case the ram is 9 in. diameter, with a
stroke of 49 ft. It consists of lengths of wrought-iron pipe screwed
together perfectly water-tight, the lower end being closed by a
cast-iron plug. The ram works in a cylinder 11 in. diameter of 9 ft.
lengths of flanged cast-iron pipe. The ram passes water-tight through
the cylinder cover, which is provided with double hat leathers to
prevent leakage outwards or inwards. As the weight of the ram and cage
is much more than sufficient to cause a descent of the cage, part of the
weight is balanced. A chain attached to the cage passes over a pulley at
the top of the lift, and carries at its free end a balance weight B,
working in T iron guides. Water is admitted to the cylinder from a 4-in.
supply pipe through a two-way slide, worked by a rack, spindle and
endless rope. The lift works under 73 ft. of head, and lifts 1350 lb at
2 ft. per second. The efficiency is from 75 to 80%.

  The principal prejudicial resistance to the motion of a ram of this
  kind is the friction of the cup leathers, which make the joint between
  the cylinder and ram. Some experiments by John Hick give for the
  friction of these leathers the following formula. Let F = the total
  friction in pounds; d = diameter of ram in ft.; p = water-pressure in
  pounds per sq. ft.; k a coefficient.

  F = k p d

  k = 0.00393 if the leathers are new or badly lubricated;
    = 0.00262 if the leathers are in good condition and well lubricated.

  Since the total pressure on the ram is P = ¼[pi]d²p, the fraction of
  the total pressure expended in overcoming the friction of the leathers
  is F/P = .005/d to .0033/d, d being in feet.

  Let H be the height of the pressure column measured from the free
  surface of the supply reservoir to the bottom of the ram in its lowest
  position, H_b the height from the discharge reservoir to the same
  point, h the height of the ram above its lowest point at any moment, S
  the length of stroke, [Omega] the area of the ram, W the weight of
  cage, R the weight of ram, B the weight of balance weight, w the
  weight of balance chain per foot run, F the friction of the cup
  leather and slides. Then, neglecting fluid friction, if the ram is
  rising the accelerating force is

    P1 = G(H - h)[Omega] - R - W + B - w(S - h) + wh - F,

  and if the ram is descending

    P2 = G(H_b - h)[Omega] + W + R - B + w(S - h) - wh - F.

  If w = ½ G[Omega], P1 and P2 are constant throughout the stroke; and
  the moving force in ascending and descending is the same, if

    B = W + R + wS - G[Omega](H - H_b)/2.

  Using the values just found for w and B,

    P1 = P2 = ½G[Omega](H - H_b) - F.

  Let W + R + wS + B = U, and let P be the constant accelerating force
  acting on the system, then the acceleration is (P/U)g. The velocity at
  the end of the stroke is (assuming the friction to be constant)

    v = [root](2PgS/U);

  and the mean velocity of ascent is ½v.

[Illustration: FIG. 172.]

§ 177. _Armstrong's Hydraulic Jigger._--This is simply a single-acting
hydraulic cylinder and ram, provided with sheaves so as to give motion
to a wire rope or chain. It is used in various forms of lift and crane.
Fig. 172 shows the arrangement. A hydraulic ram or plunger B works in a
stationary cylinder A. Ram and cylinder carry sets of sheaves over which
passes a chain or rope, fixed at one end to the cylinder, and at the
other connected over guide pulleys to a lift or crane. For each pair of
pulleys, one on the cylinder and one on the ram, the movement of the
free end of the rope is doubled compared with that of the ram. With
three pairs of pulleys the free end of the rope has a movement equal to
six times the stroke of the ram, the force exerted being in the inverse
proportion.

§ 178. _Rotative Hydraulic Engines._--Valve-gear mechanism similar in
principle to that of steam engines can be applied to actuate the
admission and discharge valves, and the pressure engine is then
converted into a continuously-acting motor.

  Let H be the available fall to work the engine after deducting the
  loss of head in the supply and discharge pipes, Q the supply of water
  in cubic feet per second, and [eta] the efficiency of the engine. Then
  the horse-power of the engine is

    H.P. = [eta]GQH/550.

  The efficiency of large slow-moving pressure engines is [eta] = .66 to
  .8. In small motors of this kind probably [eta] is not greater than
  .5. Let v be the mean velocity of the piston, then its diameter d is
  given by the relation

    Q = [pi]d²v/4 in double-acting engines,
      = [pi]d²v/8 in single-acting engines.

  If there are n cylinders put Q/n for Q in these equations.

Small rotative pressure engines form extremely convenient motors for
hoists, capstans or winches, and for driving small machinery. The
single-acting engine has the advantage that the pressure of the piston
on the crank pin is always in one direction; there is then no knocking
as the dead centres are passed. Generally three single-acting cylinders
are used, so that the engine will readily start in all positions, and
the driving effort on the crank pin is very uniform.

[Illustration: FIG. 173.]

  _Brotherhood Hydraulic Engine._--Three cylinders at angles of 120°
  with each other are formed in one casting with the frame. The
  plungers are hollow trunks, and the connecting rods abut in
  cylindrical recesses in them and are connected to a common crank pin.
  A circular valve disk with concentric segmental ports revolves at the
  same rate as the crank over ports in the valve face common to the
  three cylinders. Each cylinder is always in communication with either
  an admission or exhaust port. The blank parts of the circular valve
  close the admission and exhaust ports alternately. The fixed valve
  face is of lignum vitae in a metal recess, and the revolving valve of
  gun-metal. In the case of a small capstan engine the cylinders are 3½
  in. diameter and 3 in. stroke. At 40 revs. per minute, the piston
  speed is 31 ft. per minute. The ports are 1 in. diameter or 1/12 of
  the piston area, and the mean velocity in the ports 6.4 ft. per sec.
  With 700 lb. per sq. in. water pressure and an efficiency of 50%, the
  engine is about 3 h.p. A common arrangement is to have three parallel
  cylinders acting on a three-throw crank shaft, the cylinders
  oscillating on trunnions.

  _Hastie's Engine._--Fig. 173 shows a similar engine made by Messrs
  Hastie of Greenock. G, G, G are the three plungers which pass out of
  the cylinders through cup leathers, and act on the same crank pin. A
  is the inlet pipe which communicates with the cock B. This cock
  controls the action of the engine, being so constructed that it acts
  as a reversing valve when the handle C is in its extreme positions and
  as a brake when in its middle position. With the handle in its middle
  position, the ports of the cylinders are in communication with the
  exhaust. Two passages are formed in the framing leading from the cock
  B to the ends of the cylinders, one being in communication with the
  supply pipe A, the other with the discharge pipe Q. These passages end
  as shown at E. The oscillation of the cylinders puts them alternately
  in communication with each of these passages, and thus the water is
  alternately admitted and exhausted.

  [Illustration: FIG. 174.]

  [Illustration: FIG. 175.]

  In any ordinary rotative engine the length of stroke is invariable.
  Consequently the consumption of water depends simply on the speed of
  the engine, irrespective of the effort overcome. If the power of the
  engine must be varied without altering the number of rotations, then
  the stroke must be made variable. Messrs Hastie have contrived an
  exceedingly ingenious method of varying the stroke automatically, in
  proportion to the amount of work to be done (fig. 174). The crank pin
  I is carried in a slide H moving in a disk M. In this is a double cam
  K acting on two small steel rollers J, L attached to the slide H. If
  the cam rotates it moves the slide and increases or decreases the
  radius of the circle in which the crank pin I rotates. The disk M is
  keyed on a hollow shaft surrounding the driving shaft P, to which the
  cams are attached. The hollow shaft N has two snugs to which the
  chains RR are attached (fig. 175). The shaft P carries the spring case
  SS to which also are attached the other ends of the chains. When the
  engine is at rest the springs extend themselves, rotating the hollow
  shaft N and the frame M, so as to place the crank pin I at its nearest
  position to the axis of rotation. When a resistance has to be
  overcome, the shaft N rotates relatively to P, compressing the
  springs, till their resistance balances the pressure due to the
  resistance to the rotation of P. The engine then commences to work,
  the crank pin being in the position in which the turning effort just
  overcomes the resistance. If the resistance diminishes, the springs
  force out the chains and shorten the stroke of the plungers, and vice
  versa. The following experiments, on an engine of this kind working a
  hoist, show how the automatic arrangement adjusted the water used to
  the work done. The lift was 22 ft. and the water pressure in the
  cylinders 80 lb. per sq. in.

    Weight lifted,  Chain   427   633   745   857   969  1081  1193
      in lb.         only

    Water used, in    7½     10    14    16    17    20    21    22
      gallons

§ 179. _Accumulator Machinery._--It has already been pointed out that it
is in some cases convenient to use a steam engine to create an
artificial head of water, which is afterwards employed in driving
water-pressure machinery. Where power is required intermittently, for
short periods, at a number of different points, as, for instance, in
moving the cranes, lock gates, &c., of a dockyard, a separate steam
engine and boiler at each point is very inconvenient; nor can engines
worked from a common boiler be used, because of the great loss of heat
and the difficulties which arise out of condensation in the pipes. If a
tank, into which water is continuously pumped, can be placed at a great
elevation, the water can then be used in hydraulic machinery in a very
convenient way. Each hydraulic machine is put in communication with the
tank by a pipe, and on opening a valve it commences work, using a
quantity of water directly proportional to the work done. No attendance
is required when the machine is not working.

[Illustration: FIG. 176.]

A site for such an elevated tank is, however, seldom available, and in
place of it a beautiful arrangement termed an accumulator, invented by
Lord Armstrong, is used. This consists of a tall vertical cylinder; into
this works a solid ram through cup leathers or hemp packing, and the ram
is loaded by fixed weights, so that the pressure in the cylinder is 700
lb. or 800 lb. per sq. in. In some cases the ram is fixed and the
cylinder moves on it. The pumping engines which supply the energy that
is stored in the accumulator should be a pair coupled at right angles,
so as to start in any position. The engines pump into the accumulator
cylinder till the ram is at the top of its stroke, when by a catch
arrangement acting on the engine throttle valve the engines are stopped.
If the accumulator ram descends, in consequence of water being taken to
work machinery, the engines immediately recommence working. Pipes lead
from the accumulator to each of the machines requiring to be driven, and
do not require to be of large size, as the pressure is so great.

  Fig. 176 shows a diagrammatic way the scheme of a system of
  accumulator machinery. A is the accumulator, with its ram carrying a
  cylindrical wrought-iron tank W, in which weights are placed to load
  the accumulator. At R is one of the pressure engines or jiggers,
  worked from the accumulator, discharging the water after use into the
  tank T. In this case the pressure engine is shown working a set of
  blocks, the fixed block being on the ram cylinder, the running block
  on the ram. The chain running over these blocks works a lift cage C,
  the speed of which is as many times greater than that of the ram as
  there are plies of chain on the block tackle. B is the balance weight
  of the cage.

  [Illustration: FIG. 177.]

  In the use of accumulators on shipboard for working gun gear or
  steering gear, the accumulator ram is loaded by springs, or by steam
  pressure acting on a piston much larger than the ram.

  R. H. Tweddell has used accumulators with a pressure of 2000 lb. per
  sq. in. to work hydraulic riveting machinery.

  The amount of energy stored in the accumulator, having a ram d in. in
  diameter, a stroke of S ft., and delivering at p lb. pressure per sq.
  in., is

    [pi]
    ---- p d²S foot-pounds.
     4

  Thus, if the ram is 9 in., the stroke 20 ft., and the pressure 800 lb.
  per sq. in., the work stored in the accumulator when the ram is at the
  top of the stroke is 1,017,600 foot-pounds, that is, enough to drive a
  machine requiring one horse power for about half an hour. As, however,
  the pumping engine replaces water as soon as it is drawn off, the
  working capacity of the accumulator is very much greater than this.
  Tweddell found that an accumulator charged at 1250 lb. discharged at
  1225 lb. per sq. in. Hence the friction was equivalent to 12½ lb. per
  sq. in. and the efficiency 98%.

  When a very great pressure is required a differential accumulator
  (fig. 177) is convenient. The ram is fixed and passes through both
  ends of the cylinder, but is of different diameters at the two ends, A
  and B. Hence if d1, d2 are the diameters of the ram in inches and p
  the required pressure in lb. per sq. in., the load required is
  ¼p[pi](d1² - d2²). An accumulator of this kind used with riveting
  machines has d1 = 5½ in., d2 = 4¾ in. The pressure is 2000 lb. per sq.
  in. and the load 5.4 tons.

  [Illustration: FIG. 178.]

  Sometimes an accumulator is loaded by water or steam pressure instead
  of by a dead weight. Fig. 178 shows the arrangement. A piston A is
  connected to a plunger B of much smaller area. Water pressure, say
  from town mains, is admitted below A, and the high pressure water is
  pumped into and discharged from the cylinder C in which B works. If r
  is the ratio of the areas of A and B, then, neglecting friction, the
  pressure in the upper cylinder is r times that under the piston A.
  With a variable rate of supply and demand from the upper cylinder, the
  piston A rises and falls, maintaining always a constant pressure in
  the upper cylinder.


_Water Wheels._

§ 180. _Overshot and High Breast Wheels._--When a water fall ranges
between 10 and 70 ft. and the water supply is from 3 to 25 cub. ft. per
second, it is possible to construct a bucket wheel on which the water
acts chiefly by its weight. If the variation of the head-water level
does not exceed 2 ft., an overshot wheel may be used (fig. 179). The
water is then projected over the summit of the wheel, and falls in a
parabolic path into the buckets. With greater variation of head-water
level, a pitch-back or high breast wheel is better. The water falls over
the top of a sliding sluice into the wheel, on the same side as the head
race channel. By adjusting the height of the sluice, the requisite
supply is given to the wheel in all positions of the head-water level.

  The wheel consists of a cast-iron or wrought-iron axle C supporting
  the weight of the wheel. To this are attached two  sets of arms A of
  wood or iron, which support circular segmental plates, B, termed
  shrouds. A cylindrical sole plate dd extends between the shrouds on
  the inner side. The buckets are formed by wood planks or curved
  wrought-iron plates extending from shroud to shroud, the back of the
  buckets being formed by the sole plate.

[Illustration: FIG. 179.]

  The efficiency may be taken at 0.75. Hence, if h.p. is the effective
  horse power, H the available fall, and Q the available water supply
  per second,

    h.p. = 0.75 (GQH/550) = 0.085 QH.

  If the peripheral velocity of the water wheel is too great, water is
  thrown out of the buckets before reaching the bottom of the fall. In
  practice, the circumferential velocity of water wheels of the kind now
  described is from 4½ to 10 ft. per second, about 6 ft. being the usual
  velocity of good iron wheels not of very small size. In order that the
  water may enter the buckets easily, it must have a greater velocity
  than the wheel. Usually the velocity of the water at the point where
  it enters the wheel is from 9 to 12 ft. per second, and to produce
  this it must enter the wheel at a point 16 to 27 in. below the
  head-water level. Hence the diameter of an overshot wheel may be

    D = H - 1(1/3) to H - 2¼ ft.

  Overshot and high breast wheels work badly in backwater, and hence if
  the tail-water level varies, it is better to reduce the diameter of
  the wheel so that its greatest immersion in flood is not more than 1
  ft. The depth d of the shrouds is about 10 to 16 in. The number of
  buckets may be about

    N = [pi]D/d.

  Let v be the peripheral velocity of the wheel. Then the capacity of
  that portion of the wheel which passes the sluice in one second is

    Q1 = vb(Dd - d²)/D
       = v b d nearly,

  b being the breadth of the wheel between the shrouds. If, however,
  this quantity of water were allowed to pass on to the wheel the
  buckets would begin to spill their contents almost at the top of the
  fall. To diminish the loss from spilling, it is not only necessary to
  give the buckets a suitable form, but to restrict the water supply to
  one-fourth or one-third of the gross bucket capacity. Let m be the
  value of this ratio; then, Q being the supply of water per second,

    Q = mQ1 = mb dv.

  This gives the breadth of the wheel if the water supply is known. The
  form of the buckets should be determined thus. The outer element of
  the bucket should be in the direction of motion of the water entering
  relatively to the wheel, so that the water may enter without splashing
  or shock. The buckets should retain the water as long as possible, and
  the width of opening of the buckets should be 2 or 3 in. greater than
  the thickness of the sheet of water entering.

  For a wooden bucket (fig. 180, A), take ab = distance between two
  buckets on periphery of wheel. Make ed = ½ eb and bc = 6/5 to 5/4
  ab. Join cd. For an iron bucket (fig. 180, B), take ed = 1/3 eb; bc =
  6/5 ab. Draw cO making an angle of 10° to 15° with the radius at c.
  On Oc take a centre giving a circular arc passing near d, and round
  the curve into the radial part of the bucket de.

[Illustration: FIG. 180.]

There are two ways in which the power of a water wheel is given off to
the machinery driven. In wooden wheels and wheels with rigid arms, a
spur or bevil wheel keyed on the axle of the turbine will transmit the
power to the shafting. It is obvious that the whole turning moment due
to the weight of the water is then transmitted through the arms and axle
of the water wheel. When the water wheel is an iron one, it usually has
light iron suspension arms incapable of resisting the bending action due
to the transmission of the turning effort to the axle. In that case spur
segments are bolted to one of the shrouds, and the pinion to which the
power is transmitted is placed so that the teeth in gear are, as nearly
as may be, on the line of action of the resultant of the weight of the
water in the loaded arc of the wheel.

