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              Vol. I.         OCTOBER, 1892           No. 2.

                                  THE

                           KANSAS UNIVERSITY

                               QUARTERLY




                               CONTENTS


        UNICURSAL CURVES BY METHOD OF INVERSION,   _H. B. Newson_
        FOREIGN SETTLEMENTS IN KANSAS,             _W. H. Carruth_
        THE GREAT SPIRIT SPRING MOUND,             _R. H. S. Bailey_
        ON PASCAL’S LIMAÇON AND THE CARDIOID,      _H. C. Riggs_
        DIALECT WORD-LIST,                         _W. H. Carruth_

                      PUBLISHED BY THE UNIVERSITY
                           LAWRENCE, KANSAS

                   _Price of this number, 50 cents_

    Entered at the Post-office in Lawrence as Second-class matter.

                       COMMITTEE OF PUBLICATION

                 E. H. S. BAILEY        F. W. BLACKMAR
                 W. H. CARRUTH          C. G. DUNLAP
                 E. MILLER              S. W. WILLISTON
                    V. L. KELLOGG, MANAGING EDITOR

                       JOURNAL PUBLISHING HOUSE
                           LAWRENCE, KANSAS
                                 1892




                     KANSAS UNIVERSITY QUARTERLY.

              VOL. I.        OCTOBER, 1892.        NO. 2.




Unicursal Curves by Method of Inversion.


BY HENRY BYRON NEWSON.

This paper contains a summary of the work done during the last school
year by my class in Modern Geometry. Since many of the results were
suggested or entirely wrought out by class-room discussion, it becomes
practically impossible to assign to each member of the class his
separate portion. Many of the results were contributed by Messrs. M. E.
Rice, A. L. Candy, H. C. Riggs, and Miss Annie L. MacKinnon.

The reader who is not familiar with the method of Geometric Inversion
should read Townsend’s Modern Geometry, chapters IX and XXIV; or a
recent monograph entitled, “Das Princep der Reziproken Radien,” by C.
Wolff, of Erlangen.

When a conic is inverted from a point on the curve, the inverse curve
is a nodal, circular cubic.

This is shown analytically as follows: let the equation of the conic be
written

                  ax² + 2hxy + by² + 2gx + 2fy = 0;

which shows that the origin is a point on the curve. Substituting for

                           x              y
              x and y  —————————  and  ———————— ,
                        x² + y²       (x² + y²)

we have as the equation of the inverse curve

            ax² + 2hxy + by² + 2(gx + fy)(x² + y²) = 0.

The terms of the second degree show that the origin is a double point
on the cubic; and is a crunode, acnode, or cusp, according as the
conic is a hyperbola, ellipse, or parabola. The terms of the third
degree break up into three linear factors, viz: gx + fy, x + iy, and
x - iy, which are the equations of the three lines joining the origin
to the three points where the line at infinity cuts the cubic; thus
showing that the cubic passes through the imaginary circular points at
infinity.

Since the above transformation is rational, it follows that there
is a (1, 1) correspondence between the conic and the cubic. This
fact is also evident from the nature of the method of inversion. The
cubic has its maximum number of double points, viz: one; and hence is
unicursal. This unicursal circular cubic may be projected into the most
general form of unicursal cubic; the cuspidal variety, however, always
remaining cuspidal.

By applying the method of inversion to many of the well known theorems
of conics, new theorems are obtained for unicursal, circular cubics.
If one of these new theorems states a projective property, it may at
once by the method of projection be extended to all unicursal cubics.
Examples will be given below.

The following method of generating a unicursal cubic is often useful.
Given two projective pencils of rays having their vertices at A and
B; the locus of the intersection of corresponding rays is a conic
through A and B. Invert the whole system from A. The pencil through
A remains as a whole unchanged, while the pencil through B inverts
into a system of co-axial circles through A and B, and the generated
conic becomes a circular cubic through A and B, having a node at A.
Now project the whole figure and we have the following:—given a system
of conics through four fixed points and a pencil of rays projective
with it and having its vertex at one of the fixed points, the locus
of the intersection of corresponding elements of the two systems is
a unicursal cubic, having its node at the vertex of the pencil, and
passing through the three other fixed points.

Unicursal cubics are divisible into two distinct varieties, nodal and
cuspidal. The nodal variety is a curve of the fourth class and has
three points of inflection, one of which is always real. The cuspidal
variety is of the third class and has one point of inflection (Salmon,
H. P. C., Art. 147). Each of these varieties forms a group projective
within itself; that is to say, any nodal cubic may be projected into
every other possible nodal cubic, and the same is true with regard to
the cuspidal. But a nodal cubic can not be projected into a cuspidal
and vice versa.

In applying this method of investigation to the various forms of
unicursal cubics and quartics, only a limited number of theorems are
given in each case. It will be at once evident that many more theorems
might be added, but enough are given in each case to illustrate the
method and show the range of its application. It is not necessary to
work out all the details, as this paper is intended to be suggestive
rather than exhaustive.


NODAL CUBICS.

If an ellipse be inverted from one of its vertices, the inverse
curve is symmetrical with respect to the axis; it has one point of
inflection at infinity and the asymptote is an inflectional tangent.
This asymptote is the inverse of the circle of curvature at the vertex.
The cubic has two other points of inflection situated symmetrically
with respect to the axis. Hence the three points of inflection lie on
a right line, a projective theorem which is consequently true of all
nodal cubics. The axis is evidently the harmonic polar of the point
of inflection at infinity. Since the axis bisects the angle between
the tangents at the node, it follows that the line joining a point of
inflection to the node, the two tangents at the node, and the harmonic
polar of the point of inflection, form a harmonic pencil. There are
three such lines, one to each node, and three harmonic polars; these
form a pencil in involution, the tangents at the node being the foci.

Since the asymptote is perpendicular to the axis, we have by projection
the following theorem:—through a point of inflection I, draw any line
cutting the cubic in B and C. Through P the point of intersection of
the harmonic polar and inflectional tangent of I, draw two lines to B
and C. The four lines meeting in P form a harmonic pencil. The point
of contact of the tangent from I to the cubic is on the harmonic polar
of I. Any two inflectional tangents meet on the harmonic polar of the
third point of inflection.

The locus of the foot of the perpendicular from the focus of a conic
on a tangent is the auxiliary circle. Inverting from the vertex, there
are two points, A and B, on the axis of the curve, such that if a
circle be drawn through one of them and the node, cutting at right
angles a tangent circle through the node, their point of intersection
will be on the tangent to the curve where it is cut by the axis.
Projecting:—through a point of inflection I of a nodal cubic draw a
line cutting the cubic in P and Q; there are two determinate points
on the harmonic polar of I, which have the following property:—draw a
conic through P, Q, and the node touching the cubic; draw another conic
through one of these points, P, Q, and the node cutting the former, so
that their tangents at their point of intersection, together with the
lines from it to P and Q form a harmonic pencil; the locus of such a
point of intersection is the tangent from I to the cubic.

If three conics circumscribe the same quadrilateral, the common tangent
to any two is cut harmonically by the third. Inverting from one of the
vertices of the quadrilateral: if three nodal, circular cubics have
a common double point and pass through three other fixed points, the
common tangent circle through the common node to any two of the cubics
is cut harmonically by the third; _i. e._, so that the pencil from the
node to the two points of intersection and the points of contact is
harmonic. Projecting this:—given three nodal cubics having a common
node and passing through five other fixed points; let a conic be passed
through the common node and two of the fixed points, touching two of
the cubics. The pencil from the common node to the points of contact
and the point where the conic cuts the third cubic is harmonic.

The following theorem may be proved in similar manner:—given a system
of cubics having a common node and passing through five other fixed
points; let a conic be drawn through the common node and two of the
fixed points; the lines drawn from the points where it cuts the cubics
to the common node form a pencil in involution.

A variable chord drawn through a fixed point P to a conic subtends a
pencil in involution at any point O on the conic. Inverting from O:—a
system of circles through the double point of a nodal circular cubic
and any other fixed point P, is cut by the cubic in pairs of points
which determine at the node a pencil in involution. Projecting:—a
system of conics through the node of a unicursal cubic, two fixed
points on the curve, and any fourth fixed point, is cut by the cubic in
pairs of points which determine at the node a pencil in involution.

We give another proof of the theorem that the three points of
inflection of a nodal cubic lie on a right line. This is easily shown
by inversion and is a beautiful example of the method.

There are three points on a conic whose osculating circles pass through
a given point on the conic; these three points lie on a circle passing
through the given point.[1] (Salmon’s Conics, Art. 244, Ex. 5.) By
inverting from the given point and then projecting, we readily see that
there are three points of inflection on a nodal cubic which lie on a
right line. If the above conic be an ellipse, the three osculating
circles are all real; but if it be a hyperbola, one only is real. Hence
an acnodal cubic has three real points of inflection, while a crunodal
one has one real and two imaginary.

The reciprocals of many of the theorems of this section are of interest
and will be given under Quartics.


CUSPIDAL CUBICS.[2]

Inverting the parabola from its vertex we obtain the Cissoid of
Diocles. The focus of the parabola inverts into a point on the cuspidal
tangent which I shall call the focus of the cissoid. The circle of
curvature at the vertex of the parabola inverts into the asymptote of
the cissoid. This asymptote is also plainly the inflectional tangent,
and the point at infinity is the point of inflection. The directrix
of the parabola inverts into a circle through the cusp of the cissoid
having the cuspidal tangent for a diameter. Hall calls this the
directrix circle. The double ordinate of the parabola which is tangent
to the circle of curvature of the vertex inverts into the circle
usually called the base circle of the cissoid.[3]

The cissoid may fairly be called the simplest form of the cuspidal
cubic. Its projection and polar reciprocal are both cuspidal cubics. I
shall now deduce from the parabola a few simple propositions for the
cissoid, and then extend them to all cuspidal cubics.