The largest high breast wheels ever constructed were probably the four
wheels, each 50 ft. in diameter, and of 125 h.p., erected by Sir W.
Fairbairn in 1825 at Catrine in Ayrshire. These wheels are still
working.

[Illustration: FIG. 181.]

§ 181. _Poncelet Water Wheel._--When the fall does not exceed 6 ft., the
best water motor to adopt in many cases is the Poncelet undershot water
wheel. In this the water acts very nearly in the same way as in a
turbine, and the Poncelet wheel, although slightly less efficient than
the best turbines, in normal conditions of working, is superior to most
of them when working with a reduced supply of water. A general notion of
the action of the water on a Poncelet wheel has already been given in §
159. Fig. 181 shows its construction. The water penned back between the
side walls of the wheel pit is allowed to flow to the wheel under a
movable sluice, at a velocity nearly equal to the velocity due to the
whole fall. The water is guided down a slope of 1 in 10, or a curved
race, and enters the wheel without shock. Gliding up the curved floats
it comes to rest, falls back, and acquires at the point of discharge a
backward velocity relative to the wheel nearly equal to the forward
velocity of the wheel. Consequently it leaves the wheel deprived of
nearly the whole of its original kinetic energy.

  Taking the efficiency at 0.60, and putting H for the available fall,
  h.p. for the horse-power, and Q for the water supply per second,

    h.p. = 0.068 QH.

  The diameter D of the wheel may be taken arbitrarily. It should not be
  less than twice the fall and is more often four times the fall. For
  ordinary cases the smallest convenient diameter is 14 ft. with a
  straight, or 10 ft. with a curved, approach channel. The radial depth
  of bucket should be at least half the fall, and radius of curvature of
  buckets about half the radius of the wheel. The shrouds are usually of
  cast iron with flanges to receive the buckets. The buckets may be of
  iron 1/8 in. thick bolted to the flanges with 5/16 in. bolts.

  Let H´ be the fall measured from the free surface of the head-water to
  the point F where the mean layer enters the wheel; then the velocity
  at which the water enters is v = [root](2gH´), and the best
  circumferential velocity of the wheel is V = 0.55f to 0.6v. The number
  of rotations of the wheel per second is N = V/[pi]D. The thickness
  of the sheet of water entering the wheel is very important. The best
  thickness according to experiment is 8 to 10 in. The maximum thickness
  should not exceed 12 to 15 in., when there is a surplus water supply.
  Let e be the thickness of the sheet of water entering the wheel, and b
  its width; then

    bev = Q; or b = Q/ev.

  Grashof takes e = (1/6)H, and then

    b = 6Q/H [root](2gH).

  Allowing for the contraction of the stream, the area of opening
  through the sluice may be 1.25 be to 1.3 be. The inside width of the
  wheel is made about 4 in. greater than b.

  Several constructions have been given for the floats of Poncelet
  wheels. One of the simplest is that shown in figs. 181, 182.

  Let OA (fig. 181) be the vertical radius of the wheel. Set off OB, OD
  making angles of 15° with OA. Then BD may be the length of the close
  breasting fitted to the wheel. Draw the bottom of the head face BC at
  a slope of 1 in 10. Parallel to this, at distances ½e and e, draw EF
  and GH. Then EF is the mean layer and GH the surface layer entering
  the wheel. Join OF, and make OFK = 23°. Take FK = 0.5 to 0.7 H. Then K
  is the centre from which the bucket curve is struck and KF is the
  radius. The depth of the shrouds must be sufficient to prevent the
  water from rising over the top of the float. It is ½H to 2/3 H. The
  number of buckets is not very important. They are usually 1 ft. apart
  on the circumference of the wheel.

  The efficiency of a Poncelet wheel has been found in experiments to
  reach 0.68. It is better to take it at 0.6 in estimating the power of
  the wheel, so as to allow some margin.

  [Illustration: FIG. 182.]

  In fig. 182 v_i is the initial and v_o the final velocity of the
  water, v_r parallel to the vane the relative velocity of the water and
  wheel, and V the velocity of the wheel.


_Turbines._

§ 182. The name turbine was originally given in France to any water
motor which revolved in a horizontal plane, the axis being vertical. The
rapid development of this class of motors dates from 1827, when a prize
was offered by the Société d'Encouragement for a motor of this kind,
which should be an improvement on certain wheels then in use. The prize
was ultimately awarded to Benoît Fourneyron (1802-1867), whose turbine,
but little modified, is still constructed.

_Classification of Turbines._--In some turbines the whole available
energy of the water is converted into kinetic energy before the water
acts on the moving part of the turbine. Such turbines are termed
_Impulse or Action Turbines_, and they are distinguished by this that
the wheel passages are never entirely filled by the water. To ensure
this condition they must be placed a little above the tail water and
discharge into free air. Turbines in which part only of the available
energy is converted into kinetic energy before the water enters the
wheel are termed _Pressure or Reaction Turbines_. In these there is a
pressure which in some cases amounts to half the head in the clearance
space between the guide vanes and wheel vanes. The velocity with which
the water enters the wheel is due to the difference between the pressure
due to the head and the pressure in the clearance space. In pressure
turbines the wheel passages must be continuously filled with water for
good efficiency, and the wheel may be and generally is placed below the
tail water level.

Some turbines are designed to act normally as impulse turbines
discharging above the tail water level. But the passages are so designed
that they are just filled by the water. If the tail water rises and
drowns the turbine they become pressure turbines with a small clearance
pressure, but the efficiency is not much affected. Such turbines are
termed _Limit turbines_.

Next there is a difference of constructive arrangement of turbines,
which does not very essentially alter the mode of action of the water.
In axial flow or so-called parallel flow turbines, the water enters and
leaves the turbine in a direction parallel to the axis of rotation, and
the paths of the molecules lie on cylindrical surfaces concentric with
that axis. In radial outward and inward flow turbines, the water enters
and leaves the turbine in directions normal to the axis of rotation, and
the paths of the molecules lie exactly or nearly in planes normal to the
axis of rotation. In outward flow turbines the general direction of flow
is away from the axis, and in inward flow turbines towards the axis.
There are also mixed flow turbines in which the water enters normally
and is discharged parallel to the axis of rotation.

Another difference of construction is this, that the water may be
admitted equally to every part of the circumference of the turbine wheel
or to a portion of the circumference only. In the former case, the
condition of the wheel passages is always the same; they receive water
equally in all positions during rotation. In the latter case, they
receive water during a part of the rotation only. The former may be
termed turbines with complete admission, the latter turbines with
partial admission. A reaction turbine should always have complete
admission. An impulse turbine may have complete or partial admission.

When two turbine wheels similarly constructed are placed on the same
axis, in order to balance the pressures and diminish journal friction,
the arrangement may be termed a twin turbine.

If the water, having acted on one turbine wheel, is then passed through
a second on the same axis, the arrangement may be termed a compound
turbine. The object of such an arrangement would be to diminish the
speed of rotation.

Many forms of reaction turbine may be placed at any height not exceeding
30 ft. above the tail water. They then discharge into an air-tight
suction pipe. The weight of the column of water in this pipe balances
part of the atmospheric pressure, and the difference of pressure,
producing the flow through the turbine, is the same as if the turbine
were placed at the bottom of the fall.

         I. Impulse Turbines.         |       II. Reaction Turbines.
                                      |
    (Wheel passages not filled, and   |  (Wheel passages filled, discha-
      discharging above the tail      |    rging above or below the tail
      water.)                         |    water or into a suction-pipe.)
    (a) Complete admission. (Rare.)   |  Always with complete admission.
    (b) Partial admission. (Usual.)   |
    \_________________________________\/_______________________________/
             Axial flow, outward flow, inward flow, or mixed flow.
    \_________________________________\/_______________________________/
               Simple turbines; twin turbines; compound turbines.

  § 183. _The Simple Reaction Wheel._--It has been shown, in § 162,
  that, when water issues from a vessel, there is a reaction on the
  vessel tending to cause motion in a direction opposite to that of the
  jet. This principle was applied in a rotating water motor at a very
  early period, and the Scotch turbine, at one time much used, differs
  in no essential respect from the older form of reaction wheel.

  [Illustration: FIG. 183.]

  The old reaction wheel consisted of a vertical pipe balanced on a
  vertical axis, and supplied with water (fig. 183). From the bottom of
  the vertical pipe two or more hollow horizontal arms extended, at the
  ends of which were orifices from which the water was discharged. The
  reaction of the jets caused the rotation of the machine.

  Let H be the available fall measured from the level of the water in
  the vertical pipe to the centres cf the orifices, r the radius from
  the axis of rotation to the centres of the orifices, v the velocity of
  discharge through the jets, [alpha] the angular velocity of the
  machine. When the machine is at rest the water issues from the
  orifices with the velocity [root](2gH) (friction being neglected). But
  when the machine rotates the water in the arms rotates also, and is in
  the condition of a forced vortex, all the particles having the same
  angular velocity. Consequently the pressure in the arms at the
  orifices is H + [alpha]²r²/2g ft. of water, and the velocity of
  discharge through the orifices is v = [root](2gH + [alpha]²r²). If the
  total area of the orifices is [omega], the quantity discharged from
  the wheel per second is

    Q = [omega]v = [omega] [root](2gH + [alpha]²r²).

  While the water passes through the orifices with the velocity v, the
  orifices are moving in the opposite direction with the velocity
  [alpha]r. The absolute velocity of the water is therefore

    v - [alpha]r = [root](2gH + [alpha]²r²) - [alpha]r.

  The momentum generated per second is (GQ/g)(v - [alpha]r), which is
  numerically equal to the force driving the motor at the radius r. The
  work done by the water in rotating the wheel is therefore

    (GQ/g) (v - [alpha]r) ar foot-pounds per sec.

  The work expended by the water fall is GQH foot-pounds per second.
  Consequently the efficiency of the motor is

            (v - [alpha]r) [alpha]r   {[root]{2gH + [alpha]²r²]} - [alpha]r} [alpha]r
    [eta] = ----------------------- = -----------------------------------------------.
                      gH                                    gH

  Let

                                             gH         g²H²
    [root]{2gH + [alpha]²r²} = [alpha]r + -------- - ----------- ...
                                          [alpha]r   2[alpha]³r³

  then

    [eta] = 1 - gH/2[alpha]r + ...

  which increases towards the limit 1 as [alpha]r increases towards
  infinity. Neglecting friction, therefore, the maximum efficiency is
  reached when the wheel has an infinitely great velocity of rotation.
  But this condition is impracticable to realize, and even, at
  practicable but high velocities of rotation, the friction would
  considerably reduce the efficiency. Experiment seems to show that the
  best efficiency is reached when [alpha]r = [root](2gH). Then the
  efficiency apart from friction is

    [eta] = {[root](2[alpha]²r²) - [alpha]r} [alpha]r/gH
          = 0.414 [alpha]²r²/gH = 0.828,

  about 17% of the energy of the fall being carried away by the water
  discharged. The actual efficiency realized appears to be about 60%, so
  that about 21% of the energy of the fall is lost in friction, in
  addition to the energy carried away by the water.

  § 184. _General Statement of Hydrodynamical Principles necessary for
  the Theory of Turbines._

  (a) When water flows through any pipe-shaped passage, such as the
  passage between the vanes of a turbine wheel, the relation between the
  changes of pressure and velocity is given by Bernoulli's theorem (§
  29). Suppose that, at a section A of such a passage, h1 is the
  pressure measured in feet of water, v1 the velocity, and z1 the
  elevation above any horizontal datum plane, and that at a section B
  the same quantities are denoted by h2, v2, z2. Then

    h1 - h2 = (v2² - v1²)/2g + z2 - z1.   (1)

  If the flow is horizontal, z2 = z1; and

    h1 - h2 = (v2² - v1²)/2g.   (la)

  (b) When there is an abrupt change of section of the passage, or an
  abrupt change of section of the stream due to a contraction, then, in
  applying Bernoulli's equation allowance must be made for the loss of
  head in shock (§ 36). Let v1, v2 be the velocities before and after
  the abrupt change, then a stream of velocity v1 impinges on a stream
  at a velocity v2, and the relative velocity is v1 - v2. The head lost
  is (v1 - v2)²/2g. Then equation (1a) becomes

    h1 - h2 = (v1² - v2²)/2g - (v1 - v2)²/2g = v2(v1 - v2)/g   (2)

  [Illustration: FIG. 184.]

  To diminish as much as possible the loss of energy from irregular
  eddying motions, the change of section in the turbine passages must be
  very gradual, and the curvature without discontinuity.

  (c) _Equality of Angular Impulse and Change of Angular
  Momentum._--Suppose that a couple, the moment of which is M, acts on a
  body of weight W for t seconds, during which it moves from A1 to A2
  (fig. 184). Let v1 be the velocity of the body at A1, v2 its velocity
  at A2, and let p1, p2 be the perpendiculars from C on v1 and v2. Then
  Mt is termed the angular impulse of the couple, and the quantity

    (W/g)(v2p2 - v1p1)

  is the change of angular momentum relatively to C. Then, from the
  equality of angular impulse and change of angular momentum

    Mt = (W/g)(v2p2 - v1p1),

  or, if the change of momentum is estimated for one second,

    M = (W/g)(v2p2 - v1p1).

  Let r1, r2 be the radii drawn from C to A1, A2, and let w1, w2 be the
  components of v1, v2, perpendicular to these radii, making angles
  [beta] and [alpha] with v1, v2. Then

    v1 = w1 sec [beta]; v2 = w2 sec [alpha]

    p1 = r1 cos [beta]; p2 = r2 cos [alpha],

    .: M = (W/g) (w2r2 - w1r1),   (3)

  where the moment of the couple is expressed in terms of the radii
  drawn to the positions of the body at the beginning and end of a
  second, and the tangential components of its velocity at those points.

  Now the water flowing through a turbine enters at the admission
  surface and leaves at the discharge surface of the wheel, with its
  angular momentum relatively to the axis of the wheel changed. It
  therefore exerts a couple -M tending to rotate the wheel, equal and
  opposite to the couple M which the wheel exerts on the water. Let Q
  cub. ft. enter and leave the wheel per second, and let w1, w2 be the
  tangential components of the velocity of the water at the receiving
  and discharging surfaces of the wheel, r1, r2 the radii of those
  surfaces. By the principle above,

    -M = (GQ/g)(w2r2 - w1r1).   (4)

  If [alpha] is the angular velocity of the wheel, the work done by the
  water on the wheel is

    T = Ma = (GQ/g)(w1r1 - w2r2) [alpha] foot-pounds per second.   (5)

  § 185. _Total and Available Fall._--Let H_t be the total difference of
  level from the head-water to the tail-water surface. Of this total
  head a portion is expended in overcoming the resistances of the head
  race, tail race, supply pipe, or other channel conveying the water.
  Let [h]_p be that loss of head, which varies with the local
  conditions in which the turbine is placed. Then

    H = H_t - [h]_p

  is the available head for working the turbine, and on this the
  calculations for the turbine should be based. In some cases it is
  necessary to place the turbine above the tail-water level, and there
  is then a fall [h] from the centre of the outlet surface of
  the turbine to the tail-water level which is wasted, but which is
  properly one of the losses belonging to the turbine itself. In that
  case the velocities of the water in the turbine should be calculated
  for a head H - [h], but the efficiency of the turbine for the
  head H.

  § 186. _Gross Efficiency and Hydraulic Efficiency of a Turbine._--Let
  T_d be the useful work done by the turbine, in foot-pounds per second,
  T_t the work expended in friction of the turbine shaft, gearing, &c.,
  a quantity which varies with the local conditions in which the turbine
  is placed. Then the effective work done by the water in the turbine is

    T = T_d + T_t.

  The gross efficiency of the whole arrangement of turbine, races, and
  transmissive machinery is

    [eta]_t = T_d/CQH_t.   (6)

  And the hydraulic efficiency of the turbine alone is

    [eta] = T/GQH.   (7)

  It is this last efficiency only with which the theory of turbines is
  concerned.

  From equations (5) and (7) we get

    [eta]GQH = (GQ/g)(w1r1 - w2r2)a;

    [eta] = (w1r1 - w2r2)a/gH.   (8)

  This is the fundamental equation in the theory of turbines. In
  general,[7] w1 and w2, the tangential components of the water's motion
  on entering and leaving the wheel, are completely independent. That
  the efficiency may be as great as possible, it is obviously necessary
  that w2 = 0. In that case

    [eta] = w1r1a/gH.   (9)

  ar1 is the circumferential velocity of the wheel at the inlet surface.
  Calling this V1, the equation becomes

    [eta] = w1V1/gH.   (9a)

  This remarkably simple equation is the fundamental equation in the
  theory of turbines. It was first given by Reiche (_Turbinenbaues_,
  1877).

[Illustration: FIG. 185.]

[Illustration: FIG. 186.]

[Illustration: FIG. 187.]

[Illustration: FIG. 188.]

[Illustration: FIG. 189.]

§ 187. _General Description of a Reaction Turbine._--Professor James
Thomson's inward flow or vortex turbine has been selected as the type of
reaction turbines. It is one of the best in normal conditions of
working, and the mode of regulation introduced is decidedly superior to
that in most reaction turbines. Figs. 185 and 186 are external views of
the turbine case; figs. 187 and 188 are the corresponding sections; fig.
189 is the turbine wheel. The example chosen for illustration has
suction pipes, which permit the turbine to be placed above the
tail-water level. The water enters the turbine by cast-iron supply pipes
at A, and is discharged through two suction pipes S, S. The water
on entering the case distributes itself through a rectangular supply
chamber SC, from which it finds its way equally to the four guide-blade
passages G, G, G, G. In these passages it acquires a velocity about
equal to that due to half the fall, and is directed into the wheel at an
angle of about 10° or 12° with the tangent to its circumference. The
wheel W receives the water in equal proportions from each guide-blade
passage. It consists of a centre plate p (fig. 189) keyed on the shaft
aa, which passes through stuffing boxes on the suction pipes. On each
side of the centre plate are the curved wheel vanes, on which the
pressure of the water acts, and the vanes are bounded on each side by
dished or conical cover plates c, c. Joint-rings j, j on the cover
plates make a sufficiently water-tight joint with the casing, to
prevent leakage from the guide-blade chamber into the suction pipes. The
pressure near the joint rings is not very great, probably not one-fourth
the total head. The wheel vanes receive the water without shock, and
deliver it into central spaces, from which it flows on either side to
the suction pipes. The mode of regulating the power of the turbine is
very simple. The guide-blades are pivoted to the case at their inner
ends, and they are connected by a link-work, so that they all open and
close simultaneously and equally. In this way the area of opening
through the guide-blades is altered without materially altering the
angle or the other conditions of the delivery into the wheel. The
guide-blade gear may be variously arranged. In this example four
spindles, passing through the case, are linked to the guide-blades
inside the case, and connected together by the links l, l, l on the
outside of the case. A worm wheel on one of the spindles is rotated by a
worm d, the motion being thus slow enough to adjust the guide-blades
very exactly. These turbines are made by Messrs Gilkes & Co. of Kendal.

[Illustration: FIG. 190.]

  Fig. 190 shows another arrangement of a similar turbine, with some
  adjuncts not shown in the other drawings. In this case the turbine
  rotates horizontally, and the turbine case is placed entirely below
  the tail water. The water is supplied to the turbine by a vertical
  pipe, over which is a wooden pentrough, containing a strainer, which
  prevents sticks and other solid bodies getting into the turbine. The
  turbine rests on three foundation stones, and, the pivot for the
  vertical shaft being under water, there is a screw and lever
  arrangement for adjusting it as it wears. The vertical shaft gives
  motion to the machinery driven by a pair of bevel wheels. On the right
  are the worm and wheel for working the guide-blade gear.

  [Illustration: FIG. 191.]

  § 188. _Hydraulic Power at Niagara._--The largest development of
  hydraulic power is that at Niagara. The Niagara Falls Power Company
  have constructed two power houses on the United States side, the first
  with 10 turbines of 5000 h.p. each, and the second with 10 turbines of
  5500 h.p. The effective fall is 136 to 140 ft. In the first power
  house the turbines are twin outward flow reaction turbines with
  vertical shafts running at 250 revs. per minute and driving the
  dynamos direct. In the second power house the turbines are inward flow
  turbines with draft tubes or suction pipes. Fig. 191 shows a section
  of one of these turbines. There is a balancing piston keyed on the
  shaft, to the under side of which the pressure due to the fall is
  admitted, so that the weight of turbine, vertical shaft and part of
  the dynamo is water borne. About 70,000 h.p. is daily distributed
  electrically from these two power houses. The Canadian Niagara Power
  Company are erecting a power house to contain eleven units of 10,250
  h.p. each, the turbines being twin inward flow reaction turbines. The
  Electrical Development Company of Ontario are erecting a power house
  to contain 11 units of 12,500 h.p. each. The Ontario Power Company are
  carrying out another scheme for developing 200,000 h.p. by twin inward
  flow turbines of 12,000 h.p. each. Lastly the Niagara Falls Power and
  Manufacturing Company on the United States side have a station giving
  35,000 h.p. and are constructing another to furnish 100,000 h.p. The
  mean flow of the Niagara river is about 222,000 cub. ft. per second
  with a fall of 160 ft. The works in progress if completed will utilize
  650,000 h.p. and require 48,000 cub. ft. per second or 21½% of the
  mean flow of the river (Unwin, "The Niagara Falls Power Stations,"
  _Proc. Inst. Mech. Eng._, 1906).

  [Illustration: FIG. 192.]

  § 189. _Different Forms of Turbine Wheel._--The wheel of a turbine or
  part of the machine on which the water acts is an annular space,
  furnished with curved vanes dividing it into passages exactly or
  roughly rectangular in cross section. For radial flow turbines the
  wheel may have the form A or B, fig. 192, A being most usual with
  inward, and B with outward flow turbines. In A the wheel vanes are
  fixed on each side of a centre plate keyed on the turbine shaft. The
  vanes are limited by slightly-coned annular cover plates. In B the
  vanes are fixed on one side of a disk, keyed on the shaft, and limited
  by a cover plate parallel to the disk. Parallel flow or axial flow
  turbines have the wheel as in C. The vanes are limited by two
  concentric cylinders.


  _Theory of Reaction Turbines._

  [Illustration: FIG. 193.]

  § 190. _Velocity of Whirl and Velocity of Flow._--Let acb (fig. 193)
  be the path of the particles of water in a turbine wheel. That path
  will be in a plane normal to the axis of rotation in radial flow
  turbines, and on a cylindrical surface in axial flow turbines. At any
  point c of the path the water will have some velocity v, in the
  direction of a tangent to the path. That velocity may be resolved into
  two components, a whirling velocity w in the direction of the wheel's
  rotation at the point c, and a component u at right angles to this,
  radial in radial flow, and parallel to the axis in axial flow
  turbines. This second component is termed the velocity of flow. Let
  v_o, w_o, u_o be the velocity of the water, the whirling velocity and
  velocity of flow at the outlet surface of the wheel, and v_i, w_i, u_i
  the same quantities at the inlet surface of the wheel. Let [alpha] and
  [beta] be the angles which the water's direction of motion makes with
  the direction of motion of the wheel at those surfaces. Then

    w_o = v_o cos [beta]; u_o = v_o sin [beta]

    w_i = v_i cos [alpha]; u_i = v_i sin [alpha].   (10)

  The velocities of flow are easily ascertained independently from the
  dimensions of the wheel. The velocities of flow at the inlet and
  outlet surfaces of the wheel are normal to those surfaces. Let
  [Omega]_o, [Omega]_i be the areas of the outlet and inlet surfaces of
  the wheel, and Q the volume of water passing through the wheel per
  second; then

    v_o = Q/[Omega]_o; v_i = Q/[Omega]_i.   (11)

  Using the notation in fig. 191, we have, for an inward flow turbine
  (neglecting the space occupied by the vanes),

    [Omega]_o = 2[pi]r0d0; [Omega]_i = 2[pi]r_i d_i.   (12a)

  Similarly, for an outward flow turbine,

    [Omega]_o = 2[pi]r_o d; [Omega]_i = 2[pi]r_i d;   (12b)

  and, for an axial flow turbine,

    [Omega]_o = [Omega]_i = [pi](r2² - r1²).   (12c)

  [Illustration: FIG. 194.]

  _Relative and Common Velocity of the Water and Wheel._--There is
  another way of resolving the velocity of the water. Let V be the
  velocity of the wheel at the point c, fig. 194. Then the velocity of
  the water may be resolved into a component V, which the water has in
  common with the wheel, and a component v_r, which is the velocity of
  the water relatively to the wheel.

  _Velocity of Flow._--It is obvious that the frictional losses of head
  in the wheel passages will increase as the velocity of flow is
  greater, that is, the smaller the wheel is made. But if the wheel
  works under water, the skin friction of the wheel cover increases as
  the diameter of the wheel is made greater, and in any case the weight
  of the wheel and consequently the journal friction increase as the
  wheel is made larger. It is therefore desirable to choose, for the
  velocity of flow, as large a value as is consistent with the condition
  that the frictional losses in the wheel passages are a small fraction
  of the total head.

  The values most commonly assumed in practice are these:--

    In axial flow turbines,   u_o = u_i = 0.15 to 0.2 [root](2gH);

    In outward flow turbines, u_i = 0.25 [root]2g(H - [h]),
                              u_o = 0.21 to 0.17 [root]2g(H - [h]);

    In inward flow turbines,  u_o = u_i = 0.125 [root](2gH).

  § 191. _Speed of the Wheel._--The best speed of the wheel depends
  partly on the frictional losses, which the ordinary theory of turbines
  disregards. It is best, therefore, to assume for V_o and V_i values
  which experiment has shown to be most advantageous.

  In axial flow turbines, the circumferential velocities at the mean
  radius of the wheel may be taken

    V_o = V_i = 0.6 [root](2gH) to 0.66 [root](2gH).

  In a radial outward flow turbine,

    V_i = 0.56 [root]{2g(H - [h])}

    V_o = V_i r_o/r_i,

  where r_o, r_i are the radii of the outlet and inlet surfaces.

  In a radial inward flow turbine,

    V_i = 0.66 [root](2gH),

    V_o = V_i r_o/r_i.

  If the wheel were stationary and the water flowed through it, the
  water would follow paths parallel to the wheel vane curves, at least
  when the vanes were so close that irregular motion was prevented.
  Similarly, when the wheel is in motion, the water follows paths
  relatively to the wheel, which are curves parallel to the wheel vanes.
  Hence the relative component, v_r, of the water's motion at c is
  tangential to a wheel vane curve drawn through the point c. Let v_o,
  V_o, v_(ro) be the velocity of the water and its common and relative
  components at the outlet surface of the wheel, and v_i, V_i, v_(ri) be
  the same quantities at the inlet surface; and let [theta] and [phi] be
  the angles the wheel vanes make with the inlet and outlet surfaces;
  then

    v_o² = [root](v_(ro)² + V_o² - 2V_o v_(ro) cos [phi])

    v_i = [root](v_(ri)² + V_o² - 2V_i v_(ri) cos [theta]),   (13)

  equations which may be used to determine [phi] and [theta].

  [Illustration: FIG. 195.]

  § 192. _Condition determining the Angle of the Vanes at the Outlet
  Surface of the Wheel._--It has been shown that, when the water leaves
  the wheel, it should have no tangential velocity, if the efficiency is
  to be as great as possible; that is, w_o = 0. Hence, from (10), cos
  [beta] = 0, [beta] = 90°, U_o = V_o, and the direction of the water's
  motion is normal to the outlet surface of the wheel, radial in radial
  flow, and axial in axial flow turbines.

  Drawing v_o or u_o radial or axial as the case may be, and V_o
  tangential to the direction of motion, v_(ro) can be found by the
  parallelogram of velocities. From fig. 195,

    tan [phi] = v_o/V_o = u_o/V_o;   (14)

  but [phi] is the angle which the wheel vane makes with the outlet
  surface of the wheel, which is thus determined when the velocity of
  flow u_o and velocity of the wheel V_o are known. When [phi] is thus
  determined,

    v_(ro) = U_o cosec [phi] = V_o [root](1 + u_o²/V_o²).   (14a)

  _Correction of the Angle [phi] to allow for Thickness of Vanes._--In
  determining [phi], it is most convenient to calculate its value
  approximately at first, from a value of u_o obtained by neglecting the
  thickness of the vanes. As, however, this angle is the most important
  angle in the turbine, the value should be afterwards corrected to
  allow for the vane thickness.

  Let

    [phi]´ = tan^(-1)(u_o/V_o) = tan^(-1)(Q/[Omega]_o V_o)

  be the first or approximate value of [phi], and let t be the
  thickness, and n the number of wheel vanes which reach the outlet
  surface of the wheel. As the vanes cut the outlet surface
  approximately at the angle [phi]´, their width measured on that
  surface is t cosec [phi]´. Hence the space occupied by the vanes on
  the outlet surface is

  For

    A, fig. 192, ntd_o cosec [phi]
    B, fig. 192, ntd cosec [phi]           (15)
    C, fig. 192, nt(r2 - r1) cosec [phi].

  Call this area occupied by the vanes [omega]. Then the true value of
  the clear discharging outlet of the wheel is [Omega]_o - [omega], and
  the true value of u_o is Q/([Omega]_o - [omega]). The corrected value
  of the angle of the vanes will be

    [phi] = tan [Q/V_o ([Omega]_o - [omega]) ].   (16)

  § 193. _Head producing Velocity with which the Water enters the
  Wheel._--Consider the variation of pressure in a wheel passage, which
  satisfies the condition that the sections change so gradually that
  there is no loss of head in shock. When the flow is in a horizontal
  plane, there is no work done by gravity on the water passing through
  the wheel. In the case of an axial flow turbine, in which the flow is
  vertical, the fall d between the inlet and outlet surfaces should be
  taken into account.

  Let

     V_i, V_o be the velocities of the wheel at the inlet and outlet
                surfaces,
     v_i, v_o the velocities of the water,
     u_i, u_o the velocities of flow,
     v_(ri), v_(ro) the relative velocities,
     h_i, h_o the pressures, measured in feet of water,
     r_i, r_o the radii of the wheel,
      [alpha] the angular velocity of the wheel.

  At any point in the path of a portion of water, at radius r, the
  velocity v of the water may be resolved into a component V = [alpha]r
  equal to the velocity at that point of the wheel, and a relative
  component v_r. Hence the motion of the water may be considered to
  consist of two parts:--(a) a motion identical with that in a forced
  vortex of constant angular velocity [alpha]; (b) a flow along curves
  parallel to the wheel vane curves. Taking the latter first, and using
  Bernoulli's theorem, the change of pressure due to flow through the
  wheel passages is given by the equation

    h´_i + v_(ri)²/2g = h´_o + v_(ro)²/2g;

    h´_i - h´_o = (v_(ro)² - v_(ri)²)/2g.

  The variation of pressure due to rotation in a forced vortex is

    h´´_i - h´´_o = (V_i² - V_o²)/2g.

  Consequently the whole difference of pressure at the inlet and outlet
  surfaces of the wheel is

    h_i - h_o = h´_i + h´´_i - h´_o - h´´_o
       = (V_i² - V_o²)/2g + (v_(ro)² - v_(ri)²)/2g.   (17)

  _Case 1. Axial Flow Turbines._--V_i = V_o; and the first term on the
  right, in equation 17, disappears. Adding, however, the work of
  gravity due to a fall of d ft. in passing through the wheel,

    h_i - h_o = (v_(ro)² - v_(ri)²)/2g - d.   (17a)

  _Case 2. Outward Flow Turbines._--The inlet radius is less than the
  outlet radius, and (V_i² - V_o²)/2g is negative. The centrifugal head
  diminishes the pressure at the inlet surface, and increases the
  velocity with which the water enters the wheel. This somewhat
  increases the frictional loss of head. Further, if the wheel varies in
  velocity from variations in the useful work done, the quantity (V_i² -
  V_o²)/2g increases when the turbine speed increases, and vice versa.
  Consequently the flow into the turbine increases when the speed
  increases, and diminishes when the speed diminishes, and this again
  augments the variation of speed. The action of the centrifugal head in
  an outward flow turbine is therefore prejudicial to steadiness of
  motion. For this reason r_o : r_i is made small, generally about 5 :
  4. Even then a governor is sometimes required to regulate the speed of
  the turbine.

  _Case 3. Inward Flow Turbines._--The inlet radius is greater than
  the outlet radius, and the centrifugal head diminishes the velocity of
  flow into the turbine. This tends to diminish the frictional losses,
  but it has a more important influence in securing steadiness of
  motion. Any increase of speed diminishes the flow into the turbine,
  and vice versa. Hence the variation of speed is less than the
  variation of resistance overcome. In the so-called centre vent wheels
  in America, the ratio r_i : r_o is about 5 : 4, and then the influence
  of the centrifugal head is not very important. Professor James Thomson
  first pointed out the advantage of a much greater difference of radii.
  By making r_i : r_o = 2 : 1, the centrifugal head balances about half
  the head in the supply chamber. Then the velocity through the
  guide-blades does not exceed the velocity due to half the fall, and
  the action of the centrifugal head in securing steadiness of speed is
  considerable.

  Since the total head producing flow through the turbine is H -
  [h], of this h_i - h_o is expended in overcoming the pressure
  in the wheel, the velocity of flow into the wheel is

    v_i = c_v[root]{2g(H - [h] - (V_i² - V_o²/2g + (v{r0}² - v_(ri)²)/2g)},   (18)

  where c_v may be taken 0.96.

  From (14a),

    v{r0} = V_o [root](1 + u_o²/V_o²).

  It will be shown immediately that

    v_(ri) = u_i cosec [theta];

  or, as this is only a small term, and [theta] is on the average 90°,
  we may take, for the present purpose, v_(ri) = u_i nearly.