(1) It is known that the locus of the intersection of tangents to the
parabola which are at right angles to one another, is the directrix.
Inverting:—the locus of the intersection of tangent circles to the
cissoid through the cusp and at right angles to each other is the
directrix circle.

(2) For the parabola, two right lines O P and O Q, are drawn through
the vertex of the parabola at right angles to one another, meeting the
curve in P and Q; the line P Q cuts the axis at a fixed point, whose
abscissa is equal to its ordinate. Inverting:—two right lines, O P and
O Q, are drawn at right angles to one another through the cusp of the
cissoid, meeting the curve in P and Q; the circle O P Q passes through
the intersection of the axis and asymptote.

(3) If the normals at the points P, O, R, of a parabola meet at
a point, the circle through P O R will pass through the vertex.
Inverting:—through a fixed point and the cusp of a cissoid, three and
only three circles can be passed, cutting the cissoid at right angles;
these three points of intersection are collinear.

From the geometry of the cissoid we see that if any line be drawn
parallel to the asymptote, cutting the curve in two points, B and C,
the segment B C is bisected by the axis. Hence, projecting the curve
we have the following theorem:—any line drawn through the point of
inflection is cut harmonically by the point of inflection, the curve,
and the cuspidal tangent. Thus the cuspidal tangent is the harmonic
polar of the point of inflection. The polar reciprocal of this last
theorem reads as follows:—if from any point on the cuspidal tangent
the two other tangent lines be drawn to the curve, and a line to the
point of inflection, these four lines form a harmonic pencil. These are
fundamental propositions in the theory of cuspidal cubics.

(4) Projecting proposition (1) above, we have the generalized
theorem:—through the point of inflection draw any line cutting the
cubic in B and C; through B, C, and the cusp draw two conics tangent
to the cubic, and intersecting in a fourth point such that the two
tangents to the conics at their point of intersection, together with
the two lines from it to B and C, form a harmonic pencil; the locus of
all such intersections is a conic through B, C, and the cusp having the
point of inflection and the cuspidal tangent for pole and polar.

(5) Reciprocating (4) we have:—through any point on the cuspidal
tangent draw the two other tangents, B and C, to the cubic. Touching B,
C, and the inflectional tangent draw two conics, such that the points
of contact of their common tangent, together with the points where
their common tangent cuts the tangents B and C, form a harmonic range;
the envelope of such common tangents is a conic having the cuspidal
tangent and the point of inflection for polar and pole.

(6) Projecting (2) we obtain the following:—through the point of
inflection draw any line cutting the curve in B and C; take any other
two points on the cubic such that the pencil from the cusp, O, O (B P C
Q) is harmonic; the conic passing through O B P C Q will pass through
the intersection of the cuspidal and inflectional tangents.

(7) Reciprocating (6):—from any point on the cuspidal tangent draw two
other tangents, B and C, to the cubic; take any two other tangents, P
and Q, such that the range cut from the inflectional tangent by B, C,
P, Q, is harmonic; the conic touching B, C, P, Q, and the inflectional
tangent will also touch the line joining the point of inflection and
the cusp.

(8) Projecting (3):—through the point of inflection draw any line
cutting the cubic in B and C; through the cusp O and the points B and C
on the cubic and any other fixed point P, three, and only three, conics
can be passed, such that the tangent to the conic and cubic at their
remaining point of intersection, together with the lines from it to B
and C, form a harmonic pencil; these three points of intersection are
collinear.


SYSTEMS OF CUBICS THROUGH NINE POINTS.

Let U and V be the equations of two given cubics, then U + kV is
the equation of a system of cubics through their nine points of
intersection. Twelve cubics of this system are unicursal, and the
twelve nodes are called the twelve critic centres of the system. (See
Salmon’s H. P. C., Art. 190.)

Let the equation of the system be written briefly

    a + ka₁ + (b + kb₁) x + (c + kc₁) y + u₂ + u₃ = 0;

one, and only one, value of k makes the absolute term vanish; hence
one, and only one, curve of the system passes through the origin,
which may be any point in the plane. Make the equation of the system
homogeneous by means of z, and differentiate twice with respect to z;
we obtain thus the equations of the polar conics and polar lines of the
origin with respect to the system.

The polar conics of the origin are given by

    3(a + ka₁) + 2 { (b + kb₁) x + (c + kc₁)y } + u₂ = 0;

thus showing that the polar conics of any point, with respect to the
system of cubics, form a system through four points. The polar lines of
the origin are given by

          3(a + ka₁) + (b + kb₁) x + (c + kc₁)y = 0,

which represents a pencil of lines through a point.

Suppose now the origin to be at one of the critic centres; then for a
particular value, k₁, all terms lower than the second degree must
vanish, so that

                        ║ a    b    c  ║
                        ║              ║ = 0.
                        ║ a₁   b₁   c₁ ║

The factors of the terms of u₂, which involves k₁, represent the
tangents at the double point to the nodal cubic, and also the polar
conic of the origin with respect to this nodal cubic. Hence a critic
centre is at one of the vertices of the self-polar triangle of its
system of polar conics. The opposite side of this triangle is the
common polar line of the critic centre with respect to its system of
polar conics, and hence it is also the common polar line of the critic
centre with respect to the system of cubics. The four basal points of
the system of polar conics lie two and two upon the tangents at the
double point of the nodal cubic.

When the origin is taken at one of the nine basal points of the system
of cubics, a and a₁ both vanish. Hence it is readily seen that a basal
point of a system of cubics is also a basal point of its system of
polar conics and the vertex of its pencil of polar lines.

Suppose two of the basal points of the system of cubics to coincide,
then every cubic of the system, in order to pass through two coincident
points, must touch a common tangent at a fixed point. The common
tangent is the common polar of its point of contact, both with respect
to the system of cubics and to its system of polar conics. Hence the
union of two basal points gives rise to a critic centre. The self-polar
triangle of its system of polar conics here reduces to a limited
portion of the common tangent. This line is not a tangent to the nodal
cubic, but only passes through its double point.

Suppose three of the basal points of a system of cubics to coincide,
such a point will then be a point of inflection on each cubic of the
system. For, in the last case, if a line be drawn from the point of
contact of the common tangent to a third basal point of the system,
such a line will be a common chord of the system of cubics. Suppose,
now, this third basal point be moved along the curves until it
coincides with the other two; then the common chord becomes a common
tangent, which cuts every cubic of the system in three coincident
points, and hence is a common inflectional tangent.

Since the polar conic of a point of inflection on a cubic consists of
the inflectional tangent and the harmonic polar of the point, and since
the polar conics of a fixed point with respect to a system of cubics
pass through four fixed points, it follows that in a system of cubics
having a common point of inflection and a common inflectional tangent
the harmonic polars of the common point of inflection meet in a point.

Since the common inflectional tangent is the common polar line of the
common point of inflection, it follows that such a point is a critic
centre of the system of cubics. One cubic of the system then has a
node at the common point of inflection of the system, and forms an
exception. The line which is the common inflectional tangent to the
other cubics of the system cuts this also in three points, but is one
of the tangents at the double point; the other tangent at the double
point goes through the vertex of the pencil of harmonic polars.

It is evident that the nine basal points of a system of conics may
unite into three groups of three each. The cubics will then all have
three common points of inflection, and at these points three common
inflectional tangents. These three points all lie on a line.

When four basal points of the system of cubics coincide, such a point
is a double point on every cubic of the system. This is easily shown
as follows, using the method of inversion. Let a system of conics
through four points be inverted from one of the four points. The system
of conics inverts into a system of cubics, having a common node and
passing through three other finite fixed points and the two circular
points at infinity. Since the common node counts as four points of
intersection, it follows that any two cubics of the system, and hence
all of them, intersect in nine points. This system can be projected
into a system having a common double point and passing through any five
other fixed points.

A number of theorems concerning the system of cubics can easily be
inferred from known theorems concerning the system of conics. Since two
conics of the system are parabolas, it follows that two cubics of the
system are cuspidal. Since three conics of the system break up into
pairs of right lines, it follows that three cubics of the system break
up into a right line and a conic. Each right line and its corresponding
conic intersect in the common double point. The line at infinity cuts
the system of conics in pairs of points in involution, the points of
contact of the two parabolas of the system being the foci; it follows
on inversion that the pairs of tangents to the cubics at their common
node form a pencil in involution, the two cuspidal tangents being the
foci.

If the four basal points of the system of conics lie on a circle,
this circle inverts into a right line, and one cubic then consists of
this right line and the lines joining the centre of inversion to the
circular points at infinity. This theorem may be stated for the system
of cubics as follows: if the conic determined by the five basal points
of the system of cubics (not counting the common double points), break
up into right lines, the line passing through three of the five points,
together with the lines joining the other two points to the common
node, constitute a cubic of the system.

If three of the four basal points of the system of conics lie on a
line, the conics consist of this line and a pencil of lines through
the fourth basal point. Inverting from this fourth point and then
projecting, we have a system of cubics consisting of a pencil of lines
and a conic through the vertex and the four other fixed points. Hence,
when the five fixed points of such a system of cubics lie on a conic
through the common node, this conic is a part of every cubic of the
system. If we invert the above system of conics from one of the three
points on the right line, and then project, we obtain a system of
cubics which consists of a system of conics through four fixed points,
and a fixed right line through one of these four points. Hence, if
two of the five basal points of such a system of cubics be on a line
through the common node, this line is a part of every cubic of the
system.