  Inserting these values, and remembering that for an axial flow turbine
  V_i = V_o, [h] = 0, and the fall d in the wheel is to be
  added,
                     _                                        _
                    |     /    V_i²  /    u_o² \    u_i²    \  |
    v_i = c_v[root] | 2g ( H - ---- ( 1 + ----  ) + ---- - d ) |.
                    |_    \     2g   \    V_o² /     2g     / _|

  For an outward flow turbine,
                     _                                           _
                    |     /          V_i²  /    u_o² \    u_i² \  |
    v_i = c_v[root] | 2g ( H - [h] - ---- ( 1 + ----  ) + ----  ) |.
                    |_    \           2g   \    V_i² /     2g  / _|

  For an inward flow turbine,
                     _                                     _
                    |    {     V_i²  /    u_o² \    u_i² }  |
    v_i = c_v[root] | 2g { H - ---- ( 1 + ----  ) + ---- }  |.
                    |_   {      2g   \    V_i² /     2g  } _|

  § 194. _Angle which the Guide-Blades make with the Circumference of
  the Wheel._--At the moment the water enters the wheel, the radial
  component of the velocity is u_i, and the velocity is v_i. Hence, if
  [gamma] is the angle between the guide-blades and a tangent to the
  wheel

    [gamma] = sin^(-1) (u_i/v_i).

  This angle can, if necessary, be corrected to allow for the thickness
  of the guide-blades.

  [Illustration: FIG. 196.]

  § 195. _Condition determining the Angle of the Vanes at the Inlet
  Surface of the Wheel._--The single condition necessary to be satisfied
  at the inlet surface of the wheel is that the water should enter the
  wheel without shock. This condition is satisfied if the direction of
  relative motion of the water and wheel is parallel to the first
  element of the wheel vanes.

  Let A (fig. 196) be a point on the inlet surface of the wheel, and let
  v_i represent in magnitude and direction the velocity of the water
  entering the wheel, and V_i the velocity of the wheel. Completing the
  parallelogram, v_(ri) is the direction of relative motion. Hence the
  angle between v_(ri) and V_i is the angle [theta] which the vanes
  should make with the inlet surface of the wheel.

  § 196. _Example of the Method of designing a Turbine. Professor James
  Thomson's Inward Flow Turbine._--

  Let

    H = the available fall after deducting loss of head in pipes and
          channels from the gross fall;
    Q = the supply of water in cubic feet per second; and
    [eta] = the efficiency of the turbine.

  The work done per second is [eta]GQH, and the horse-power of the
  turbine is h.p. = [eta]GQH/550. If [eta] is taken at 0.75, an
  allowance will be made for the frictional losses in the turbine, the
  leakage and the friction of the turbine shaft. Then h.p. = 0.085QH.

  The velocity of flow through the turbine (uncorrected for the space
  occupied by the vanes and guide-blades) may be taken

    u_i = u_i = 0.125 [root](2gH),

  in which case about (1/64)th of the energy of the fall is carried away
  by the water discharged.

  The areas of the outlet and inlet surface of the wheel are then

    2[pi]r_o d_o = 2[pi]r_i d_i = Q/0.125 [root](2gH).

  If we take r_o, so that the axial velocity of discharge from the
  central orifices of the wheel is equal to u_o, we get

    r_o = 0.3984 [root](Q/[root]H),

    d_o = r_o.

  If, to obtain considerable steadying action of the centrifugal head,
  r_i = 2r_o, then d_i = ½d_o.

  _Speed of the Wheel._--Let V_i = 0.66 [root](2gH), or the speed due to
  half the fall nearly. Then the number of rotations of the turbine per
  second is

    N = V_i/2[pi]r_i = 1.0579 [root](H[root]H/Q);

  also

    V_o = V_i r_o/r_i = 0.33 [root](2gH).

  _Angle of Vanes with Outlet Surface._

    Tan[phi] = u_o/V_o = 0.125/0.33 = .3788;

    [phi] = 21º nearly.

  If this value is revised for the vane thickness it will ordinarily
  become about 25º.

  _Velocity with which the Water enters the Wheel._--The head producing
  the velocity is

    H - (V_i²/2g) (1 + u_o²/V_i²) + u_i²/2g
      = H {1 - .4356 (1 + 0.0358) + .0156}
      = 0.5646H.

  Then the velocity is

    V_i = .96 [root](2g(.5646H)) = 0.721 [root](2gH).

  _Angle of Guide-Blades._

    Sin [gamma] = u_i/v_i = 0.125/0.721 = 0.173;

    [gamma] = 10° nearly.

  _Tangential Velocity of Water entering Wheel._

    w_i = v_i cos [gamma] = 0.7101 [root](2gH).

  _Angle of Vanes at Inlet Surface._

    Cot [theta] = (w_i - V_i)/u_i = (.7101 - .66)/.125 = .4008;

    [theta] = 68° nearly.

  _Hydraulic Efficiency of Wheel._

    [eta] = w_iV_i/gH = .7101 × .66 × 2
          = 0.9373.

  This, however, neglects the friction of wheel covers and leakage. The
  efficiency from experiment has been found to be 0.75 to 0.80.


_Impulse and Partial Admission Turbines._

§ 197. The principal defect of most turbines with complete admission is
the imperfection of the arrangements for working with less than the
normal supply. With many forms of reaction turbine the efficiency is
considerably reduced when the regulating sluices are partially
closed, but it is exactly when the supply of water is deficient that it
is most important to get out of it the greatest possible amount of work.
The imperfection of the regulating arrangements is therefore, from the
practical point of view, a serious defect. All turbine makers have
sought by various methods to improve the regulating mechanism. B.
Fourneyron, by dividing his wheel by horizontal diaphragms, virtually
obtained three or more separate radial flow turbines, which could be
successively set in action at their full power, but the arrangement is
not altogether successful, because of the spreading of the water in the
space between the wheel and guide-blades. Fontaine similarly employed
two concentric axial flow turbines formed in the same casing. One was
worked at full power, the other regulated. By this arrangement the loss
of efficiency due to the action of the regulating sluice affected only
half the water power. Many makers have adopted the expedient of erecting
two or three separate turbines on the same waterfall. Then one or more
could be put out of action and the others worked at full power. All
these methods are rather palliatives than remedies. The movable
guide-blades of Professor James Thomson meet the difficulty directly,
but they are not applicable to every form of turbine.

[Illustration: FIG. 197.]

C. Callon, in 1840, patented an arrangement of sluices for axial or
outward flow turbines, which were to be closed successively as the water
supply diminished. By preference the sluices were closed by pairs, two
diametrically opposite sluices forming a pair. The water was thus
admitted to opposite but equal arcs of the wheel, and the forces driving
the turbine were symmetrically placed. As soon as this arrangement was
adopted, a modification of the mode of action of the water in the
turbine became necessary. If the turbine wheel passages remain full of
water during the whole rotation, the water contained in each passage
must be put into motion each time it passes an open portion of the
sluice, and stopped each time it passes a closed portion of the sluice.
It is thus put into motion and stopped twice in each rotation. This
gives rise to violent eddying motions and great loss of energy in shock.
To prevent this, the turbine wheel with partial admission must be placed
above the tail water, and the wheel passages be allowed to clear
themselves of water, while passing from one open portion of the sluices
to the next.

But if the wheel passages are free of water when they arrive at the open
guide passages, then there can be no pressure other than atmospheric
pressure in the clearance space between guides and wheel. The water must
issue from the sluices with the whole velocity due to the head; received
on the curved vanes of the wheel, the jets must be gradually deviated
and discharged with a small final velocity only, precisely in the same
way as when a single jet strikes a curved vane in the free air. Turbines
of this kind are therefore termed turbines of free deviation. There is
no variation of pressure in the jet during the whole time of its action
on the wheel, and the whole energy of the jet is imparted to the wheel,
simply by the impulse due to its gradual change of momentum. It is clear
that the water may be admitted in exactly the same way to any fraction
of the circumference at pleasure, without altering the efficiency of the
wheel. The diameter of the wheel may be made as large as convenient, and
the water admitted to a small fraction of the circumference only. Then
the number of revolutions is independent of the water velocity, and may
be kept down to a manageable value.

[Illustration: FIG. 198.]

[Illustration: FIG. 199.]

  § 198. _General Description of an Impulse Turbine or Turbine with Free
  Deviation._--Fig. 197 shows a general sectional elevation of a Girard
  turbine, in which the flow is axial. The water, admitted above a
  horizontal floor, passes down through the annular wheel containing the
  guide-blades G, G, and thence into the revolving wheel WW. The
  revolving wheel is fixed to a hollow shaft suspended from the pivot p.
  The solid internal shaft ss is merely a fixed column supporting the
  pivot. The advantage of this is that the pivot is accessible for
  lubrication and adjustment. B is the mortise bevel wheel by which the
  power of the turbine is given off. The sluices are worked by the hand
  wheel h, which raises them successively, in a way to be described
  presently. d, d are the sluice rods. Figs. 198, 199 show the sectional
  form of the guide-blade chamber and wheel and the curves of the wheel
  vanes and guide-blades, when drawn on a plane development of the
  cylindrical section of the wheel; a, a, a are the sluices for cutting
  off the water; b, b, b are apertures by which the entrance or exit of
  air is facilitated as the buckets empty and fill. Figs. 200, 201 show
  the guide-blade gear. a, a, a are the sluice rods as before. At the
  top of each sluice rod is a small block c, having a projecting tongue,
  which slides in the groove of the circular cam plate d, d. This
  circular plate is supported on the frame e, and revolves on it by
  means of the flanged rollers f. Inside, at the top, the cam plate is
  toothed, and gears into a spur pinion connected with the hand wheel h.
  At gg is an inclined groove or shunt. When the tongues of the blocks
  c, c arrive at g, they slide up to a second groove, or the reverse,
  according as the cam plate is revolved in one direction or in the
  other. As this operation takes place with each sluice successively,
  any number of sluices can be opened or closed as desired. The turbine
  is of 48 horse power on 5.12 ft. fall, and the supply of water varies
  from 35 to 112 cub. ft. per second. The efficiency in normal working
  is given as 73%. The mean diameter of the wheel is 6 ft., and the
  speed 27.4 revolutions per minute.

  [Illustration: FIG. 200.]

  [Illustration: FIG. 201.]

  [Illustration: FIG. 202.]

  As an example of a partial admission radial flow impulse turbine, a
  100 h.p. turbine at Immenstadt may be taken. The fall varies from 538
  to 570 ft. The external diameter of the wheel is 4½ ft., and its
  internal diameter 3 ft. 10 in. Normal speed 400 revs. per minute.
  Water is discharged into the wheel by a single nozzle, shown in fig.
  202 with its regulating apparatus and some of the vanes. The water
  enters the wheel at an angle of 22° with the direction of motion, and
  the final angle of the wheel vanes is 20°. The efficiency on trial was
  from 75 to 78%.

  § 199. _Theory of the Impulse Turbine._--The theory of the impulse
  turbine does not essentially differ from that of the reaction turbine,
  except that there is no pressure in the wheel opposing the discharge
  from the guide-blades. Hence the velocity with which the water enters
  the wheel is simply

    v_i = 0.96 [root]{2g(H - [h])},

  where [heta] is the height of the top of the wheel above the tail
  water. If the hydropneumatic system is used, then [h] = 0. Let
  Q_m be the maximum supply of water, r1, r2 the internal and external
  radii of the wheel at the inlet surface; then

    u_i = Q_m/{[pi](r2² - r1²)}.

  The value of u_i may be about 0.45 [root]{2g(H - [eta][h])},
  whence r1, r2 can be determined.

  The guide-blade angle is then given by the equation

    sin [gamma] = u_i/v_i = 0.45/0.94 = .48;

    [gamma] = 29°.

  The value of u_i should, however, be corrected for the space occupied
  by the guide-blades.

  The tangential velocity of the entering water is

    w_i = v_i cos [gamma] = 0.82 [root]{2g(H - [h])}.

  The circumferential velocity of the wheel may be (at mean radius)

    V_i = 0.5 [root]{2g(H - [h])}.

  Hence the vane angle at inlet surface is given by the equation

    cot [theta] = (w_i - V_i)/u_i = (0.82 - 0.5)/0.45 = .71;

    [theta] = 55°.

  The relative velocity of the water striking the vane at the inlet edge
  is v_(ri) = u_i cosec[theta] = 1.22 u_i. This relative velocity remains
  unchanged during the passage of the water over the vane; consequently
  the relative velocity at the point of discharge is v_(ro) = 1.22 u_i.
  Also in an axial flow turbine V_o = V_i.

  If the final velocity of the water is axial, then

    cos [phi] = V_o/v_(ro) = V_i/v_(ri) = 0.5/(1.22 × 0.45) = cos 24º 23´.

  This should be corrected for the vane thickness. Neglecting this, u_o
  = v_(ro) sin [phi] = v_(ri) sin [phi] = u_i cosec [theta] sin [phi] =
  0.5u_i. The discharging area of the wheel must therefore be greater
  than the inlet area in the ratio of at least 2 to 1. In some actual
  turbines the ratio is 7 to 3. This greater outlet area is obtained by
  splaying the wheel, as shown in the section (fig. 199).

  [Illustration: FIG. 203.]

  § 200. _Pelton Wheel._--In the mining district of California about
  1860 simple impulse wheels were used, termed hurdy-gurdy wheels. The
  wheels rotated in a vertical plane, being supported on a horizontal
  axis. Round the circumference were fixed flat vanes which were struck
  normally by a jet from a nozzle of size varying with the head and
  quantity of water. Such wheels have in fact long been used. They are
  not efficient, but they are very simply constructed. Then attempts
  were made to improve the efficiency, first by using hemispherical cup
  vanes, and then by using a double cup vane with a central dividing
  ridge, an arrangement invented by Pelton. In this last form the water
  from the nozzle passes half to each side of the wheel, just escaping
  clear of the backs of the advancing buckets. Fig. 203 shows a Pelton
  vane. Some small modifications have been made by other makers, but
  they are not of any great importance. Fig. 204 shows a complete Pelton
  wheel with frame and casing, supply pipe and nozzle. Pelton wheels
  have been very largely used in America and to some extent in Europe.
  They are extremely simple and easy to construct or repair and on falls
  of 100 ft. or more are very efficient. The jet strikes tangentially to
  the mean radius of the buckets, and the face of the buckets is not
  quite radial but at right angles to the direction of the jet at the
  point of first impact. For greatest efficiency the peripheral velocity
  of the wheel at the mean radius of the buckets should be a little less
  than half the velocity of the jet. As the radius of the wheel can be
  taken arbitrarily, the number of revolutions per minute can be
  accommodated to that of the machinery to be driven. Pelton wheels have
  been made as small as 4 in. diameter, for driving sewing machines, and
  as large as 24 ft. The efficiency on high falls is about 80%. When
  large power is required two or three nozzles are used delivering on
  one wheel. The width of the buckets should be not less than seven
  times the diameter of the jet.

  [Illustration: FIG. 204.]

  At the Comstock mines, Nevada, there is a 36-in. Pelton wheel made of
  a solid steel disk with phosphor bronze buckets riveted to the rim.
  The head is 2100 ft. and the wheel makes 1150 revolutions per minute,
  the peripheral velocity being 180 ft. per sec. With a ½-in. nozzle the
  wheel uses 32 cub. ft. of water per minute and develops 100 h.p. At
  the Chollarshaft, Nevada, there are six Pelton wheels on a fall of
  1680 ft. driving electrical generators. With 5/8-in. nozzles each
  develops 125 h.p.

  [Illustration: FIG. 205]

  § 201. _Theory of the Pelton Wheel._--Suppose a jet with a velocity v
  strikes tangentially a curved vane AB (fig. 205) moving in the same
  direction with the velocity u. The water will flow over the vane with
  the relative velocity v - u and at B will have the tangential
  relative velocity v - u making an angle [alpha] with the direction of
  the vane's motion. Combining this with the velocity u of the vane, the
  absolute velocity of the water leaving the vane will be w = Bc. The
  component of w in the direction of motion of the vane is Ba = Bb - ab
  = u - (v - u) cos [alpha]. Hence if Q is the quantity of water
  reaching the vane per second the change of momentum per second in the
  direction of the vane's motion is (GQ/g)[v - {u - (v - u) cos
  [alpha]}] = (GQ/g)(v - u)(1 + cos [alpha]). If a = 0°, cos [alpha] =
  1, and the change of momentum per second, which is equal to the effort
  driving the vane, is P = 2(GQ/g)(v - u). The work done on the vane is
  Pu = 2(GQ/g)(v - u)u. If a series of vanes are interposed in
  succession, the quantity of water impinging on the vanes per second is
  the total discharge of the nozzle, and the energy expended at the
  nozzle is GQv²/2g. Hence the efficiency of the arrangement is, when
  [alpha] = 0°, neglecting friction,

    [eta] = 2Pu/GQv² = 4(v - u)u/v²,

  which is a maximum and equal to unity if u = ½v. In that case the
  whole energy of the jet is usefully expended in driving the series of
  vanes. In practice [alpha] cannot be quite zero or the water leaving
  one vane would strike the back of the next advancing vane. Fig. 203
  shows a Pelton vane. The water divides each way, and leaves the vane
  on each side in a direction nearly parallel to the direction of motion
  of the vane. The best velocity of the vane is very approximately half
  the velocity of the jet.

  § 202. _Regulation of the Pelton Wheel._--At first Pelton wheels were
  adjusted to varying loads merely by throttling the supply. This method
  involves a total loss of part of the head at the sluice or throttle
  valve. In addition as the working head is reduced, the relation
  between wheel velocity and jet velocity is no longer that of greatest
  efficiency. Next a plan was adopted of deflecting the jet so that only
  part of the water reached the wheel when the load was reduced, the
  rest going to waste. This involved the use of an equal quantity of
  water for large and small loads, but it had, what in some cases is an
  advantage, the effect of preventing any water hammer in the supply
  pipe due to the action of the regulator. In most cases now regulation
  is effected by varying the section of the jet. A conical needle in the
  nozzle can be advanced or withdrawn so as to occupy more or less of
  the aperture of the nozzle. Such a needle can be controlled by an
  ordinary governor.

§ 203. _General Considerations on the Choice of a Type of Turbine._--The
circumferential speed of any turbine is necessarily a fraction of the
initial velocity of the water, and therefore is greater as the head is
greater. In reaction turbines with complete admission the number of
revolutions per minute becomes inconveniently great, for the diameter
cannot be increased beyond certain limits without greatly reducing the
efficiency. In impulse turbines with partial admission the diameter can
be chosen arbitrarily and the number of revolutions kept down on high
falls to any desired amount. Hence broadly reaction turbines are better
and less costly on low falls, and impulse turbines on high falls. For
variable water flow impulse turbines have some advantage, being more
efficiently regulated. On the other hand, impulse turbines lose
efficiency seriously if their speed varies from the normal speed due to
the head. If the head is very variable, as it often is on low falls, and
the turbine must run at the same speed whatever the head, the impulse
turbine is not suitable. Reaction turbines can be constructed so as to
overcome this difficulty to a great extent. Axial flow turbines with
vertical shafts have the disadvantage that in addition to the weight of
the turbine there is an unbalanced water pressure to be carried by the
footstep or collar bearing. In radial flow turbines the hydraulic
pressures are balanced. The application of turbines to drive dynamos
directly has involved some new conditions. The electrical engineer
generally desires a high speed of rotation, and a very constant speed at
all times. The reaction turbine is generally more suitable than the
impulse turbine. As the diameter of the turbine depends on the quantity
of water and cannot be much varied without great inefficiency, a
difficulty arises on low falls. This has been met by constructing four
independent reaction turbines on the same shaft, each having of course
the diameter suitable for one-quarter of the whole discharge, and having
a higher speed of rotation than a larger turbine. The turbines at
Rheinfelden and Chevres are so constructed. To ensure constant speed of
rotation when the head varies considerably without serious inefficiency,
an axial flow turbine is generally used. It is constructed of three or
four concentric rings of vanes, with independent regulating sluices,
forming practically independent turbines of different radii. Any one of
these or any combination can be used according to the state of the
water. With a high fall the turbine of largest radius only is used, and
the speed of rotation is less than with a turbine of smaller radius. On
the other hand, as the fall decreases the inner turbines are used either
singly or together, according to the power required. At the Zürich
waterworks there are turbines of 90 h.p. on a fall varying from 10½ ft.
to 4¾ ft. The power and speed are kept constant. Each turbine has three
concentric rings. The outermost ring gives 90 h.p. with 105 cub. ft. per
second and the maximum fall. The outer and middle compartments give the
same power with 140 cub. ft. per second and a fall of 7 ft. 10 in. All
three compartments working together develop the power with about 250
cub. ft. per second. In some tests the efficiency was 74% with the outer
ring working alone, 75.4% with the outer and middle ring working and a
fall of 7 ft., and 80.7% with all the rings working.

[Illustration: FIG. 206.]

§ 204. _Speed Governing._--When turbines are used to drive dynamos
direct, the question of speed regulation is of great importance. Steam
engines using a light elastic fluid can be easily regulated by governors
acting on throttle or expansion valves. It is different with water
turbines using a fluid of great inertia. In one of the Niagara penstocks
there are 400 tons of water flowing at 10 ft. per second, opposing
enormous resistance to rapid change of speed of flow. The sluices of
water turbines also are necessarily large and heavy. Hence relay
governors must be used, and the tendency of relay governors to
hunt must be overcome. In the Niagara Falls Power House No. 1, each
turbine has a very sensitive centrifugal governor acting on a ratchet
relay. The governor puts into gear one or other of two ratchets driven
by the turbine itself. According as one or the other ratchet is in gear
the sluices are raised or lowered. By a subsidiary arrangement the
ratchets are gradually put out of gear unless the governor puts them in
gear again, and this prevents the over correction of the speed from the
lag in the action of the governor. In the Niagara Power House No. 2, the
relay is an hydraulic relay similar in principle, but rather more
complicated in arrangement, to that shown in fig. 206, which is a
governor used for the 1250 h.p. turbines at Lyons. The sensitive
governor G opens a valve and puts into action a plunger driven by oil
pressure from an oil reservoir. As the plunger moves forward it
gradually closes the oil admission valve by lowering the fulcrum end f
of the valve lever which rests on a wedge w attached to the plunger. If
the speed is still too high, the governor reopens the valve. In the case
of the Niagara turbines the oil pressure is 1200 lb. per sq. in. One
millimetre of movement of the governor sleeve completely opens the relay
valve, and the relay plunger exerts a force of 50 tons. The sluices can
be completely opened or shut in twelve seconds. The ordinary variation
of speed of the turbine with varying load does not exceed 1%. If all the
load is thrown off, the momentary variation of speed is not more than
5%. To prevent hydraulic shock in the supply pipes, a relief valve is
provided which opens if the pressure is in excess of that due to the
head.

[Illustration: FIG. 207.]

§ 205. _The Hydraulic Ram._--The hydraulic ram is an arrangement by
which a quantity of water falling a distance h forces a portion of the
water to rise to a height h1, greater than h. It consists of a supply
reservoir (A, fig. 207), into which the water enters from some natural
stream. A pipe s of considerable length conducts the water to a lower
level, where it is discharged intermittently through a self-acting
pulsating valve at d. The supply pipe s may be fitted with a flap valve
for stopping the ram, and this is attached in some cases to a float, so
that the ram starts and stops itself automatically, according as the
supply cistern fills or empties. The lower float is just sufficient to
keep open the flap after it has been raised by the action of the upper
float. The length of chain is adjusted so that the upper float opens the
flap when the level in the cistern is at the desired height. If the
water-level falls below the lower float the flap closes. The pipe s
should be as long and as straight as possible, and as it is subjected to
considerable pressure from the sudden arrest of the motion of the water,
it must be strong and strongly jointed. a is an air vessel, and e the
delivery pipe leading to the reservoir at a higher level than A, into
which water is to be pumped. Fig. 208 shows in section the construction
of the ram itself. d is the pulsating discharge valve already mentioned,
which opens inwards and downwards. The stroke of the valve is regulated
by the cotter through the spindle, under which are washers by which the
amount of fall can be regulated. At o is a delivery valve, opening
outwards, which is often a ball-valve but sometimes a flap-valve. The
water which is pumped passes through this valve into the air vessel a,
from which it flows by the delivery pipe in a regular stream into the
cistern to which the water is to be raised. In the vertical chamber
behind the outer valve a small air vessel is formed, and into this
opens an aperture ¼ in. in diameter, made in a brass screw plug b. The
hole is reduced to 1/16 in. in diameter at the outer end of the plug
and is closed by a small valve opening inwards. Through this, during the
rebound after each stroke of the ram, a small quantity of air is sucked
in which keeps the air vessel supplied with its elastic cushion of air.

[Illustration: FIG. 208.]

During the recoil after a sudden closing of the valve d, the pressure
below it is diminished and the valve opens, permitting outflow. In
consequence of the flow through this valve, the water in the supply pipe
acquires a gradually increasing velocity. The upward flow of the water,
towards the valve d, increases the pressure tending to lift the valve,
and at last, if the valve is not too heavy, lifts and closes it. The
forward momentum of the column in the supply pipe being destroyed by the
stoppage of the flow, the water exerts a pressure at the end of the pipe
sufficient to open the delivery valve o, and to cause a portion of the
water to flow into the air vessel. As the water in the supply pipe comes
to rest and recoils, the valve d opens again and the operation is
repeated. Part of the energy of the descending column is employed in
compressing the air at the end of the supply pipe and expanding the pipe
itself. This causes a recoil of the water which momentarily diminishes
the pressure in the pipe below the pressure due to the statical head.
This assists in opening the valve d. The recoil of the water is
sufficiently great to enable a pump to be attached to the ram body
instead of the direct rising pipe. With this arrangement a ram working
with muddy water may be employed to raise clear spring water. Instead of
lifting the delivery valve as in the ordinary ram, the momentum of the
column drives a sliding or elastic piston, and the recoil brings it
back. This piston lifts and forces alternately the clear water through
ordinary pump valves.


PUMPS

§ 206. The different classes of pumps correspond almost exactly to the
different classes of water motors, although the mechanical details of
the construction are somewhat different. They are properly reversed
water motors. Ordinary reciprocating pumps correspond to water-pressure
engines. Chain and bucket pumps are in principle similar to water wheels
in which the water acts by weight. Scoop wheels are similar to undershot
water wheels, and centrifugal pumps to turbines.

_Reciprocating Pumps_ are single or double acting, and differ from
water-pressure engines in that the valves are moved by the water instead
of by automatic machinery. They may be classed thus:--

1. _Lift Pumps._--The water drawn through a foot valve on the ascent of
the pump bucket is forced through the bucket valve when it descends, and
lifted by the bucket when it reascends. Such pumps give an intermittent
discharge.

2. _Plunger or Force Pumps_, in which the water drawn through the foot
valve is displaced by the descent of a solid plunger, and forced through
a delivery valve. They have the advantage that  the friction is less
than that of lift pumps, and the packing round the plunger is easily
accessible, whilst that round a lift pump bucket is not. The flow is
intermittent.

3. _The Double-acting Force Pump_ is in principle a double plunger pump.
The discharge fluctuates from zero to a maximum and back to zero each
stroke, but is not arrested for any appreciable time.

4. _Bucket and Plunger Pumps_ consist of a lift pump bucket combined
with a plunger of half its area. The flow varies as in a double-acting
pump.

5. _Diaphragm Pumps_ have been used, in which the solid plunger is
replaced by an elastic diaphragm, alternately depressed into and raised
out of a cylinder.

As single-acting pumps give an intermittent discharge three are
generally used on cranks at 120°. But with all pumps the variation of
velocity of discharge would cause great waste of work in the delivery
pipes when they are long, and even danger from the hydraulic ramming
action of the long column of water. An air vessel is interposed between
the pump and the delivery pipes, of a volume from 5 to 100 times the
space described by the plunger per stroke. The air in this must be
replenished from time to time, or continuously, by a special air-pump.
At low speeds not exceeding 30 ft. per minute the delivery of a pump is
about 90 to 95% of the volume described by the plunger or bucket, from 5
to 10% of the discharge being lost by leakage. At high speeds the
quantity pumped occasionally exceeds the volume described by the
plunger, the momentum of the water keeping the valves open after the
turn of the stroke.

The velocity of large mining pumps is about 140 ft. per minute, the
indoor or suction stroke being sometimes made at 250 ft. per minute.
Rotative pumping engines of large size have a plunger speed of 90 ft.
per minute. Small rotative pumps are run faster, but at some loss of
efficiency. Fire-engine pumps have a speed of 180 to 220 ft. per minute.

The efficiency of reciprocating pumps varies very greatly. Small
reciprocating pumps, with metal valves on lifts of 15 ft., were found by
Morin to have an efficiency of 16 to 40%, or on the average 25%. When
used to pump water at considerable pressure, through hose pipes, the
efficiency rose to from 28 to 57%, or on the average, with 50 to 100 ft.
of lift, about 50%. A large pump with barrels 18 in. diameter, at speeds
under 60 ft. per minute, gave the following results:--

  Lift in feet    14½   34    47
  Efficiency     .46   .66   .70

The very large steam-pumps employed for waterworks, with 150 ft. or more
of lift, appear to reach an efficiency of 90%, not including the
friction of the discharge pipes. Reckoned on the indicated work of the
steam-engine the efficiency may be 80%.

Many small pumps are now driven electrically and are usually three-throw
single-acting pumps driven from the electric motor by gearing. It is not
convenient to vary the speed of the motor to accommodate it to the
varying rate of pumping usually required. Messrs Hayward Tyler have
introduced a mechanism for varying the stroke of the pumps (Sinclair's
patent) from full stroke to nil, without stopping the pumps.

§ 207. _Centrifugal Pump._--For large volumes of water on lifts not
exceeding about 60 ft. the most convenient pump is the centrifugal pump.
Recent improvements have made it available also for very high lifts. It
consists of a wheel or fan with curved vanes enclosed in an annular
chamber. Water flows in at the centre and is discharged at the
periphery. The fan may rotate in a vertical or horizontal plane and the
water may enter on one or both sides of the fan. In the latter case
there is no axial unbalanced pressure. The fan and its casing must be
filled with water before it can start, so that if not drowned there must
be a foot valve on the suction pipe. When no special attention needs to
be paid to efficiency the water may have a velocity of 6 to 7 ft. in the
suction and delivery pipes. The fan often has 6 to 12 vanes. For a
double-inlet fan of diameter D, the diameter of the inlets is D/2. If Q
is the discharge in cub. ft. per second D = about 0.6 [root]Q in average
cases. The peripheral speed is a little greater than the velocity due
to the lift. Ordinary centrifugal pumps will have an efficiency of 40 to
60%.

The first pump of this kind which attracted notice was one exhibited by
J. G. Appold in 1851, and the special features of his pump have been
retained in the best pumps since constructed. Appold's pump raised
continuously a volume of water equal to 1400 times its own capacity per
minute. It had no valves, and it permitted the passage of solid bodies,
such as walnuts and oranges, without obstruction to its working. Its
efficiency was also found to be good.

[Illustration: FIG. 209.]

Fig. 209 shows the ordinary form of a centrifugal pump. The pump disk
and vanes B are cast in one, usually of bronze,

and the disk is keyed on the driving shaft C. The casing A has a
spirally enlarging discharge passage into the discharge pipe K. A cover
L gives access to the pump. S is the suction pipe which opens into the
pump disk on both sides at D.

Fig. 210 shows a centrifugal pump differing from ordinary centrifugal
pumps in one feature only. The water rises through a suction pipe S,
which divides so as to enter the pump wheel W at the centre on each
side. The pump disk or wheel is very similar to a turbine wheel. It is
keyed on a shaft driven by a belt on a fast and loose pulley arrangement
at P. The water rotating in the pump disk presses outwards, and if the
speed is sufficient a continuous flow is maintained through the pump and
into the discharge pipe D. The special feature in this pump is that the
water, discharged by the pump disk with a whirling velocity of not
inconsiderable magnitude, is allowed to continue rotation in a chamber
somewhat larger than the pump. The use of this whirlpool chamber was
first suggested by Professor James Thomson. It utilizes the energy due
to the whirling velocity of the water which in most pumps is wasted in
eddies in the discharge pipe. In the pump shown guide-blades are also
added which have the direction of the stream lines in a free vortex.
They do not therefore interfere with the action of the water when
pumping the normal quantity, but only prevent irregular motion. At A is
a plug by which the pump case is filled before starting. If the pump is
above the water to be pumped, a foot valve is required to permit the
pump to be filled. Sometimes instead of the foot valve a delivery valve
is used, an air-pump or steam jet pump being employed to exhaust the air
from the pump case.

[Illustration: FIG. 210.]

  § 208. _Design and Proportions of a Centrifugal Pump._--The design of
  the pump disk is very simple. Let r_i, r_o be the radii of the inlet
  and outlet surfaces of the pump disk, d_i, d_o the clear axial width
  at those radii. The velocity of flow through the pump may be taken
  the same as for a turbine. If Q is the quantity pumped, and H the
  lift,

    u_i = 0.25 [root](2gH).   (1)

    2[pi]r_i d_i = Q/u_i.

  Also in practice

    d_i = 1.2 r_i ....

  Hence,

    r_i = .2571 [root](Q/[root]H).   (2)

  Usually

    r_o = 2r_i,

  and

    d_o = d_i or ½d_i

  according as the disk is parallel-sided or coned. The water enters the
  wheel radially with the velocity u_i, and

    u_o = Q/2[pi]r_o d_o.   (3)

  [Illustration: FIG. 211.]

  Fig. 211 shows the notation adopted for the velocities. Suppose the
  water enters the wheel with the velocity v_i, while the velocity of
  the wheel is V_i. Completing the parallelogram, v_(ri) is the relative
  velocity of the water and wheel, and is the proper direction of the
  wheel vanes. Also, by resolving, u_i and w_i are the component
  velocities of flow and velocities of whir of the velocity v_i of the
  water. At the outlet surface, v_o is the final velocity of discharge,
  and the rest of the notation is similar to that for the inlet surface.

  Usually the water flows equally in all directions in the eye of the
  wheel, in that case v_i is radial. Then, in normal conditions of
  working, at the inlet surface,

    v_i = u_i                                        \
    w_i = 0                                           > (4)
    tan[theta] = u_i/V_i                              |
    v_(ri) = u_i cosec [theta] = [root](u_i² + V_i²) /

  If the pump is raising less or more than its proper quantity, [theta]
  will not satisfy the last condition, and there is then some loss of
  head in shock.