If a system of conics having one basal point at infinity be inverted
from one of the remaining basal points, this point at infinity inverts
to the center of inversion, and we obtain a system of cubics having
five coincident basal points and hence passing through only four
others. The system of cubics is now so arranged that one tangent at
their common double point is common to all. Only one cubic of the
system is cuspidal. As before three cubics break up into a right line
and conic.

If two of three basal points of the system of conics be at infinity,
the system of cubics obtained by projection and inversion has six
coincident basal points and hence only three others. This system has
both tangents at the common node common to all cubics of the system. If
the two basal points at infinity in the system of conics be coincident,
all the conics are parabolas, and hence all the cubics of the system
are cuspidal and have a common cuspidal tangent.

If three of the basal points of the system of conics be at infinity,
the conics consist of the line at infinity and a pencil of lines
through the finite basal point. Inverting from the latter, we obtain
a system of cubics with seven coincident basal points. This system
is made up of a pencil of lines meeting in the seven coincident basal
points together with the two lines joining this to the other two basal
points of the system. These two lines are part of every cubic of the
system.

If one of the remaining basal points be moved up to join the seven
coincident ones, one of these fixed lines becomes indeterminate, and
the system of cubics through eight coincident points consists of a
fixed line through the eight coincident points and the ninth fixed
point together with any two lines of the pencil through the eight
points. If the nine basal points coincide, any three lines through it
form a cubic of the system.


UNICURSAL QUARTICS.

The inverse of a conic from any point not on the curve is a nodal
bicircular quartic. This is shown by inverting the general equation of
the conic

                ax² + 2hxy + by² + 2gx + 2fy + c = 0;

                                 x               y
by substituting for x and y, —————————  and  —————————— ,
                             (x² + y²)        (x² + y²)

we get the equation

   ax² + 2hxy + by² + 2(gx + fy)(x² + y²) + c(x² + y²)² = 0.

The origin is evidently a double point on the curve, and is a crunode,
acnode, or cusp according as the conic is a hyperbola, ellipse, or
parabola. The factors of the terms of the fourth degree, viz:
(x + iy) (x + iy) (x - iy) (x - iy), show that the two imaginary
circular points at infinity are double points on the quartic, which is
thus trinodal. Hence this nodal, bicircular quartic can be projected
into the most general form of the trinodal quartic. Trinodal quartics
are unicursal.

If the conic which we invert be a parabola, the quartic has two nodes
and one cusp. If the conic be inverted from a focus, the quartic has
the two circular points at infinity for cusps. This is best shown
analytically as follows: let the equation of the conic, origin being at
the focus, be written

                      x²      y²     2aex      b²
                     ———  +  ———  +  ————  -  ——— = 0.
                      a²      b²      a²       a²

Inverting this we have

             x²      y²       2aex(x² + y²)        b²(x² + y²)²
            ———  +  ———  +  ———————————————  -  ————————————————  = 0.
             a²      b²           a²                  a²

Now transform this equation so that the lines joining the origin to the
circular points at infinity shall be the axes of reference. To do this

let x + iy = x₁ and x - iy = y₁;

         x₁ + y₁             x₁ - y₁
 ∴ x =  ————————  and  y =  ———————— .
             2                 2i.

Making these substitutions and reducing we have (dropping the
subscripts),

    (x² + 2xy + y²)    (x² - 2xy + y²)    4aexy(x + y)     b²x²y²
  ————————————————— - ————————————————— - ———————————— - ————————— = 0.
         a²                 b²                 a²           a²

Making this equation homogeneous by means of z, we have

     ┌                                     ┐
     │  (x² + 2xy + y²)   (x² - 2xy + y²)  │   4aexyz(x + y)     b²x²y²
  z² │ ———————————————— - ———————————————— │ - ————————————— - —————————
     └        a²                b²         ┘        a²            a²

                                   4aexyz(x + y)     b²x²y²
                                -  ————————————— - ————————— = 0,
                                        a²             a²

which is the equation of the quartic referred to the triangle formed by
the three nodes. We are now able to determine the nature of the node at
the vertex (y, z). Factor x² out of all the terms which contain it; and
arrange thus:

       ┌                              ┐      ┌                         ┐
       │ z²      z²    4aeyz     b²y² │      │yz²      yz²      2aey²z │
    x² │———  -  ———  - —————  - ——————│ + 2x │————  +  ————  -  ———————│
       │ a²      b²     a²        a²  │      │ a²       b²         a²  │
       └                              ┘      └                         ┘

                           y²z²       y²z²
                        + ——————  -  —————— = 0.
                            a²         b²

The quantity which multiplies x² represents the two tangents at the
double point (y, z); but this quantity is a perfect square and hence
we have a cusp. In this way the point (x, z) may be shown to be a
cusp. Lastly, when a parabola is inverted from the focus, we obtain a
tricuspidal quartic.

The trinodal quartic can be generated in a manner analogous to that
shown for the nodal cubic. Let two projective pencils of rays have
their vertices at A and B, the locus of intersection of corresponding
rays is a conic through A and B. Invert from any point O in the plane,
and we obtain two systems of co-axial circles, O A being the axis of
one and O B of the other. The locus of intersection of corresponding
circles is a bicircular quartic having a node at O. Projecting the
whole figure we have the following theorem:—two projective systems of
conics through O P Q A and O P Q B generate by their corresponding
intersections a trinodal quartic having its nodes at O, P, and Q, and
passing through A and B.

It is evident that the quartic generated in this way may have three
nodes, one node and two cusps, two nodes and one cusp, or three cusps,
depending upon the nature of the conic inverted and the centre of
inversion. Making this the basis of classification we thus distinguish
four varieties of unicursal quartics. To these must be added a fifth
variety, viz: the quartic with a triple point. Each of these varieties
will be considered separately.

The method of treating unicursal quartics given in this and the next
four sections is in some respects similar to that suggested by Cayley
in Salmon’s Higher Plane Curves. But the method here sketched out is
very different in its point of view and much wider in its application,
yielding a multitude of new theorems not suggested by Cayley’s method.


TRINODAL QUARTICS.

The quartic with three double points is a curve of the sixth class
having four double tangents and six cusps (Salmon’s H. P. C. Art. 243).
Hence its reciprocal is of the sixth degree with four double points,
six cusps, three double tangents, and no points of inflection.

The locus of intersection of tangents to a conic at right angles to
one another is a circle. Inverting:—the locus of intersection of
circles through the node and tangent a nodal, bicircular quartic and
at right angles to one another is a circle. Projecting:—through the
three nodes of a quartic draw two conics, each touching the quartic
and intersecting so that the two tangents to the conics at their point
of intersection, together with the lines from it to two of the nodes,
form a harmonic pencil; the locus of all such intersections is a conic
through these two nodes. Whenever the two tangents to the quartic from
the third node, together with the lines from it to the other two nodes,
form a harmonic pencil, this last conic breaks up into two right lines.

Any chord of a conic through O is cut harmonically by the conic and the
polar of O. Inverting from O and projecting:—from one of the nodes of
a trinodal quartic draw the two tangents to the quartic (not tangents
at the node); draw the conic through these two points of contact and
the three nodes; any line through the first mentioned node is cut
harmonically by this conic, the quartic and the line joining the other
two nodes.

If a triangle circumscribe a conic, the three lines from the angular
points of the triangle to the points of contact of the opposite sides
intersect in a point. Inverting and projecting:—through the three nodes
of a quartic draw three conics touching the quartic; through the point
of intersection of two of these conics, the point of contact of the
third, and the three nodes draw a conic; three such conics can be drawn
and they pass through a fixed point.

The eight points of contact of two conics with their four common
tangents lie on a conic, which is the locus of a point, the pairs of
tangents from which to the two given conics form a harmonic pencil.
Inverting and projecting:—two connodal trinodal quartics have four
common tangent conics through the three nodes; their eight points
of contact lie on another connodal trinodal quartic; if from any
point on the last quartic four conics be drawn through the nodes and
tangent in pairs to the first quartics, any line through a node is cut
harmonically by these four conics.

The eight common tangents to two conics at their common points all
touch a conic. Inverting and projecting:—two connodal trinodal quartics
intersect in four other points; eight conics can be drawn through the
three nodes tangent to the quartics at these points of intersection;
these eight conics all touch another connodal trinodal quartic.

A series of conics through four fixed points is cut by any transversal
in a range of points in involution. Inverting and projecting:—a series
of connodal trinodal quartics can be passed through four other fixed
points; any conic through the three nodes cuts the series of quartics
in pairs of points which determine at a node a pencil in involution.
The conic touches two of the quartics and the lines to the points of
contact are the foci of the pencil.

If the sides of two triangles touch a given conic, their six angular
points will lie on another conic. Inverting and projecting:—if two
groups of three conics each be passed through three nodes and tangent
to the quartic, their six points of intersection (three of each group)
lie on another connodal trinodal quartic.

If the two triangles are inscribed in a conic, their six sides touch
another conic. Inverting and projecting:—if two groups of three conics
each be passed through the three nodes of a quartic so that the three
points of intersection of each group lie on the quartic, these six
conics all touch another connodal trinodal quartic.

A triangle is circumscribed about one conic, and two of its angular
points are on a second conic; the locus of its third angular point is a
conic.—Inverting and projecting:—if three conics be drawn through the
three nodes of two connodal trinodal quartics so that they all touch
one of the quartics and two of their points of intersection are on the
other quartic, the locus of their third point of intersection is a
connodal trinodal quartic.

A triangle is inscribed in one conic and two of its sides touch a
second conic; the envelope of its third side is a conic. Inverting and
projecting:—if three conics be drawn through the three nodes of two
connodal trinodal quartics so that their three points of intersection
lie on one of the quartics and two of them touch the other quartic, the
envelope of the third conic is another connodal trinodal quartic.

The theorems of this section are stated in the most general terms and
are still true when one or more of the nodes are changed into cusps. It
is therefore not necessary to give separate theorems for the case of
one cusp and two nodes.


NODAL BICUSPIDAL QUARTICS.

A quartic with one node and two cusps is a curve of the fourth class,
having one double tangent and two points of inflection (see Salmon).
Hence its reciprocal is also a nodal bicuspidal quartic, a fact of
which frequent note will be made in this section.

The inverse of a conic with respect to a focus is a curve called
Pascal’s Limaçon. From the polar equation of a conic, the focus being
the pole, it is evident that the polar equation of the limaçon may be
written in the form:

                           e             1
                    r  =  —— cos_x_  +  —— ;
                           p             p

where e and p are constants, being respectively the eccentricity and
semi-latus rectum of the conic.

From the above equation it is readily seen that the curve may be traced
by drawing from a fixed point O on a circle any number of chords
and laying off a constant length on each of these lines, measured
from the circumference of the circle. The point O is the node of the
limaçon; and the fixed circle, which I shall call the base circle, is
the inverse of the directrix of the conic. This is readily shown as
follows:—the polar equation of the directrix is r = p/(e cos_x_). Hence
the equation of its inverse is r = (e cos_x_)/p, which is the equation
of the base circle of the limaçon.

The envelope of circles on the focal radii of a conic as diameters
is the auxiliary circle. Inverting:—the envelope of perpendiculars
at the extremities of the nodal radii of a limaçon is a circle with
its centre on the axis and having double contact with the limaçon.
Projecting:—from any point on a nodal bicuspidal quartic draw lines to
the three nodes and a fourth line forming with them a harmonic pencil;
the envelope of all such lines is a conic through the two cusps and
having double contact with the quartic; the chord of contact passes
through the node and cuts the line joining the cusps so that this
point of intersection, the two cusps, and intersection of the double
tangent with the cuspidal line form a harmonic range. Reciprocating:—on
any tangent to a nodal bicircular quartic take the three points where
it cuts the two inflectional tangents and the double tangent, and a
fourth point forming with these a harmonic range; the locus of all such
points is a conic touching the two inflectional tangents and having
double contact with the quartic; the pole of the chord of contact is
on the double tangent; join this last point to the intersection of the
inflectional tangents and join the node with the same intersection;
these four lines form a harmonious pencil.

If the tangent at any point P of a conic meet the directrix in Q, the
line P Q will subtend a right angle at the focus O; the circle P O Q
has P Q for a diameter and hence cuts the conic at P at right angles.
Inverting:—from any point P on the limaçon draw O P to the node O;
draw O Q perpendicular to O P meeting the base circle in Q; P Q is
normal to the limaçon at P. Projecting:—from any point P on a nodal
bicuspidal quartic draw lines to the three nodes and a fourth harmonic
to these three; from O draw lines to the two cusps and a fourth
harmonic to these two and the line O P; the locus of the intersection
of the fourth line of each pencil is a conic through the three nodes.
Call this the basal conic of the quartic. Reciprocating:—on any given
tangent to a nodal bicuspidal quartic take its points of intersection
with the double tangent and the inflectional tangents, and a fourth
point harmonic with these; on the double tangent take its points of
intersection with the given tangent and the inflectional tangents, and
a fourth point harmonic with these; the envelope of the lines joining
the fourth point of these two ranges is a conic touching the double and
inflectional tangents.

The locus of the foot of the perpendicular from the focus on the
tangent to a conic is the auxiliary circle. Inverting:—draw a circle
through the node tangent to a limaçon; draw the diameter O P of this
circle; the locus of P is a circle having double contact with the
limaçon, the axis being the chord of contact. Cor.; the locus of the
centre of the tangent circle is also a circle. Projecting:—through the
three nodes of a nodal bicuspidal quartic draw any conic touching the
quartic; the locus of the pole with respect to this conic of the line
joining the two cusps is a conic; draw the chord O P of the first conic
through the node O and the pole of the line joining the two cusps; the
locus of P is a conic through the cusps, having double contact with the
quartic.

If chords of a conic subtend a constant angle at the focus, the
tangents at the ends of the chords will meet on a fixed conic, and
the chords will envelope another fixed conic; both these conics will
have the same focus and directrix as the given conic. Inverting:—draw
two nodal radii of a limaçon O P and O Q, making a given angle at O;
the envelope of the circle P O Q is another limaçon; the locus of the
intersection of circles through O tangent to the limaçon at P and Q is
another limaçon. These two limaçons have the same node and base circle
as the given one. Projecting:—through the node O of a nodal bicuspidal
quartic draw a pencil of radii in involution; let O P and O Q be a
conjugate pair of these nodal radii; the envelope of the conic through
P, Q, and the three nodes, is another quartic of the same kind: also
draw conics through the three nodes tangent to the quartic at P and
Q; the locus of their point of intersection is another quartic of the
same kind. These three quartics all have the same node, cusps, and base
conic.

Every focal chord of a conic is cut harmonically by the curve, the
focus, and directrix. Inverting:—every nodal chord of a limaçon is
bisected by the base circle. Projecting:—every nodal chord of a nodal
bicuspidal quartic is cut harmonically by the quartic, the base conic,
and the line joining the two cusps. Reciprocating:—from any point
on the double tangent of a nodal bicuspidal quartic draw the other
two tangents to the quartic and a line to the intersection of the
inflectional tangents; the fourth harmonic to these lines envelopes a
conic.

Since the limaçon is symmetrical with respect to the axis, it
follows that the two points of inflection are situated symmetrically
with respect to the axis. Hence the line joining the two points of
inflection is parallel to the double tangent. Therefore by projection
we infer the following general theorem for the nodal bicuspidal
quartic: the line joining the two cusps, the line joining the two
points of inflection, and the double tangent meet in a point. Also the
fourth harmonic points on each of these lines lie on a line through
the node. Reciprocating:—the point of intersection of the cuspidal
tangents, the point of intersection of inflectional tangents, and the
node all lie on a right line. From the node draw a fourth harmonic
to this right line and the tangents at the node; draw a fourth line
harmonic to this right line and the inflectional tangents; draw a
fourth harmonic to the cuspidal tangents and this right line; these
three lines all meet in a point on the double tangent.


TRICUSPIDAL QUARTICS.

A tricuspidal quartic is a curve of the third class with one double
tangent and no inflection. Its reciprocal is therefore a nodal cubic.

We shall begin by reciprocating some of the simpler properties of
nodal cubics. Since the three points of inflection of a nodal cubic
lie on a right line, it follows that the three cuspidal tangents of a
tricuspidal quartic meet in a point. The reciprocal of the harmonic
polar of a point of inflection is a point on the double tangent, found
by drawing through the point of intersection of the three cuspidal
tangents a line forming with them a harmonic pencil. Three such lines
can be drawn and it is not difficult to distinguish them. All six lines
form a pencil in involution, the lines to the points of contact of
the double tangent being the foci. I shall call such a point on the
double tangent the harmonic point of the cuspidal tangent. Since any
two inflectional tangents of a nodal cubic meet on the harmonic polar
of the third point of inflection, it follows that any two cusps of a
trinodal quartic and the harmonic point of the third cuspidal tangent
lie on a right line. Since the point of contact of the tangents from a
point of inflection of a nodal cubic is on the harmonic polar of the
point, it follows that the tangent to the tricuspidal quartic at the
point where it is cut by a cuspidal tangent passes through the harmonic
point of that cuspidal tangent.

The inverse of the parabola from a focus is the cardioid; and the
inverse of the corresponding directrix is the base circle of the
cardioid. The cardioid projects into a tricuspidal quartic and its base
circle projects into a conic through the three cusps which has the same
general properties as the base conic of the nodal bicuspidal quartic.

The circle circumscribing the triangle formed by the three tangents
to a parabola passes through the focus. Inverting:—three circles
through the cusp, and tangent to a cardioid, intersect in three
collinear points. Projecting:—three conics through the three cusps
of a tricuspidal quartic and touching the quartic intersect in three
collinear points. Reciprocating:—if three conics touch the three
inflectional tangents of a nodal cubic and the cubic itself, their
three other common tangents intersect in a point.

Circles described on the focal radii of a parabola as diameters touch
the tangent through the vertex. Inverting and projecting:—from a point
on a tricuspidal quartic lines are drawn to the three cusps and a
fourth line forming a harmonic pencil; the envelope of this fourth line
is a conic through the three cusps and touching the quartic at the
point where the latter is cut by one of the cuspidal tangents. There
are three such conics, one corresponding to each cusp. At any cusp the
tangent to its corresponding base conic, the cuspidal tangent, and the
lines to the other two cusps form a harmonic pencil. Reciprocating:—on
any tangent to a nodal cubic take the three points of intersection
with the inflectional tangents and a fourth point forming with these a
harmonic range; the locus of this fourth point is a conic touching the
three inflectional tangents and the cubic. The tangent to the cubic
where it is touched by the conic goes through a point of inflection.
On any inflectional tangent the point of contact of this conic, the
point of inflection, and the points of intersection of the other two
inflectional tangents form a harmonic range.