  At the outer circumference of the wheel or outlet surface,

    v_(ro) = u_o cosec [phi]                       \
    w_o = V_o - u_o cot [phi]                       > (5)
    v_o = [root]{u_o² + (V - _o - u_o cot [phi])²} /

  _Variation of Pressure in the Pump Disk._--Precisely as in the case of
  turbines, it can be shown that the variation of pressure between the
  inlet and outlet surfaces of the pump is

    h_o - h_i = (V_o² - V_i²)/2g - (v_(ro)² - v_(ri)²)/2g.

  Inserting the values of v_(ro), v_(ri) in (4) and (5), we get for
  normal conditions of working

    h_o -h_i = (V_o² - V_i²)/2g - u_o² cosec² [phi]/2g + (u_i² + V_i²)/2g
      = V_o²/2g - u_o² cosec² [phi]/2g + u_i²/2g.   (6)

  _Hydraulic Efficiency of the Pump._--Neglecting disk friction, journal
  friction, and leakage, the efficiency of the pump can be found in the
  same way as that of turbines (§ 186). Let M be the moment of the
  couple rotating the pump, and [alpha] its angular velocity; w_o, r_o
  the tangential velocity of the water and radius at the outlet surface;
  w_i, r_i the same quantities at the inlet surface. Q being the
  discharge per second, the change of angular momentum per second is

    (GQ/g)(w_o r_o - w_i r_i).

  Hence

    M = (GQ/g)(w_o r_o - w_i r_i).

  In normal working, w_i = 0. Also, multiplying by the angular velocity,
  the work done per second is

    M[alpha] = (GQ/g)w_o r_o[alpha].

  But the useful work done in pumping is GQH. Therefore the efficiency
  is

    [eta] = GQH/M[alpha] = gH/w_o r_o[alpha] = gH/w_o V_o.   (7)

  § 209. Case 1. _Centrifugal Pump with no Whirlpool Chamber._--When no
  special provision is made to utilize the energy of motion of the water
  leaving the wheel, and the pump discharges directly into a chamber in
  which the water is flowing to the discharge pipe, nearly the whole of
  the energy of the water leaving the disk is wasted. The water leaves
  the disk with the more or less considerable velocity v_o, and impinges
  on a mass flowing to the discharge pipe at the much slower velocity
  v_s. The radial component of v_o is almost necessarily wasted. From
  the tangential component there is a gain of pressure

    (w_o² - v_s²)/2g - (w_o - v_s)²/2g
      = v_s(w_o - v_s)g,

  which will be small, if v_s is small compared with w_o. Its greatest
  value, if v_s = ½w_o, is ½w_o²/2g, which will always be a small part
  of the whole head. Suppose this neglected. The whole variation of
  pressure in the pump disk then balances the lift and the head u_i²/2g
  necessary to give the initial velocity of flow in the eye of the
  wheel.

    u_i²/2g + H = V_o²/2g - u_o² cosec² [phi]/2g + u_i²/2g,

    H = V_o²/2g - u_o² cosec² [phi]/2g

  or

    V_o = [root](2gH + u_o² cosec² [phi]).   (8)

  and the efficiency of the pump is, from (7),

    [eta] = gH/V_o w_o = gH/{V (V_o - n_o cot [phi])},

    = (V_o² - u_o² cosec² [phi])/{2V_o (V_o - u_o cot [phi]) },   (9).

  For [phi] = 90°,

    [eta] = (V_o² - u_o²)/2V_o²,

  which is necessarily less than ½. That is, half the work expended in
  driving the pump is wasted. By recurving the vanes, a plan introduced
  by Appold, the efficiency is increased, because the velocity v_o of
  discharge from the pump is diminished. If [phi] is very small,

    cosec [phi] = cot [phi];

  and then

    [eta] = (V_o, + u_o cosec [phi])/2V_o,

  which may approach the value 1, as [phi] tends towards 0. Equation (8)
  shows that u_o cosec [phi] cannot be greater than V_o. Putting u_o =
  0.25 [root](2gH) we get the following numerical values of the
  efficiency and the circumferential velocity of the pump:--

    [phi]    [eta]      V_o

     90°      0.47     1.03 [root](2gH)
     45°      0.56     1.06       "
     30°      0.65     1.12       "
     20°      0.73     1.24       "
     10°      0.84     1.75       "

  [phi] cannot practically be made less than 20°; and, allowing for the
  frictional losses neglected, the efficiency of a pump in which [phi] =
  20° is found to be about .60.

  § 210. Case 2. _Pump with a Whirlpool Chamber_, as in fig.
  210.--Professor James Thomson first suggested that the energy of the
  water after leaving the pump disk might be utilized, if a space were
  left in which a free vortex could be formed. In such a free vortex the
  velocity varies inversely as the radius. The gain of pressure in the
  vortex chamber is, putting r_o, r_w for the radii to the outlet
  surface of wheel and to outside of free vortex,

    v_o²  /    r_o² \    v_o²  /      \
    ---- ( 1 - ----  ) = ---- ( 1 - k² ),
     2g   \    r_w² /     2g   \      /

  if

    k = r_o/r_w.

  The lift is then, adding this to the lift in the last case,

    H = {V_o² - u_o² cosec² [phi] + v_o²(1 - k²)}/2g.

  But

    v_o² = V_o² - 2V_o u_o cot [phi] + u_o² cosec² [phi];

    .: H = {(2 - k²)V_o² - 2kV_o u_o cot [phi] - k²u_o² cosec² [phi]}/2g.   (10)

  Putting this in the expression for the efficiency, we find a
  considerable increase of efficiency. Thus with

    [phi] = 90° and         k = ½, [eta] = 7/8 nearly,

    [phi] a small angle and k = ½, [eta] = 1 nearly.

  With this arrangement of pump, therefore, the angle at the outer ends
  of the vanes is of comparatively little importance. A moderate angle
  of 30° or 40° may very well be adopted. The following numerical values
  of the velocity of the circumference of the pump have been obtained by
  taking k = ½, and u_o = 0.25 [root](2gH).

    [phi]      V_o

     90°      .762 [root](2gH)
     45°      .842      "
     30°      .911      "
     20°     1.023      "

  The quantity of water to be pumped by a centrifugal pump necessarily
  varies, and an adjustment for different quantities of water cannot
  easily be introduced. Hence it is that the average efficiency of pumps
  of this kind is in practice less than the efficiencies given above.
  The advantage of a vortex chamber is also generally neglected. The
  velocity in the supply and discharge pipes is also often made greater
  than is consistent with a high degree of efficiency. Velocities of 6
  or 7 ft. per second in the discharge and suction pipes, when the lift
  is small, cause a very sensible waste of energy; 3 to 6 ft. would be
  much better. Centrifugal pumps of very large size have been
  constructed. Easton and Anderson made pumps for the North Sea canal in
  Holland to deliver each 670 tons of water per minute on a lift of 5
  ft. The pump disks are 8 ft. diameter. J. and H. Gwynne constructed
  some pumps for draining the Ferrarese Marshes, which together deliver
  2000 tons per minute. A pump made under Professor J. Thomson's
  direction for drainage works in Barbados had a pump disk 16 ft. in
  diameter and a whirlpool chamber 32 ft. in diameter. The efficiency of
  centrifugal pumps when delivering less or more than the normal
  quantity of water is discussed in a paper in the _Proc. Inst. Civ.
  Eng._ vol. 53.

§ 211. _High Lift Centrifugal Pumps._--It has long been known that
centrifugal pumps could be worked in series, each pump overcoming a part
of the lift. This method has been perfected, and centrifugal pumps for
very high lifts with great efficiency have been used by Sulzer and
others. C. W. Darley (_Proc. Inst. Civ. Eng._, supplement to vol. 154,
p. 156) has described some pumps of this new type driven by Parsons
steam turbines for the water supply of Sydney, N.S.W. Each pump was
designed to deliver 1½ million gallons per twenty-four hours against a
head of 240 ft. at 3300 revs. per minute. Three pumps in series give
therefore a lift of 720 ft. The pump consists of a central double-sided
impeller 12 in. diameter. The water entering at the bottom divides and
enters the runner at each side through a bell-mouthed passage. The shaft
is provided with ring and groove glands which on the suction side keep
the air out and on the pressure side prevent leakage. Some water from
the pressure side leaks through the glands, but beyond the first grooves
it passes into a pocket and is returned to the suction side of the pump.
For the glands on the suction side water is supplied from a low-pressure
service. No packing is used in the glands. During the trials no water
was seen at the glands. The following are the results of tests made at
Newcastle:--

  +-------------------------------------+-------+-------+-------+-------+
  |                                     |   I.  |  II.  |  III. |  IV.  |
  +-------------------------------------+-------+-------+-------+-------+
  | Duration of test              hours |   2   |  1.54 |  1.2  | 1.55  |
  | Steam pressure      lb. per sq. in. |  57   |  57   |  84   |  55   |
  | Weight of steam per water           |       |       |       |       |
  |     h.p. hour                   lb. | 27.93 | 30.67 | 28.83 | 27.89 |
  | Speed in revs, per min.             |  3300 |  3330 |  3710 | 3340  |
  | Height of suction               ft. |   11  |   11  |   11  |  11   |
  | Total lift                      ft. |  762  |  744  |  917  |  756  |
  | Million galls. per day pumped--     |       |       |       |       |
  |   By Ventun meter                   | 1.573 | 1.499 | 1.689 | 1.503 |
  |   By orifice                        | 1.623 | 1.513 | 1.723 | 1.555 |
  | Water h.p.                          |  252  |  235  |  326  |  239  |
  +-------------------------------------+-------+-------+-------+-------+

In trial IV. the steam was superheated 95° F. From other trials under
the same conditions as trial I. the Parsons turbine uses 15.6 lb. of
steam per brake h.p. hour, so that the combined efficiency of turbine
and pumps is about 56%, a remarkably good result.

[Illustration: FIG. 212.]

§ 212. _Air-Lift Pumps._--An interesting and simple method of pumping by
compressed air, invented by Dr J. Pohlé of Arizona, is likely to be very
useful in certain cases. Suppose a rising main placed in a deep bore
hole in which there is a considerable depth of water. Air compressed to
a sufficient pressure is conveyed by an air pipe and introduced at the
lower end of the rising main. The air rising In the main diminishes the
average density of the contents of the main, and their aggregate weight
no longer balances the pressure at the lower end of the main due to its
submersion. An upward flow is set up, and if the air supply is
sufficient the water in the rising main is lifted to any required
height. The higher the lift above the level in the bore hole the deeper
must be the point at which air is injected. Fig. 212 shows an airlift
pump constructed for W. H. Maxwell at the Tunbridge Wells waterworks.
There is a two-stage steam air compressor, compressing air to from 90 to
100 lb. per sq. in. The bore hole is 350 ft. deep, lined with steel
pipes 15 in. diameter for 200 ft. and with perforated pipes 13½ in.
diameter for the lower 150 ft. The rest level of the water is 96 ft.
from the ground-level, and the level when pumping 32,000 gallons per
hour is 120 ft. from the ground-level. The rising main is 7 in.
diameter, and is carried nearly to the bottom of the bore hole and to 20
ft. above the ground-level. The air pipe is 2½ in. diameter. In a trial
run 31,402 gallons per hour were raised 133 ft. above the level in the
well. Trials of the efficiency of the system made at San Francisco with
varying conditions will be found in a paper by E. A. Rix (_Journ. Amer.
Assoc. Eng. Soc._ vol. 25, 1900). Maxwell found the best results
when the ratio of immersion to lift was 3 to 1 at the start and 2.2 to 1
at the end of the trial. In these conditions the efficiency was 37%
calculated on the indicated h.p. of the steam-engine, and 46% calculated
on the indicated work of the compressor. 2.7 volumes of free air were
used to 1 of water lifted. The system is suitable for temporary
purposes, especially as the quantity of water raised is much greater
than could be pumped by any other system in a bore hole of a given size.
It is useful for clearing a boring of sand and may be advantageously
used permanently when a boring is in sand or gravel which cannot be kept
out of the bore hole. The initial cost is small.

§ 213. _Centrifugal Fans._--Centrifugal fans are constructed similarly
to centrifugal pumps, and are used for compressing air to pressures not
exceeding 10 to 15 in. of water-column. With this small variation of
pressure the variation of volume and density of the air may be neglected
without sensible error. The conditions of pressure and discharge for
fans are generally less accurately known than in the case of pumps, and
the design of fans is generally somewhat crude. They seldom have
whirlpool chambers, though a large expanding outlet is provided in the
case of the important Guibal fans used in mine ventilation.

  It is usual to reckon the difference of pressure at the inlet and
  outlet of a fan in inches of water-column. One inch of water-column =
  64.4 ft. of air at average atmospheric pressure = 5.2lb. per sq. ft.

  Roughly the pressure-head produced in a fan without means of utilizing
  the kinetic energy of discharge would be v²/2g ft. of air, or 0.00024
  v² in. of water, where v is the velocity of the tips of the fan blades
  in feet per second. If d is the diameter of the fan and t the width at
  the external circumference, then [pi]dt is the discharge area of the
  fan disk. If Q is the discharge in cub. ft. per sec., u = Q/[pi]dt is
  the radial velocity of discharge which is numerically equal to the
  discharge per square foot of outlet in cubic feet per second. As both
  the losses in the fan and the work done are roughly proportional to u²
  in fans of the same type, and are also proportional to the gauge
  pressure p, then if the losses are to be a constant percentage of the
  work done u may be taken proportional to [root]p. In ordinary cases u
  = about 22[root]p. The width t of the fan is generally from 0.35 to
  0.45d. Hence if Q is given, the diameter of the fan should be:--

    For t = 0.35d,     d = 0.20 [root](Q/[root]p)
    For t = 0.45d,     d = 0.18 [root](Q/[root]p)

  If p is the pressure difference in the fan in inches of water, and N
  the revolutions of fan,

    v = [pi]dN/60         ft. per sec.
    N = 1230 [root]p/d    revs. per min.

  As the pressure difference is small, the work done in compressing the
  air is almost exactly 5.2pQ foot-pounds per second. Usually, however,
  the kinetic energy of the air in the discharge pipe is not
  inconsiderable compared with the work done in compression. If w is the
  velocity of the air where the discharge pressure is measured, the air
  carries away w²/2g foot-pounds per lb. of air as kinetic energy. In Q
  cubic feet or 0.0807 Qlb. the kinetic energy is 0.00125 Qw²
  foot-pounds per second.

  The efficiency of fans is reckoned in two ways. If B.H.P. is the
  effective horse-power applied at the fan shaft, then the efficiency
  reckoned on the work of compression is

    [eta] = 5.2 pQ/550 B.H.P.

  On the other hand, if the kinetic energy in the delivery pipe is taken
  as part of the useful work the efficiency is

    [eta]2 = (5.2 pQ + 0.00125 Qw²)/550 B.H.P.

  Although the theory above is a rough one it agrees sufficiently with
  experiment, with some merely numerical modifications.

  An extremely interesting experimental investigation of the action of
  centrifugal fans has been made by H. Heenan and W. Gilbert (_Proc.
  Inst. Civ. Eng._ vol. 123, p. 272). The fans delivered through an air
  trunk in which different resistances could be obtained by introducing
  diaphragms with circular apertures of different sizes. Suppose a fan
  run at constant speed with different resistances and the compression
  pressure, discharge and brake horse-power measured. The results plot
  in such a diagram as is shown in fig. 213. The less the resistance to
  discharge, that is the larger the opening in the air trunk, the
  greater the quantity of air discharged at the given speed of the fan.
  On the other hand the compression pressure diminishes. The curve
  marked total gauge is the compression pressure + the velocity head in
  the discharge pipe, both in inches of water. This curve falls, but not
  nearly so much as the compression curve, when the resistance in the
  air trunk is diminished. The brake horse-power increases as the
  resistance is diminished because the volume of discharge increases
  very much. The curve marked efficiency is the efficiency calculated
  on the work of compression only. It is zero for no discharge, and zero
  also when there is no resistance and all the energy given to the air
  is carried away as kinetic energy. There is a discharge for which this
  efficiency is a maximum; it is about half the discharge which there is
  when there is no resistance and the delivery pipe is full open. The
  conditions of speed and discharge corresponding to the greatest
  efficiency of compression are those ordinarily taken as the best
  normal conditions of working. The curve marked total efficiency gives
  the efficiency calculated on the work of compression and kinetic
  energy of discharge. Messrs Gilbert and Heenan found the efficiencies
  of ordinary fans calculated on the compression to be 40 to 60% when
  working at about normal conditions.

  [Illustration: FIG. 213.]

  Taking some of Messrs Heenan and Gilbert's results for ordinary fans
  in normal conditions, they have been found to agree fairly with the
  following approximate rules. Let p_c be the compression pressure and q
  the volume discharged per second per square foot of outlet area of
  fan. Then the total gauge pressure due to pressure of compression and
  velocity of discharge is approximately: p = p_c + 0.0004 q² in. of
  water, so that if p_c is given, p can be found approximately. The
  pressure p depends on the circumferential speed v of the fan disk--

    p = 0.00025 v² in. of water

    v = 63 [root]p ft. per sec.

  The discharge per square foot of outlet of fan is--

    q = 15 to 18 [root]p cub. ft. per sec.