The circle described on any focal chord of a parabola as diameter will
touch the directrix. Inverting:—the circle described on any cuspidal
chord of a cardioid will touch the base circle. Projecting:—through a
cusp C draw any chord of a tricuspidal quartic meeting the quartic in
P and O; draw a conic through P, O, and the other two cusps so that
the pencil at P formed by the tangent to the conic and the lines to
the cusps is harmonic; all such conics will touch the base conic of the
cusp C. Reciprocating:—from O, on any inflectional tangent of a nodal
cubic, draw two tangents P and Q to the cubic; draw a conic touching
the tangents P and Q and the other two inflectional tangents so that
the range on one of these tangents formed by the point of contact of
the conic and the intersection of the three inflectional tangent is
harmonic; the envelope of all such conics is a conic touching the three
inflectional tangents.

The directrix of a parabola is the locus of the intersection of tangents
at right angles to one another. Inverting and projecting:—through any
point P on the base conic of a cusp C of the tricuspidal quartic, two
conics can be drawn through the three cusps and touching the quartic;
their two tangents at P and the lines to the other two cusps form a
harmonic pencil; their two points of contact lie on a line through
C. Reciprocating:—from any point on one of the inflectional tangents
to a nodal cubic draw the two tangents P and Q; draw two conics each
touching the cubic and the three inflectional tangents, one touching P
and the other Q; the envelope of their other common tangent is a conic
touching the three inflectional tangents; the two points of contact of
any one of these common tangents and the points where it cuts the other
two inflectional tangents form a harmonic range.

Any two parabolas which have a common focus and their axes in opposite
directions cut at right angles. Inverting:—any two cardioids having a
common cusp and their axes in opposite directions cut at right angles.
Projecting:—two tricuspidal quartics having common cusps and at one
of the cusps the same cuspidal tangent, but the cusps pointed in
opposite directions, cut at such an angle that the tangents at a point
of intersection and the lines to the other two cusps form a harmonic
pencil. Reciprocating:—two nodal cubics have common inflectional
tangents and on one of them the points of inflection common, but the
branches of the curve on opposite sides of the line; any common tangent
to the two curves is cut harmonically by the points of contact and the
other two inflectional tangents.

Circles are described on any two focal chords of a parabola as
diameters; their common chord goes through the vertex of the parabola.
Inverting:—circles are described on any two cuspidal chords of a
cardioid; the circle through their points of intersection and the cusp
goes also through the vertex of the cardioid. Projecting:—through one
of the cusps of a tricuspidal quartic draw two chords; draw conics
through the other two cusps and the extremities of each of these
chords so that the pole of the line joining the other two cusps with
respect to each of these conics is on the corresponding chord; the
conic through the points of intersection of these two conics and the
cusps passes also through the point where the cuspidal tangent of
the first mentioned cusp cuts the quartic. Reciprocating:—on one of
the inflectional tangents, of a nodal cubic take two points P and Q;
draw a pair of tangents from each of these points to the cubic; draw
two conics each touching a pair of these tangents and the other two
inflectional tangents, so that the polars of the point of intersection
of the other two inflectional tangents with respect to each of those
conics pass respectively through P and Q; the conic touching the common
tangents to these two conics and the three inflectional tangents
touches also the tangent from the first mentioned point of inflection
to the cubic.


QUARTICS WITH A TRIPLE POINT.

Since a triple point is analytically equivalent to three double points,
a quartic with a triple point is unicursal. Such a quartic is obtained
by inverting a unicursal cubic from its node. The equation of such
a cubic may be written u₂ + u₃ = 0, where u₂ and u₃ are homogeneous
functions of the second and third degree respectively in x and y. Hence
the equation of the inverse curve is u₃ + u₂ (x² + y²), which shows
that the origin is a triple point and the quartic circular. By
projecting this all other forms may be obtained.

The nature of the triple point depends upon the relation of the line at
infinity to the cubic before inversion. Thus the line at infinity may
cut the cubic in three distinct points all real, or one real and two
imaginary, in one real and two coincident points (an ordinary tangent),
or in three coincident points (an inflectional tangent). Hence the
quartic may have at the triple point three distinct tangents all real,
or one real and two imaginary, one real and two coincident, or all
coincident.

This quartic may be generated in a manner similar to that used for the
curves already discussed. We showed in the section on nodal cubics that
a system of conics through A, B, C, D, and a projective pencil of rays
with its vertex at A generate by the intersection of corresponding
elements a cubic with a node at A. Invert the whole figure from A
and then project:—the pencil of rays remains a pencil; the system of
conics becomes a system of unicursal cubics having a common node at A
and passing through five other common points; the cubic inverts and
projects into a quartic with a triple point at A, passing through the
five other common points of the system of cubics.

The three points of inflection of a nodal cubic lie on a right line.
Inverting:—there are three points on a circular quartic with a triple
point whose osculating circles pass through the triple point, and
these three points lie on a circle through the triple point. Let these
three points be designated by A, B, and C. The lines from the triple
point O to the points A, B, C, and the common chord of the osculating
circles at two of them form a harmonic pencil. Through one of these
points, A, and the triple point draw a circle touching the quartic; the
point of contact is on the common chord of the osculating circles at B
and C.

From theorems which we have already proved for a system of cubics
having a common node and passing through five others fixed points,
we can infer other theorems for a system of quartics having a common
triple point and passing through seven other fixed points. For example,
any conic through the common double point and two of the fixed points
is cut by the cubics in pairs of points which determine at the node a
pencil in involution. Hence any cubic having its node at the common
triple point and passing through any four of the fixed points is cut
by the quartics in pairs of points which determine at the common
triple point a pencil in involution. Again, the pairs of tangents to
the cubics at the common double point form a pencil in involution, the
two cuspidal tangents being the foci of the pencil. Inverting:—the
line at infinity (which passes through two of the fixed points, i. e.
the circular points) cuts the system of circular quartics in pairs
of points in involution. Projecting:—a line through any two of the
seven fixed points cuts the system of quartics in pairs of points
in involution. Since the line at infinity touches the inverse of a
cuspidal cubic, it follows that any line through two of the fixed
points will touch two of the quartics of the system; these points of
contact are therefore the foci of the involution.

Other theorems on such a system of quartics will be given in the next
section.


SYSTEMS OF QUARTICS THROUGH SIXTEEN POINTS.

Let U and V represent a system of quartics through sixteen points.
Since the discriminant of quartic is of the 27th degree in the
coefficients it follows that there are 27 values of k for which the
discriminant vanishes, and hence 27 quartics of the system which
have double points. As in case of cubics these 27 points are called
the critic centres of the system. Let the equation of the system of
quartics be written

              u₄ + u₃ + u₂ + u₁ + u₀ = 0.

In a manner similar to that employed for cubics, we find the equation
of the polar cubics of the origin with respect to the system to be

                 u₃ + 2u₂ + 3u₁ + 4u₀ = 0.

The polar conics of the origin are given by

                     u₂ + 3u₁ + 6u₀ = 0;

and the polar lines of the origin, by u₁ + 4u₀ = 0.

The origin may be any point in the plane and hence we conclude that
only one quartic of the system passes through a given point and that
the polar cubics of any point form a system through nine points. The
polar conics of any point form a system through four points and the
polar lines meet in a point.

If one of the critic centres be taken for origin, we can readily see
that such a point is also a critic centre on each of its systems of
polar curves. It is thus at a vertex of the self-polar triangle of
its system of polar conics and the opposite side of the triangle is
the common polar line of the critic centre with respect to each of
the systems of curves. The tangents at the node of the nodal quartic
coincide with those of its polar cubic and these we know coincide with
the lines which constitute its polar conic.

If two of the sixteen basal points coincide, such a point is a critic
centre. The argument is the same as for a system of cubics. We can also
see that two of the basal points of each of its systems of polar curves
coincide at the critic centre. The sixteen basal points of the system
of quartics may unite two and two so that it is possible to draw a
system of quartics touching eight given lines each at a fixed point.

If three of the basal points of our system of quartics coincide, all
the quartics have at such a point a common point of inflection and a
common inflectional tangent. The demonstration is the same as that
already given for cubics. The system of polar cubics of such a point
also have this point for a common point if inflection and the same
tangent for a common inflectional tangent. I prefer to show this
analytically for the sake of the method. The equation of the system of
quartics having the origin for a common point of inflection and the
axis of y for a common inflectional tangent may be written

    u₄ + u₃ + {(B + kB₁)xy + (C + kC₁)y²} + (A + kA₁)y = 0.

The equation of the polar cubics of the origin is therefore,

    u₃ + 2{(B + kB₁)xy + (C + kC₁)y²} + 3(A + kA₁)y = 0,

which proves the proposition. The properties of the system of polar
conics of such a point are therefore the same as those already proved
for cubics. One quartic of the system has a double point at the common
point of inflection of the others.

When four basal points coincide they give rise either to a common point
of undulation or a common double point on all the quartics of the
system. The equation of the system having a common point of undulation
may be written

     u₄ + (A + kA₁)x²y + (B + kB₁)xy² + (C + kC₁)y³

                 + (D + kD₁)xy + (E + kE₁)y² + (F + kF₁)y = 0.

There is one value of k for which the last term vanishes, and hence the
origin is a critic centre. The polar cubics of the point of undulation
break up into a system of conics through four points and the common
tangent at the common point of undulation. For the equation of the
polar cubics is

     y{(A + kA₁)x² + (B + kK₁)xy + (C + kC₁)y²

                 + 2(D + kD₁)x + 2(E + kE₁)y + (F + kF₁)} = 0.

The system of polar conics of the origin consequently breaks up into
the line y = 0 and a pencil meeting in a point. The common tangent at
the common point of undulation is also the common polar line of the
point of undulation.

When the four coincident basal points form a common double point on
the quartic, it is not difficult to show that two of the quartics are
cuspidal at this point. The polar cubics of the common double point
form a system having the same point for common double point. The
tangents to the quartics at the common node constitute the system of
polar conics and form a pencil in involution. Twelve of the sixteen
basal points may unite in three groups of four each and the system of
quartics is then trinodal and passes through four other fixed points.
This is the system obtained by inverting a system of conics through
four points and then projecting.

A few special cases should be noticed here. If the four fixed points
and two of the nodes lie on a conic, this conic together with the two
lines from the third node to the first two constitute a quartic of
the system. If the four fixed points lie on a line, the quartic then
consists of this line and the sides of the triangle formed by the
nodes. If the three nodes and three of the fixed points lie on a conic,
the system of quartics then consists of this conic and a system of
conics through the three nodes and the fourth fixed point. A special
case of a system of quartics with three nodes is a system of cubics
having a common node and passing through five other fixed points
together with a line through two of them.

If a fifth basal point be moved up to join the four at the common node,
the quartics have one tangent at the common node common to all. If
six basal points coincide they have both tangents at the node common
to all. In this case one of the quartics has a triple point at the
common node of the others. If seven basal points coincide, one of these
tangents is an inflectional tangent as well. If eight points coincide,
both are inflectional tangents.

When nine of the basal points of a system of quartics coincide, the
quartics have a common triple point. This is nicely shown by inverting
a system of nodal cubics from the common node. The inverse curves form
a system of quartics having a triple point and passing through seven
other fixed points. The common triple point on two quartics counts for
nine points of intersection and the seven others make the requisite
sixteen. From our knowledge of a system of cubics having a common node
it is readily inferred that three of the quartics must each break up
into a nodal cubic and a right line through the node. If the seven
fixed points of the system of quartics lie on a cubic having a node
at the common triple point, the system of quartics then consists of
this cubic and a pencil of lines through the node. If two of the seven
fixed points lie on a line through the common triple point, the system
of quartics then consists of this right line and a system of cubics
through the other five points and having a common node at the common
triple point.

The system of cubics having a common node may have one, two, or three
of the other basal points at infinity; and these may be all distinct
or two or three of them coincident. Whence we infer that if the system
of quartics have ten coincident basal points, one of the tangents at
the triple point is common to all the quartics of the system. If eleven
basal points coincide, two of the triple-point tangents are common to
all the quartics. If twelve coincide, all three triple-point tangents
are common. These triple-point tangents may be all distinct, two
coincident, or all three coincident.

If thirteen basal points coincide, the system of quartics then consists
of the three fixed lines joining the multiple point to the other three,
together with a pencil of lines through the multiple point. If fourteen
points coincide, two lines are fixed and these with any two lines of
the pencil form a quartic of the system. If fifteen points coincide,
only one line is fixed and each quartic consists of this line and any
other three of the pencil. When all sixteen points coincide, any four
lines through it form a quartic of the system.

In this paper cubic and quartic curves only are considered. I expect
in a future paper to extend the methods herein developed to curves of
still higher degrees. Many of the present results can be generalized
and stated for a unicursal curve of the nth degree. I have purposely
omitted all consideration of focal properties of these curves. There
are also many special forms of interest which do not properly belong to
a general treatment of the subject.


NOTE A.

The theorem concerning the three points on a conic A, B, and C, whose
osculating circles pass through a fourth point O on the conic, is due
to Steiner. From the properties of the harmonic polars of the points of
inflection on a nodal cubic we may infer many other theorems concerning
the points A, B, and C on a conic. Let the cubic be projected into
a circular cubic and then inverted from the node. Its points of
inflection A₁, B₁, C₁ invert into the points A, B, and C. The harmonic
polar of A₁ inverts into the common chord O P of the circles osculating
the conic at B and C; and similarly for the other harmonic polars.

The pencil O {A B P C} is harmonic. Any circle through A and O meets
the conic in S and T so that the pencil O {A S P T} is harmonic. The
two circles through O and tangent to the conic at S and T intersect on
O P. If two circles be drawn through O and A intersecting the conic
one in S and T and the other in U and V, the circles O S U and O T V
intersect on O P; so also the circles O S V and O T U. But one circle
can be drawn through O and A and tangent to the conic; its point of
contact is on O P. Let l, m, and n be three points on the conic on a
circle through O. Draw the circles O A l, O A m, and O A n intersecting
the conic again in l₁, m₁, n₁; l₁, m₁, n₁, are also on a circle through
O, and the circles through l, m, n and l₁, m₁, n₁ intersect on O P.


NOTE B.

From the fundamental property of the Cissoid of Diocles we can obtain
by inversion an interesting theorem concerning the parabola. In the
figure of the Cissoid given in Salmon’s H. P. C. Art. 214, A M₁ = M R,
whence A M₁ = A R - A M; or A R = A M + A M₁. Inverting from the cusp
and representing the inverse points by the same letters, we have for
the parabola

                          1       1        1
                         ———  =  ———  +  ————— .
                         A R     A M      A M₁

This result is interpreted as follows:—draw the circle of curvature at
the vertex of a parabola; this circle is tangent to the ordinate B D
which is equal to the abscissa A D; draw a line through A cutting the
circle in R, the ordinate B D in M, and the parabola in M₁; then

                          1        1        1
                         ———  =   ———  +  ————— .
                         A R      A M      A M₁

Draw the circle with centre at D and radius A D; any chord of the
parabola through the vertex is cut harmonically by the parabola, the
circle, and the double ordinate through D.

Footnotes:

[1] See note A.

[2] A few of the results of this section are due to the late Mr. H. B.
Hall.

[3] See note B.




Foreign Settlements in Kansas.


A CONTRIBUTION TO DIALECT STUDY IN THE STATE.

Explanatory.—Some years ago when the subject of dialect study in
Kansas, or rather of Kansas dialect, was mentioned, Mr. Noble Prentis,
a gentleman who is warranted in speaking with authority on Kansas,
was inclined to think that he settled the question in short order by
declaring that there is no Kansas dialect. Probably the majority of
intelligent citizens of the state would turn off the subject with the
same reply. In the sense of a mode of speech common to the inhabitants
of Kansas and peculiar to them, Mr. Prentis was indeed right. There is
no vocabulary, at least no extensive vocabulary, by which the native of
Kansas may be recognized in the American Babel. We have no distinctive
pronunciation by which we may be known from the inhabitants of Nebraska
or set apart from the citizens of Missouri. The verb fails to agree
with its subject and the participle is deprived of its final ‘g’ with
about equal frequency in Western Kansas and Eastern Colorado.

But in the same sense it is true that there is no Kansas flora, no
Kansas fauna; that is, there is no plant and there is no animal found
quite generally in Kansas and found nowhere outside of Kansas. The
remark that there is no such thing as a Kansas dialect rests upon a
misapprehension of what is meant by the term. In just the same way
that we speak of the flora and the fauna of Kansas we may speak of
the dialect of Kansas. Yet to avoid popular misapprehension it may be
better to speak of dialect in Kansas, rather than of Kansas dialect.

Dialect study involves the observation and description of all facts
concerning the natural living speech of men, and especially those
points in which the speech of individuals or groups differs from that
of the standard literary language as represented in classic writers
and classic speakers. Standard literary English is always a little
behind the times. It is the stuffed and mounted specimen in the
museum. Dialect is the live animal on its native heath. Most people,
indeed, will think that their speech does not differ materially from
standard English. They say, “We speak near enough alike ‘for practical
purposes’. But a thousand years hence the pronunciation of our country
may have changed so much that it will seem like another language, and
our descendants will write learned theses to prove that we pronounced
‘cough’ like cow or like cuff. A new language will have grown out of an
old one, but no one know how it came about. Careful dialect study will
help explain it.”

Kansas is a peculiarly favorable field for dialect study. We have here
side by side representatives from nearly every state in the Union, and
from a dozen foreign countries. The observer has here what elsewhere
he must travel over half the world to find. In a district where the
people are all natives, the speech is so nearly homogeneous that it
is difficult to find any one who recognizes the peculiarities of his
own language, but here the contrast of strange tongues strikes us
immediately and we become conscious early of the fact that all men do
not speak alike.

Study of dialect may be classified under the heads of pronunciation,
grammar and vocabulary. Of these the last two are the easiest, and may
be carried on by almost any one with pleasure and valuable results.
Pronunciation is the most difficult of these matters to study, as
competent observation and reports can be made only by one who has made
a thorough study of Phonetics. To those who might wish to take up the
study of this branch of the subject, Sweet’s Primer of Phonetics, and
Grandgent’s “Vowel Measurements” and “German and English Sounds” are
recommended.

In the study of dialect vocabularies it may become of the greatest
importance to establish the exact locality of a word and the origin
of the persons by whom it is used. For instance, in a family of my
acquaintance the word ‘slandering’ = sauntering was familiar. It
was a great puzzle to me until I learned that some of the children
had been in the care of a German maid. The German word ‘schlendern’
suggested the unquestionable source of the peculiar word. As a source
of information regarding the origin of the foreign elements of our
population when their native speech shall have been forgotten, but
when the influence of it will be left in vocabulary and pronunciation
I have thought that a map of the state with the location of all the
foreign settlements of even quite small size would be of interest and
in time of great value. In the following pages I transmit the results
of my inquiries so far as received. It is my intention to make the
report complete and to publish the map, when as complete as it can be
made, in colors. Unexpected difficulties have delayed the work and
prevented its being complete. I depended for my information upon the
County Superintendents of the State, a class of unusually intelligent
and well-informed men and women. But in not a few cases there seems to
have been a suspicion in the mind of my correspondent that I might
be a special officer of the state trying to locate violations of the
law requiring district schools to be conducted in English, and hence
information regarding schools in foreign tongue was withheld or given
but partially. And in some cases my informants were not well posted. A
superintendent by the name of Schauermann in a county containing a town
called Suabia, tells me that there are no foreigners in his county. In
such cases time must be taken to secure a correct result.