  The total discharge is

    Q = [pi] dt q = 47 to 56 dt [root]p

  For

    t = .35d, d = 0.22 to 0.25 [root](Q/[root]p) ft.

    t = .45d, d = 0.20 to 0.22 [root](Q/[root]p) ft.

    N = 1203 [root]p/d.

  These approximate equations, which are derived purely from experiment,
  do not differ greatly from those obtained by the rough theory given
  above. The theory helps to explain the reason for the form of the
  empirical results.     (W. C. U.)


FOOTNOTES:

  [1] Except where other units are given, the units throughout this
    article are feet, pounds, pounds per sq. ft., feet per second.

  [2] _Journal de M. Liouville_, t. xiii. (1868); _Mémoires de
    l'Académie, des Sciences de l'Institut de France_, t. xxiii., xxiv.
    (1877).

  [3] The following theorem is taken from a paper by J. H. Cotterill,
    "On the Distribution of Energy in a Mass of Fluid in Steady Motion,"
    _Phil. Mag._, February 1876.

  [4] The discharge per second varied from .461 to .665 cub. ft. in two
    experiments. The coefficient .435 is derived from the mean value.

  [5] "Formulae for the Flow of Water in Pipes," _Industries_
    (Manchester, 1886).

  [6] Boussinesq has shown that this mode of determining the corrective
    factor [alpha] is not satisfactory.

  [7] In general, because when the water leaves the turbine wheel it
    ceases to act on the machine. If deflecting vanes or a whirlpool are
    added to a turbine at the discharging side, then v1 may in part
    depend on v2, and the statement above is no longer true.




HYDRAZINE (DIAMIDOGEN), N2H4 or H2 N·NH2, a compound of hydrogen and
nitrogen, first prepared by Th. Curtius in 1887 from diazo-acetic ester,
N2CH·CO2C2H5. This ester, which is obtained by the action of potassium
nitrate on the hydrochloride of amidoacetic ester, yields on hydrolysis
with hot concentrated potassium hydroxide an acid, which Curtius
regarded as C3H3N6(CO2H)3, but which A. Hantzsch and O. Silberrad
(_Ber._, 1900, 33, p. 58) showed to be C2H2N4(CO2H)2, bisdiazoacetic
acid. On digestion of its warm aqueous solution with warm dilute
sulphuric acid, hydrazine sulphate and oxalic acid are obtained. C. A.
Lobry de Bruyn (_Ber._, 1895, 28, p. 3085) prepared free hydrazine by
dissolving its hydrochloride in methyl alcohol and adding sodium
methylate; sodium chloride was precipitated and the residual liquid
afterwards fractionated under reduced pressure. It can also be prepared
by reducing potassium dinitrososulphonate in ice cold water by means of
sodium amalgam:--

  KSO3 \           KSO3 \
        > N·NO -->       > N·NH2 --> K2SO4 + N2H4.
    KO /              H /

P. J. Schestakov (_J. Russ. Phys. Chem. Soc._, 1905, 37, p. 1) obtained
hydrazine by oxidizing urea with sodium hypochlorite in the presence of
benzaldehyde, which, by combining with the hydrazine, protected it from
oxidation. F. Raschig (German Patent 198307, 1908) obtained good yields
by oxidizing ammonia with sodium hypochlorite in solutions made viscous
with glue. Free hydrazine is a colourless liquid which boils at 113.5°
C., and solidifies about 0° C. to colourless crystals; it is heavier
than water, in which it dissolves with rise of temperature. It is
rapidly oxidized on exposure, is a strong reducing agent, and reacts
vigorously with the halogens. Under certain conditions it may be
oxidized to azoimide (A. W. Browne and F. F. Shetterly, _J. Amer. C.S._,
1908, p. 53). By fractional distillation of its aqueous solution
hydrazine hydrate N2H4·H2O (or perhaps H2N·NH3OH), a strong base, is
obtained, which precipitates the metals from solutions of copper and
silver salts at ordinary temperatures. It dissociates completely in a
vacuum at 143°, and when heated under atmospheric pressure to 183° it
decomposes into ammonia and nitrogen (A. Scott, _J. Chem. Soc._, 1904,
85, p. 913). The sulphate N2H4·H2SO4, crystallizes in tables which are
slightly soluble in cold water and readily soluble in hot water; it is
decomposed by heating above 250° C. with explosive evolution of gas and
liberation of sulphur. By the addition of barium chloride to the
sulphate, a solution of the hydrochloride is obtained, from which the
crystallized salt may be obtained on evaporation.

  Many organic derivatives of hydrazine are known, the most important
  being phenylhydrazine, which was discovered by Emil Fischer in 1877.
  It can be best prepared by V. Meyer and Lecco's method (_Ber._, 1883,
  16, p. 2976), which consists in reducing phenyldiazonium chloride in
  concentrated hydrochloric acid solution with stannous chloride also
  dissolved in concentrated hydrochloric acid. Phenylhydrazine is
  liberated from the hydrochloride so obtained by adding sodium
  hydroxide, the solution being then extracted with ether, the ether
  distilled off, and the residual oil purified by distillation under
  reduced pressure. Another method is due to E. Bamberger. The diazonium
  chloride, by the addition of an alkaline sulphite, is converted into a
  diazosulphonate, which is then reduced by zinc dust and acetic acid to
  phenylhydrazine potassium sulphite. This salt is then hydrolysed by
  heating it with hydrochloric acid--

    C6H5N2Cl + K2SO3 = KCl + C6H5N2·SO3K,

    C6H5N2·SO3K + 2H = C6H5·NH·NH·SO3K,

    C6H5NH·NH·SO3K + HCl + H2O = C6H5·NH·NH2·HCl + KHSO4.

  Phenylhydrazine is a colourless oily liquid which turns brown on
  exposure. It boils at 241° C., and melts at 17.5° C. It is slightly
  soluble in water, and is strongly basic, forming well-defined salts
  with acids. For the detection of substances containing the carbonyl
  group (such for example as aldehydes and ketones) phenylhydrazine is a
  very important reagent, since it combines with them with elimination
  of water and the formation of well-defined hydrazones (see ALDEHYDES,
  KETONES and SUGARS). It is a strong reducing agent; it precipitates
  cuprous oxide when heated with Fehling's solution, nitrogen and
  benzene being formed at the same time--C6H5·NH·NH2 + 2CuO = Cu2O + N2
  + H2O + C6H5. By energetic reduction of phenylhydrazine (e.g. by use
  of zinc dust and hydrochloric acid), ammonia and aniline are
  produced--C6H5NH·NH2 + 2H = C6H5NH2 + NH3. It is also a most important
  synthetic reagent. It combines with aceto-acetic ester to form
  phenylmethylpyrazolone, from which antipyrine (q.v.) may be obtained.
  Indoles (q.v.) are formed by heating certain hydrazones with anhydrous
  zinc chloride; while semicarbazides, pyrrols (q.v.) and many other
  types of organic compounds may be synthesized by the use of suitable
  phenylhydrazine derivatives.




HYDRAZONE, in chemistry, a compound formed by the condensation of a
hydrazine with a carbonyl group (see ALDEHYDES; KETONES).




HYDROCARBON, in chemistry, a compound of carbon and hydrogen. Many occur
in nature in the free state: for example, natural gas, petroleum and
paraffin are entirely composed of such bodies; other natural sources are
india-rubber, turpentine and certain essential oils. They are also
revealed by the spectroscope in stars, comets and the sun. Of artificial
productions the most fruitful and important is provided by the
destructive or dry distillation of many organic substances; familiar
examples are the distillation of coal, which yields ordinary lighting
gas, composed of gaseous hydrocarbons, and also coal tar, which, on
subsequent fractional distillations, yields many liquid and solid
hydrocarbons, all of high industrial value. For details reference should
be made to the articles wherein the above subjects are treated. From the
chemical point of view the hydrocarbons are of fundamental importance,
and, on account of their great number, and still greater number of
derivatives, they are studied as a separate branch of the science,
namely, organic chemistry.

  See CHEMISTRY for an account of their classification, &c.




HYDROCELE (Gr. [Greek: hydôr], water, and [Greek: kêlê], tumour), the
medical term for any collection of fluid other than pus or blood in the
neighbourhood of the testis or cord. The fluid is usually serous.
Hydrocele may be congenital or arise in the middle-aged without apparent
cause, but it is usually associated with chronic orchitis or with
tertiary syphilitic enlargements. The hydrocele appears as a rounded,
fluctuating translucent swelling in the scrotum, and when greatly
distended causes a dragging pain. Palliative treatment consists in
tapping aseptically and removing the fluid, the patient afterwards
wearing a suspender. The condition frequently recurs and necessitates
radical treatment. Various substances may be injected; or the hydrocele
is incised, the tunica partly removed and the cavity drained.




HYDROCEPHALUS (Gr. [Greek: hydôr], water, and [Greek: kephalê], head), a
term applied to disease of the brain which is attended with excessive
effusion of fluid into its cavities. It exists in two forms--_acute_ and
_chronic hydrocephalus_. Acute hydrocephalus is another name for
tuberculous meningitis (see MENINGITIS).

_Chronic hydrocephalus_, or "water on the brain," consists in an
effusion of fluid into the lateral ventricles of the brain. It is not
preceded by tuberculous deposit or acute inflammation, but depends upon
congenital malformation or upon chronic inflammatory changes affecting
the membranes. When the disease is congenital, its presence in the
foetus is apt to be a source of difficulty in parturition. It is however
more commonly developed in the first six months of life; but it
occasionally arises in older children, or even in adults. The chief
symptom is the gradual increase in size of the upper part of the head
out of all proportion to the face or the rest of the body. Occurring at
an age when as yet the bones of the skull have not become welded
together, the enlargement may go on to an enormous extent, the Spaces
between the bones becoming more and more expanded. In a well-marked case
the deformity is very striking; the upper part of the forehead projects
abnormally, and the orbital plates of the frontal bone being inclined
forwards give a downward tilt to the eyes, which have also peculiar
rolling movements. The face is small, and this, with the enlarged head,
gives a remarkable aged expression to the child. The body is
ill-nourished, the bones are thin, the hair is scanty and fine and the
teeth carious or absent.

The average circumference of the adult head is 22 in., and in the normal
child it is of course much less. In chronic hydrocephalus the head of an
infant three months old has measured 29 in.; and in the case of the man
Cardinal, who died in Guy's Hospital, the head measured 33 in. In such
cases the head cannot be supported by the neck, and the patient has to
keep mostly in the recumbent posture. The expansibility of the skull
prevents destructive pressure on the brain, yet this organ is materially
affected by the presence of the fluid. The cerebral ventricles are
distended, and the convolutions are flattened. Occasionally the fluid
escapes into the cavity of the cranium, which it fills, pressing down
the brain to the base of the skull. As a consequence, the functions of
the brain are interfered with, and the mental condition is impaired. The
child is dull, listless and irritable, and sometimes imbecile. The
special senses become affected as the disease advances; sight is often
lost, as is also hearing. Hydrocephalic children generally sink in a few
years; nevertheless there have been instances of persons with this
disease living to old age. There are, of course, grades of the
affection, and children may present many of the symptoms of it in a
slight degree, and yet recover, the head ceasing to expand, and becoming
in due course firmly ossified.

Various methods of treatment have been employed, but the results are
unsatisfactory. Compression of the head by bandages, and the
administration of mercury with the view of promoting absorption of the
fluid, are now little resorted to. Tapping the fluid from time to time
through one of the spaces between the bones, drawing off a little, and
thereafter employing gentle pressure, has been tried, but rarely with
benefit. Attempts have also been made to establish a permanent drainage
between the interior of the lateral ventricle and the sub-dural space,
and between the lumbar region of the spine and the abdomen, but without
satisfactory results. On the whole, the plan of treatment which aims at
maintaining the patient's nutrition by appropriate food and tonics is
the most rational and successful.     (E. O.*)




HYDROCHARIDEAE, in botany, a natural order of Monocotyledons, belonging
to the series Helobieae. They are water-plants, represented in Britain
by frog-bit (_Hydrocharis Morsusranae_) and water-soldier (_Stratiotes
aloïdes_). The order contains about fifty species in fifteen genera,
twelve of which occur in fresh water while three are marine: and
includes both floating and submerged forms. _Hydrocharis_ floats on the
surface of still water, and has rosettes of kidney-shaped leaves, from
among which spring the flower-stalks; stolons bearing new leaf-rosettes
are sent out on all sides, the plant thus propagating itself on the same
way as the strawberry. _Stratiotes aloïdes_ has a rosette of stiff
sword-like leaves, which when the plant is in flower project above the
surface; it is also stoloniferous, the young rosettes sinking to the
bottom at the beginning of winter and rising again to the surface in the
spring. _Vallisneria_ (eel-grass) contains two species, one native of
tropical Asia, the other inhabiting the warmer parts of both hemispheres
and reaching as far north as south Europe. It grows in the mud at the
bottom of fresh water, and the short stem bears a cluster of long,
narrow grass-like leaves; new plants are formed at the end of horizontal
runners. Another type is represented by _Elodea canadensis_ or
water-thyme, which has been introduced into the British Isles from North
America. It is a small, submerged plant with long, slender branching
stems bearing whorls of narrow toothed leaves; the flowers appear at the
surface when mature. _Halophila_, _Enhalus_ and _Thalassia_ are
submerged maritime plants found on tropical coasts, mainly in the Indian
and Pacific oceans; _Halophila_ has an elongated stem rooting at the
nodes; _Enhalus_ a short, thick rhizome, clothed with black threads
resembling horse-hair, the persistent hard-bast strands of the leaves;
_Thalassia_ has a creeping rooting stem with upright branches bearing
crowded strap-shaped leaves in two rows. The flowers spring from, or are
enclosed in, a spathe, and are unisexual and regular, with generally a
calyx and corolla, each of three members; the stamens are in whorls of
three, the inner whorls are often barren; the two to fifteen carpels
form an inferior ovary containing generally numerous ovules on often
large, produced, parietal placentas. The fruit is leathery or fleshy,
opening irregularly. The seeds contain a large embryo and no endosperm.
In _Hydrocharis_ (fig. 1), which is dioecious, the flowers are borne
above the surface of the water, have conspicuous white petals, contain
honey and are pollinated by insects. _Stratiotes_ has similar flowers
which come above the surface only for pollination, becoming submerged
again during ripening of the fruit. In _Vallisneria_ (fig. 2), which is
also dioecious, the small male flowers are borne in large numbers in
short-stalked spathes; the petals are minute and scale-like, and only
two of the three stamens are fertile; the flowers become detached before
opening and rise to the surface, where the sepals expand and form a
float bearing the two projecting semi-erect stamens. The female flowers
are solitary and are raised to the surface on a long, spiral stalk; the
ovary bears three broad styles, on which some of the large, sticky
pollen-grains from the floating male flowers get deposited, (fig. 3).
After pollination the female flower becomes drawn below the surface by
the spiral contraction of the long stalk, and the fruit ripens near the
bottom. _Elodea_ has polygamous flowers (that is, male, female and
hermaphrodite), solitary, in slender, tubular spathes; the male flowers
become detached and rise to the surface; the females are raised to the
surface when mature, and receive the floating pollen from the male. The
flowers of _Halophila_ are submerged and apetalous.

[Illustration: FIG. 1.--_Hydrocharis Morsusranae_--Frog-bit--male plant.

  1, Female flower.
  2, Stamens, enlarged.
  3, Barren pistil of male flower, enlarged.
  4, Pistil of female flower.
  5, Fruit.
  6, Fruit cut transversely.
  7, Seed.
  8, 9, Floral diagrams of male and female flowers respectively.
  s, Rudimentary stamens.]

[Illustration: FIG. 2.--_Vallisneria spiralis_--Eel grass--about ¼
natural size. A, Female plant; B, Male plant.]

[Illustration: FIG. 3.]

The order is a widely distributed one; the marine forms are tropical or
subtropical, but the fresh-water genera occur also in the temperate
zones.




HYDROCHLORIC ACID, also known in commerce as "spirits of salts" and
"muriatic acid," a compound of hydrogen and chlorine. Its chemistry is
discussed under CHLORINE, and its manufacture under ALKALI MANUFACTURE.




HYDRODYNAMICS (Gr. [Greek: hydôr], water, [Greek: dynamis], strength),
the branch of hydromechanics which discusses the motion of fluids (see
HYDROMECHANICS).




HYDROGEN [symbol H, atomic weight 1.008 (o = 16)], one of the chemical
elements. Its name is derived from Gr. [Greek: hydôr], water, and
[Greek: gennaein], to produce, in allusion to the fact that water is
produced when the gas burns in air. Hydrogen appears to have been
recognized by Paracelsus in the 16th century; the combustibility of the
gas was noticed by Turquet de Mayenne in the 17th century, whilst in
1700 N. Lémery showed that a mixture of hydrogen and air detonated on
the application of a light. The first definite experiments concerning
the nature of hydrogen were made in 1766 by H. Cavendish, who showed
that it was formed when various metals were acted upon by dilute
sulphuric or hydrochloric acids. Cavendish called it "inflammable air,"
and for some time it was confused with other inflammable gases, all of
which were supposed to contain the same inflammable principle,
"phlogiston," in combination with varying amounts of other substances.
In 1781 Cavendish showed that water was the only substance produced when
hydrogen was burned in air or oxygen, it having been thought previously
to this date that other substances were formed during the reaction, A.
L. Lavoisier making many experiments with the object of finding an acid
among the products of combustion.