The questions asked were: Locate, and give origin, date and approximate
numbers of any settlements—six or more families—of foreigners in
your county. Do they still use their language to any extent? Do they
have church service and schools conducted in their native tongue? In
many replies one or more of these points was neglected so that the
information is not yet by any means what I desire to make it. However,
for the purpose of dialect study approximate correctness in location is
of chief importance, and accuracy as to numbers quite secondary.

Through the aid of ministers and others to whom I have been referred
by the superintendents I hope to make this report complete in the
following respects: The more exact limits of the settlement; the
numbers of those foreign-born; the province as well as land from
which they came; the number of churches; the number of schools and
the length of time the same are conducted. I solicit the co-operation
of everyone interested in this work, and also in the whole subject of
dialect study. As intimated above, interested observers can without
especial training do a service to science and at the same time find a
fascinating pastime for themselves by making collections of words and
constructions which they believe to be unusual or new. If any such
are sent to the writer they will be duly acknowledged. They should in
every case be accompanied by a statement of the age, condition and
birth-place of the person using them.

I wish here to call attention to the work of the American Dialect
Society which exists to promote this study. It desires as wide a
membership as possible, and membership is open to all interested in
the subject. The publication of the Society, Dialect Notes, contains
reports of word-lists and other studies, and will be an aid to any
who wish to undertake similar work. Subscriptions and membership fees
should be sent to Mr. C. H. Grandgent, Treas., Cambridge, Mass.


REPORTS BY COUNTIES.

    ATCHISON.—Reports no foreigners, by John
        Klopfenstein, Supt.

    ALLEN.—Swedes and Danes, from 600 to 700, settled
        from 1873 to 1880. Have church service, and four to five
        months school in Swedish. Grove and Elsmore townships.
        Germans in and around Humboldt.

    ANDERSON.—Irish in Reeder township, 1860 and 1874.
        Germans, 1860 in Putman township, 1880 in Westphalia
        township. Have both church and schools in German.

    BARBER.—Reports “no foreigners worth making
        account of”, by J. O. Hahn, Sup’t.

    BARTON.—No report.

    BOURBON.—Reports no foreigners.

    BROWN.—No report.

    BUTLER.—Germans (Prussians), speaking Low German,
        in Fairmount and Milton townships. Hold church services
        but no schools in German.

    CHASE.—Russian Mennonites, speaking both Russian
        and German, in Diamond Creek township, no church, but a
        portion of schooling in German. Germans at Strong City,
        with both church and schools in their native tongue.

    CHATAUQUA.—Some Norwegians and Swedes, 1870, no location
        given. Neither schools nor churches in native tongue.
        One colony of ‘Russians’ (Mennonites?), who have also
        given up their language.

    CHEROKEE.—Weir City, French and Italians, number
        considerable. Scammon, Scotch, also in large numbers.
        The French and Italians have neither schools nor church
        in the native tongue. Germans in Ross, twenty families;
        with church originally Lutheran, now Mennonite; school
        irregularly during past ten years. Swedes, a few
        families in Cherokee township, have entirely given up
        Swedish language. The Scotch, French and Italians in
        mines or mining industries.

    CHEYENNE.—Germans settled in 1885-86 on Hackberry
        Creek, 160 persons; in the northeast corner of the
        county, 100; on west border of county, north of
        Republican river, 120; all with churches and the last
        two with occasional schools. Swedes are across the
        Republican adjoining last named German settlement,
        120, entered 1886, having neither church nor school in
        Swedish.

    CLARKE.—Reports no foreigners.

    CLAY.—No report.

    CLOUD.—Canadian French are scattered over much of
        the county, with considerable settlements in and around
        the towns of Concordia, Clyde, St. Joseph and Aurora. In
        all there are churches, in the first three schools also
        conducted in French. Norwegians occupy portions of Sibley
        and Lincoln townships with two churches in their own
        tongue. They number about three hundred. Irish occupy
        portions of Solomon and Lyon, the south part of Meredith
        and the southeast corner of Grant townships.

    COFFEY.—Germans in Liberty and on border of Leroy
        and Avon township. Have church service in German.

    COMANCHE.—Germans. A few scattered families.

    COWLEY.—A few Swedes and Germans, widely scattered.

    CRAWFORD.—Irish and French in Grant township;
        Swedes in west part of Sherman township, have all
        given up their language. Italians, Austrians and other
        nationalities in south part of Washington, southeast
        part of Sheridan and all over Baker township, especially
        in Pittsburg, employed in mining and smelting.

    DECATUR.—Swedes in Oberlin township; Mennonites
        in Prairie Dog township; Germans in and around Dresden,
        with Catholic church; Bohemians in Jennings and Garfield
        townships.

    DICKINSON.—Germans, 500 in number settled in
        1860 in Liberty, Union and Lyon townships. Have three
        churches and two schools in German. Also in Wheatland,
        Jefferson, Bonner and Ridge townships, one church and a
        school. Swedes, 100 settled in 1860 in Center and Hayes
        township, with two churches and one school in Swedish.
        Irish, several hundred in south part of Banner township.

    DONIPHAN.—Germans in Wayne, Marion and southern
        part of Center, Burr Oak and Washington townships, with
        church service in native tongue. Norwegians in eastern
        part of Wolf River township.

    DOUGLAS.—There are German settlements in Eudora
        township (300), Marion township (600), and Big Springs
        township (100), with churches in all and school in
        the first. There are about five hundred Germans in
        Lawrence, with three German churches. There are smaller
        settlements of Germans and Scandinavians at several
        points in the county.

    EDWARDS.—Germans and Swedes in Kinsley, Jackson
        and Trenton townships, have church service in their
        mother tongue.

    ELK.—Swedes in Painter and Hood townships; Irish
        in Falls township; Germans on the border of Elk and Wild
        Cat townships. None of these have church or school in
        the native tongue, but all use it at home.

    ELLIS.—Germans from Russia, settled about 1876
        in Catherine, Hartsook, Lookout, Wheatland and Freedom
        townships, about 3000 in number—one third of the
        population of the county in 1891. They are Catholics,
        and have both churches and parochial schools conducted
        in German. They are large wheat-growers.

    ELLSWORTH.—Germans, (Methodists) in south part of
        Valley township; Germans (Lutherans) in north part of
        Columbia and Ellsworth townships; Germans (Baptists)
        from Prussia, in Green Garden and south west corner of
        Empire townships. These all have church service, and
        the Lutherans schools in their own tongue. Bohemians in
        Valley and Noble townships.

    FINNEY.—Reports no foreigners.

    FORD.—Germans in Wheatland and Speareville
        townships. Have church, and one school conducted in
        German.

    FRANKLIN.—No report.

    GARFIELD.—A few scattered families of Germans.

    GEARY.—Irish (Connaught) came into Jackson,
        Jefferson and Liberty townships 1855, about 1500 in
        number. Germans (Anhalt) about 1500 came into Jefferson,
        Milford and Lyon townships in 1862. About 300 English
        from Sussex settled in Lyon township in 1870. The
        Germans maintain both churches and schools in German.

    GOVE.—Swedes in Lewis and south part of Grinnell
        and south west corner of Gove townships.

    GRAHAM.—A settlement of Canadian French (600)
        was made in adjacent parts of Wild Horse and Morelan
        townships about 1880. They conduct church service but no
        schools in French.

    GRANT.—Reports no foreigners.

    GRAY.—Reports no foreigners.

    GREELEY.—Swedes in the north west part of the
        county, have church service and summer school in Swedish.

    GREENWOOD.—Norwegians, about 200, in south part of
        Salem township, have church in their own tongue. Germans
        in Shell Rock township, about 300, also have church in
        their own language.

    HAMILTON.—Reports no foreigners.

    HARPER.—Germans about the town of Harper.
        Hungarians south of Bluff City. Both have church service
        in German. About 100 French in Odell and Stohrville
        townships.

    HARVEY.—Germans (Russian Mennonites) from Odessa,
        a few from Prussia, the latter speaking Low German. They
        settled from 1874 to 1876 in Alta and Garden townships,
        in Pleasant and the eastern part of Newton townships, and
        about Halstead. They have church and school in German,
        but speak Russian also. French in north part of Emma
        township, engaged in raising silk worms.

    HASKELL.—Reports no foreigners.

    HODGMAN.—Germans, about 30 families, settled about
        1884 in south east corner of Sterling township; have
        preaching in German. Swedes in north west corner of
        Marena township.

    JACKSON.—Danes in Netawaca and Whiting townships;
        Irish in Washington township; neither continue to use
        their native tongue.

    JEFFERSON.—Germans (Swiss) in Delaware, Jefferson
        and Kentucky townships, maintaining church but no
        schools in German.

    JEWELL.—Swedes, widely scattered in Sinclair,
        Allen, Ewing and Ezbon townships.

    JOHNSON.—No report.

    KEARNEY.—No report.

    KINGMAN.—A small settlement of Germans in Peters
        township, not using German to any extent. A few Irish in
        Union township.

    KIOWA.—No report.

    LABETTE.—Swedes and Norwegians settled in Valley
        and Canada townships about 1869. Still speak their
        language, but have neither church nor school in it.

    LANE.—Reports no foreigners.