Hydrogen is found in the free state in some volcanic gases, in
fumaroles, in the carnallite of the Stassfurt potash mines (H. Precht,
_Ber._, 1886, 19, p. 2326), in some meteorites, in certain stars and
nebulae, and also in the envelopes of the sun. In combination it is
found as a constituent of water, of the gases from certain mineral
springs, in many minerals, and in most animal and vegetable tissues. It
may be prepared by the electrolysis of acidulated water, by the
decomposition of water by various metals or metallic hydrides, and by
the action of many metals on acids or on bases. The alkali metals and
alkaline earth metals decompose water at ordinary temperatures;
magnesium begins to react above 70° C., and zinc at a dull red heat. The
decomposition of steam by red hot iron has been studied by H.
Sainte-Claire Deville (_Comptes rendus_, 1870, 70, p. 1105) and by H.
Debray (ibid., 1879, 88, p. 1341), who found that at about 1500° C. a
condition of equilibrium is reached. H. Moissan (_Bull. soc. chim._,
1902, 27, p. 1141) has shown that potassium hydride decomposes cold
water, with evolution of hydrogen, KH + H2O = KOH + H2. Calcium hydride
or hydrolite, prepared by passing hydrogen over heated calcium,
decomposes water similarly, 1 gram giving 1 litre of gas; it has been
proposed as a commercial source (Prats Aymerich, _Abst. J.C.S._, 1907,
ii. p. 543), as has also aluminium turnings moistened with potassium
cyanide and mercuric chloride, which decomposes water regularly at 70°,
1 gram giving 1.3 litres of gas (Mauricheau-Beaupré, _Comptes rendus_,
1908, 147, p. 310). Strontium hydride behaves similarly. In preparing
the gas by the action of metals on acids, dilute sulphuric or
hydrochloric acid is taken, and the metals commonly used are zinc or
iron. So obtained, it contains many impurities, such as carbon dioxide,
nitrogen, oxides of nitrogen, phosphoretted hydrogen, arseniuretted
hydrogen, &c., the removal of which is a matter of great difficulty (see
E. W. Morley, _Amer. Chem. Journ._, 1890, 12, p. 460). When prepared by
the action of metals on bases, zinc or aluminium and caustic soda or
caustic potash are used. Hydrogen may also be obtained by the action of
zinc on ammonium salts (the nitrate excepted) (Lorin, _Comptes rendus_,
1865, 60, p. 745) and by heating the alkali formates or oxalates with
caustic potash or soda, Na2C2O4 + 2NaOH = H2 + 2Na2CO3. Technically it
is prepared by the action of superheated steam on incandescent coke (see
F. Hembert and Henry, _Comptes rendus_, 1885, 101, p. 797; A. Naumann
and C. Pistor, _Ber._, 1885, 18, p. 1647), or by the electrolysis of a
dilute solution of caustic soda (C. Winssinger, _Chem. Zeit._, 1898,
22, p. 609; "Die Elektrizitäts-Aktiengesellschaft," _Zeit. f.
Elektrochem._, 1901, 7, p. 857). In the latter method a 15% solution of
caustic soda is used, and the electrodes are made of iron; the cell is
packed in a wooden box, surrounded with sand, so that the temperature is
kept at about 70° C.; the solution is replenished, when necessary, with
distilled water. The purity of the gas obtained is about 97%.

Pure hydrogen is a tasteless, colourless and odourless gas of specific
gravity 0.06947 (air = 1) (Lord Rayleigh, _Proc. Roy. Soc._, 1893, p.
319). It may be liquefied, the liquid boiling at -252.68° C. to -252.84°
C., and it has also been solidified, the solid melting at -264° C. (J.
Dewar, _Comptes rendus_, 1899, 129, p. 451; _Chem. News_, 1901, 84, p.
49; see also LIQUID GASES). The specific heat of gaseous hydrogen (at
constant pressure) is 3.4041 (water = 1), and the ratio of the specific
heat at constant pressure to the specific heat at constant volume is
1.3852 (W. C. Röntgen, _Pogg. Ann._, 1873, 148, p. 580). On the spectrum
see SPECTROSCOPY. Hydrogen is only very slightly soluble in water. It
diffuses very rapidly through a porous membrane, and through some metals
at a red heat (T. Graham, _Proc. Roy. Soc._, 1867, 15, p. 223; H.
Sainte-Claire Deville and L. Troost, _Comptes rendus_, 1863, 56, p.
977). Palladium and some other metals are capable of absorbing large
volumes of hydrogen (especially when the metal is used as a cathode in a
water electrolysis apparatus). L. Troost and P. Hautefeuille (_Ann.
chim. phys._, 1874, (5) 2, p. 279) considered that a palladium hydride
of composition Pd2H was formed, but the investigations of C. Hoitsema
(_Zeit. phys. Chem._, 1895, 17, p. 1), from the standpoint of the phase
rule, do not favour this view, Hoitsema being of the opinion that the
occlusion of hydrogen by palladium is a process of continuous
absorption. Hydrogen burns with a pale blue non-luminous flame, but will
not support the combustion of ordinary combustibles. It forms a highly
explosive mixture with air or oxygen, especially when in the proportion
of two volumes of hydrogen to one volume of oxygen. H. B. Baker (_Proc.
Chem. Soc._, 1902, 18, p. 40) has shown that perfectly dry hydrogen will
not unite with perfectly dry oxygen. Hydrogen combines with fluorine,
even at very low temperatures, with great violence; it also combines
with carbon, at the temperature of the electric arc. The alkali metals
when warmed in a current of hydrogen, at about 360° C., form hydrides of
composition RH (R = Na, K, Rb, Cs), (H. Moissan, _Bull. soc. chim._,
1902, 27, p. 1141); calcium and strontium similarly form hydrides CaH2,
SrH2 at a dull red heat (A. Guntz, _Comptes rendus_, 1901, 133, p.
1209). Hydrogen is a very powerful reducing agent; the gas occluded by
palladium being very active in this respect, readily reducing ferric
salts to ferrous salts, nitrates to nitrites and ammonia, chlorates to
chlorides, &c.

  For determinations of the volume ratio with which hydrogen and oxygen
  combine, see J. B. Dumas, _Ann. chim. phys._, 1843 (3), 8, p. 189; O.
  Erdmann and R. F. Marchand, ibid., p. 212; E. H. Keiser, _Ber._, 1887,
  20, p. 2323; J. P. Cooke and T. W. Richards, _Amer. Chem. Journ._,
  1888, 10, p. 191; Lord Rayleigh, _Chem. News_, 1889, 59, p. 147; E. W.
  Morley, _Zeit. phys. Chem._, 1890, 20, p. 417; and S. A. Leduc,
  _Comptes rendus_, 1899, 128, p. 1158.

Hydrogen combines with oxygen to form two definite compounds, namely,
water (q.v.), H2O, and hydrogen peroxide, H2O2, whilst the existence of
a third oxide, ozonic acid, has been indicated.

_Hydrogen peroxide_, H2O2, was discovered by L. J. Thénard in 1818
(_Ann. chim. phys._, 8, p. 306). It occurs in small quantities in the
atmosphere. It may be prepared by passing a current of carbon dioxide
through ice-cold water, to which small quantities of barium peroxide are
added from time to time (F. Duprey, _Comptes rendus_, 1862, 55, p. 736;
A. J. Balard, ibid., p. 758), BaO2 + CO2 + H2O = H2O2 + BaCO3. E. Merck
(_Abst. J.C.S._, 1907, ii., p. 859) showed that barium percarbonate,
BaCO4, is formed when the gas is in excess; this substance readily
yields the peroxide with an acid. Or barium peroxide may be decomposed
by hydrochloric, hydrofluoric, sulphuric or silicofluoric acids (L.
Crismer, _Bull. soc. chim._, 1891 (3), 6, p. 24; Hanriot, _Comptes
rendus_, 1885, 100, pp. 56, 172), the peroxide being added in
small quantities to a cold dilute solution of the acid. It is necessary
that it should be as pure as possible since the commercial product
usually contains traces of ferric, manganic and aluminium oxides,
together with some silica. To purify the oxide, it is dissolved in
dilute hydrochloric acid until the acid is neatly neutralized, the
solution is cooled, filtered, and baryta water is added until a faint
permanent white precipitate of hydrated barium peroxide appears; the
solution is now filtered, and a concentrated solution of baryta water is
added to the filtrate, when a crystalline precipitate of hydrated barium
peroxide, BaO2·H2O, is thrown down. This is filtered off and well washed
with water. The above methods give a dilute aqueous solution of hydrogen
peroxide, which may be concentrated somewhat by evaporation over
sulphuric acid in vacuo. H. P. Talbot and H. R. Moody (_Jour. Anal.
Chem._, 1892, 6, p. 650) prepared a more concentrated solution from the
commercial product, by the addition of a 10% solution of alcohol and
baryta water. The solution is filtered, and the barium precipitated by
sulphuric acid. The alcohol is removed by distillation _in vacuo_, and
by further concentration _in vacuo_ a solution may be obtained which
evolves 580 volumes of oxygen. R. Wolffenstein (_Ber._, 1894, 27, p.
2307) prepared practically anhydrous hydrogen peroxide (containing 99.1%
H2O2) by first removing all traces of dust, heavy metals and alkali from
the commercial 3% solution. The solution is then concentrated in an open
basis on the water-bath until it contains 48% H2O2. The liquid so
obtained is extracted with ether and the ethereal solution distilled
under diminished pressure, and finally purified by repeated
distillations. W. Staedel (_Zeit. f. angew. Chem._, 1902, 15, p. 642)
has described solid hydrogen peroxide, obtained by freezing concentrated
solutions.

Hydrogen peroxide is also found as a product in many chemical actions,
being formed when carbon monoxide and cyanogen burn in air (H. B.
Dixon); by passing air through solutions of strong bases in the presence
of such metals as do not react with the bases to liberate hydrogen; by
shaking zinc amalgam with alcoholic sulphuric acid and air (M. Traube,
_Ber._, 1882, 15, p. 659); in the oxidation of zinc, lead and copper in
presence of water, and in the electrolysis of sulphuric acid of such
strength that it contains two molecules of water to one molecule of
sulphuric acid (M. Berthelot, _Comptes rendus_, 1878, 86, p. 71).

The anhydrous hydrogen peroxide obtained by Wolffenstein boils at
84-85°C. (68 mm.); its specific gravity is 1.4996 (1.5° C.). It is very
explosive (W. Spring, _Zeit. anorg. Chem._, 1895, 8, p. 424). The
explosion risk seems to be most marked in the preparations which have
been extracted with ether previous to distillation, and J. W. Brühl
(_Ber._, 1895, 28, p. 2847) is of opinion that a very unstable, more
highly oxidized product is produced in small quantity in the process.
The solid variety prepared by Staedel forms colourless, prismatic
crystals which melt at -2° C.; it is decomposed with explosive violence
by platinum sponge, and traces of manganese dioxide. The dilute aqueous
solution is very unstable, giving up oxygen readily, and decomposing
with explosive violence at 100° C. An aqueous solution containing more
than 1.5% hydrogen peroxide reacts slightly acid. Towards lupetidin [aa'
dimethyl piperidine, C5H9N(CH3)2] hydrogen peroxide acts as a dibasic
acid (A. Marcuse and R. Wolffenstein, _Ber._, 1901, 34, p. 2430; see
also G. Bredig, _Zeit. Electrochem._, 1901, 7, p. 622). Cryoscopic
determinations of its molecular weight show that it is H2O2. [G.
Carrara, _Rend. della Accad. dei Lincei_, 1892 (5), 1, ii. p. 19; W. R.
Orndorff and J. White, _Amer. Chem. Journ._, 1893, 15, p. 347.] Hydrogen
peroxide behaves very frequently as a powerful oxidizing agent; thus
lead sulphide is converted into lead sulphate in presence of a dilute
aqueous solution of the peroxide, the hydroxides of the alkaline earth
metals are converted into peroxides of the type MO2·8H2O, titanium
dioxide is converted into the trioxide, iodine is liberated from
potassium iodide, and nitrites (in alkaline solution) are converted into
acid-amides (B. Radziszewski, _Ber._, 1884, 17, p. 355). In many cases
it is found that hydrogen peroxide will only act as an oxidant when in
the presence of a catalyst; for example, formic, glycollic, lactic,
tartaric, malic, benzoic and other organic acids are readily oxidized in
the presence of ferrous sulphate (H. J. H. Fenton, _Jour. Chem. Soc._,
1900, 77, p. 69), and sugars are readily oxidized in the presence of
ferric chloride (O. Fischer and M. Busch, _Ber._, 1891, 24, p. 1871). It
is sought to explain these oxidation processes by assuming that the
hydrogen peroxide unites with the compound undergoing oxidation to form
an addition compound, which subsequently decomposes (J. H. Kastle and A.
S. Loevenhart, _Amer. Chem. Journ._, 1903, 29, pp. 397, 517). Hydrogen
peroxide can also react as a reducing agent, thus silver oxide is
reduced with a rapid evolution of oxygen. The course of this reaction
can scarcely be considered as definitely settled; M. Berthelot considers
that a higher oxide of silver is formed, whilst A. Baeyer and V.
Villiger are of opinion that reduced silver is obtained [see _Comptes
rendus_, 1901, 133, p. 555; _Ann. Chim. Phys._, 1897 (7), 11, p. 217,
and Ber., 1901, 34, p. 2769]. Potassium permanganate, in the presence of
dilute sulphuric acid, is rapidly reduced by hydrogen peroxide, oxygen
being given off, 2KMnO4 + 3H2SO4 + 5H2O2 = K2SO4 + 2MnSO4 + 8H2O + 5O2.
Lead peroxide is reduced to the monoxide. Hypochlorous acid and its
salts, together with the corresponding bromine and iodine compounds,
liberate oxygen violently from hydrogen peroxide, giving hydrochloric,
hydrobromic and hydriodic acids (S. Tanatar, _Ber._, 1899, 32, p. 1013).

  On the constitution of hydrogen peroxide see C. F. Schönbein, _Jour.
  prak. Chem._, 1858-1868; M. Traube, _Ber._, 1882-1889; J. W. Brühl,
  _Ber._, 1895, 28, p. 2847; 1900, 33, p. 1709; S. Tanatar, _Ber._,
  1903, 36, p. 1893.

  Hydrogen peroxide finds application as a bleaching agent, as an
  antiseptic, for the removal of the last traces of chlorine and sulphur
  dioxide employed in bleaching, and for various quantitative
  separations in analytical chemistry (P. Jannasch, _Ber._, 1893, 26, p.
  2908). It may be estimated by titration with potassium permanganate in
  acid solution; with potassium ferricyanide in alkaline solution,
  2K3Fe(CN)6 + 2KOH + H2O2 = 2K4Fe(CN)6 + 2H2O + O2; or by oxidizing
  arsenious acid in alkaline solution with the peroxide and back
  titration of the excess of arsenious acid with standard iodine (B.
  Grützner, _Arch. der Pharm._, 1899, 237, p. 705). It may be recognized
  by the violet coloration it gives when added to a very dilute solution
  of potassium bichromate in the presence of hydrochloric acid; by the
  orange-red colour it gives with a solution of titanium dioxide in
  concentrated sulphuric acid; and by the precipitate of Prussian blue
  formed when it is added to a solution containing ferric chloride and
  potassium ferricyanide.

  _Ozonic Acid_, H2O4. By the action of ozone on a 40% solution of
  potassium hydroxide, placed in a freezing mixture, an orange-brown
  substance is obtained, probably K2O4, which A. Baeyer and V. Villiger
  (_Ber._, 1902, 35, p. 3038) think is derived from ozonic acid,
  produced according to the reaction O3 + H2O = H2O4.




HYDROGRAPHY (Gr. [Greek: hydôr], water, and [Greek: graphein], to
write), the science dealing with all the waters of the earth's surface,
including the description of their physical features and conditions; the
preparation of charts and maps showing the position of lakes, rivers,
seas and oceans, the contour of the sea-bottom, the position of
shallows, deeps, reefs and the direction and volume of currents; a
scientific description of the position, volume, configuration, motion
and condition of all the waters of the earth. See also SURVEYING
(Nautical) and OCEAN AND OCEANOGRAPHY. The Hydrographic Department of
the British Admiralty, established in 1795, undertakes the making of
charts for the admiralty, and is under the charge of the hydrographer to
the admiralty (see CHART).




HYDROLYSIS (Gr. [Greek: hydôr], water, [Greek: luein], to loosen), in
chemistry, a decomposition brought about by water after the manner shown
in the equation R·X + H·OH = R·H + X·OH. Modern research has proved that
such reactions are not occasioned by water acting as H2O, but really by
its ions (hydrions and hydroxidions), for the velocity is proportional
(in accordance with the law of chemical mass action) to the
concentration of these ions. This fact explains the so-called
"catalytic" action of acids and bases in decomposing such compounds as
the esters. The term "saponification" (Lat. _sapo_, soap) has the same
meaning, but it is more properly restricted to the hydrolysis of the
fats, i.e. glyceryl esters of organic acids, into glycerin and a soap
(see CHEMICAL ACTION).