    LEAVENWORTH.—German, in 1873 in Easton township;
        in Fair township in 1876; about 600 in each place. They
        have church service and schools in German.

    LINCOLN.—Danes settled in Grant township in 1869
        and since, 400 in number. Germans settled in Pleasant
        township in 1872, with 300, and in Indiana township in
        1869 and later with about 375. Danes and Germans have
        good schools and churches in native tongue. Bohemians
        in Highland township in 1878 with thirty families.
        They speak their native tongue, but have no schools or
        churches.

    LINN.—Reports no foreigners.

    LOGAN.—Swedes, about 200, about Page City, in
        north part of county. Have church and school both in
        Swedish.

    LYON.—Welsh, between 1000 and 1500 are located
        in and about Emporia, with three churches conducted in
        Welsh. There is a settlement of Scandinavians near Olpe
        in Centre township.

    MARION.—Germans (Russian Mennonites), settled in
        Logan, Durham, Lehigh, Risley, Menno, West Branch and
        Liberty townships, from 1870 to 1875, some 5000 in number.
        They speak both Russian and German, and have church
        service and schools in the latter tongue. Bohemians,
        about 500 in number are settled in Clark township. They
        speak Czech and have church service in that language.
        French to the number of 200 settled soon after 1870 on
        the border of Grant and Doyle townships. They speak
        French still, but have no schools or church service in
        the language.

    MARSHALL.—Germans (Pommeranians, Hanoverians,
        Frisians) to the number of 2000, settled in the west
        part of Marysville township from before 1860 to 1870.
        They have both church and school in their own tongue.
        In the north part of Murray and the south half of
        Vermillion townships are 1200 Irish, who use only
        English in church and school. They came before 1870.
        Bohemians in small numbers occupy the north part of
        Guittard, the north west corner of Waterville and the
        south part of Blue Rapids townships; Swedes a portion of
        the south part of Waterville township. No report as to
        their language.

    MEADE.—No report.

    MIAMI.—Germans occupy the north part of Wea and
        the west part of Valley townships, about 200 in each
        settlement; the first has a Catholic, the second a
        Lutheran church. Irish occupy the north part of Osage
        township, also about 200 in number.

    MITCHELL.—Germans to the number of 1200 occupy
        Pittsburg, Blue Hill, and Carr Creek townships. In the
        first there is a church, and a well-attended school
        (Catholic) at Tipton.

    MONTGOMERY.—Germans to the number of 100 are
        settled in and about Independence. They have church
        service in German (Lutheran).

    MORRIS.—Swedes occupy Diamond Valley, the west
        part of Creek, and the north part of Parker townships.
        They have several churches and occasionally a school
        conducted in Swedish.

    MORTON.—Reports no foreigners.

    MCPHERSON.—Swedes settled, about 1870, in Union,
        Smoky Hill, Harper, New Gottland, Delmore, and portions
        of other townships, in large numbers, several thousand.
        They have several churches and excellent schools
        conducted in Swedish. Germans (Russian Mennonites)
        occupy Superior, Turkey Creek, Mound, Lone Tree, King
        City, and portions of McPherson and other southern
        townships, with several churches and schools. The
        Mennonites number about 5000 and settled after 1876.

    NEMAHA.—Germans (Swiss) occupy Nemaha and
        Washington, and a portion of Richmond townships, with
        German churches and schools. Irish are in Clear Creek
        and north east corner of Neuchatel townships. Most of
        Neuchatel township is occupied by French (Swiss).

    NEOSHO.—Germans have a considerable settlement in
        the south east corner of Tioga township, with church
        service (Lutheran) in German; another in the east part
        of Lincoln township, where the language is spoken, but
        without church or school. Swedes have settlements in the
        north west part of Tioga and the east part of Big Creek
        townships; church in the first only, though in both
        Swedish is spoken almost exclusively.

    NESS.—No report.

    NORTON.—Germans to the number of 100 settled about
        1880 in Grant township. They have church service in
        German.

    OSAGE.—Swedes, (700 in number,) settled in
        Grant township in 1871, where they have four churches
        conducted in Swedish. Welsh settled in 1869 in Arvonia
        township, and others in the north part of Superior
        township, 700 in number. They have six churches with
        services in Welsh. Germans are in the north part of
        Scranton and Ridgway townships, 200 in number; French
        in the central part of Superior township, 200 strong;
        Danes, 200, in north part of Melvern and Olivet
        townships; a considerable number of Scotch and Irish in
        and near Scranton. Most of these latter are engaged in
        coal mining. None of the foreigners have schools—except
        Sunday schools—in their native tongue.

    OSBORNE.—Germans settled in Bloom township, where
        they have both church and school in their mother tongue.

    OTTAWA.—Bohemians are located about the border
        of Sheridan and Fountain townships; Danes in the south
        part of Buckeye township; Irish, arrived about 1885, in
        the south part of Chapman township. None of these have
        church or school in a foreign tongue.

    PAWNEE.—Swedes settled about 1877 in the west part
        of Garfield and the north part of Walnut townships, about
        500 in all. They speak their native language at home
        almost exclusively, and have preaching in it.

    PHILLIPS.—Germans occupy Mound and south part of
        Dayton townships, with preaching and parochial school
        in German. Dutch occupy east part of Prairie View with
        adjacent portions of Long Island, Dayton, and Beaver
        townships, with preaching in Dutch. Some Danes and Swedes
        in Crystal township, and some scattered Poles.

    POTTAWATOMIE.—Germans, about 2500, in west half of
        Mill Creek and adjacent portions of Sherman and Vienna
        townships, also in Pottawatomie and adjacent portions of
        Union, Louisville, and St. George townships. There are a
        few families in Wamego and St. Mary’s Mission. They have
        several schools and churches conducted in German. Swedes
        occupy the whole of Blue Valley and the west border of
        Greene townships, and have a small settlement in St.
        Mary township, numbering in all 1200. They have church
        service and a parochial school conducted in Swedish.
        Irish, to the number of 2000 occupy Clear Creek, Emmet,
        St. Mary and the border of St. Clere townships. French
        (Canadian), numbering 200, are found in the north part
        of Mill Creek and in Union townships, also a few about
        St. Mary’s Mission.

    PRATT.—Reports no foreigners.

    RAWLINS.—Germans in north east part of county with
        church and school in German. Swedes in east part of
        county, Bohemians and Hungarians in north and north east
        portion.

    RENO.—Germans, about 300, came in 1880 to north
        east corner of Little River township, and about 200 to
        south east corner of Sumner township; also a settlement
        in the west part of Hayes township; Dutch, about 350,
        came 1878 into Haven township; Russians are settled
        in Salt Creek and Medford townships. All have church
        service and schools in their native tongue. There are
        also a few French and Danes in the county.

    REPUBLIC.—No report.

    RICE.—There is a considerable settlement of
        Germans in Valley township, also Pennsylvania Germans in
        the west part of Sterling township, with German churches
        in both. There are also some Germans in the town of
        Lyons, with a German church.

    RILEY.—Swedes, about 2500, occupy Jackson, Swede
        Creek and adjacent portions of Mayday, Center, Fancy
        Creek and Sherman townships. They have church services
        and summer schools in their own tongue. Bohemians and
        Germans, about 500 together, occupy the north east part
        of Swede Creek township.

    ROOKS.—Germans, 10 families, settled 1880 in north
        part of Northhampton township. Bohemians, 10 families,
        located in north part of Logan township in 1879. French,
        about 30 families, south west corner of Logan, and same
        number in Twin Mound township, settled in 1878, speak
        French and have preaching in that tongue. The Germans
        have church service in German.

    RUSH.—Germans (Russian Mennonites) are located as
        follows: in Big Timber township 75 families, in Illinois
        township 25 families, in Pioneer township 50 families,
        in Lone Star township 50 families, in Banner township
        25 families, in Garfield township 25 families, in Belle
        Prairie township 30 families. In each township there is
        one church or more, but no German schools (?). Bohemians
        are found in Banner and Garfield townships, about 25
        families in each.

    RUSSELL.—No report.

    SALINE.—Germans, (Bavarians and Swabians) about
        200, in Gypsum and south part of Ohio townships; Swedes,
        3000 to 4000, in Washington, Smolan, Falun, Liberty and
        Smoky View, and adjacent parts of Spring Creek, Summit
        and Walnut townships, also in Salina. The Swedes came
        in 1868. Both Germans and Swedes have preaching and the
        latter have schools in their tongue.

    SCOTT.—No report.

    SEDGWICK.—Germans, 3000 to 4000, settled from
        1874-82 in Sherman, Grand River, Garden Plain, Attica
        and Union townships. Also about 2000 Germans in the
        city of Wichita. In both places schools and churches in
        German. Russians, Italians, French and Scandinavians
        are represented, a few hundred each, in Wichita. In the
        country townships a few Dutch and Swedes.

    SEWARD.—Reports no foreigners.

    SHAWNEE.—Germans (Moravians) in Rossville
        township, speak their native tongue almost exclusively,
        but have neither school nor preaching in German.

    SHERIDAN.—No reports.

    SHERMAN.—Germans, 20 families about the center of
        the county. Swedes, 10 families in north east corner
        and 25 families in south west corner. Both Germans and
        Swedes have schools and preaching in their native tongue.

    SMITH.—Germans in west part of Swan and Cedar
        townships, and on border of Harvey and Banner townships,
        in both churches, and in the first schools, in German.
        Dutch, in the south half of Lincoln township, have
        church but no schools.

    STAFFORD.—Germans in Hayes and Cooper townships,
        three hundred in number, with two churches having
        service in German.

    STANTON.—A few scattered Germans.

    STEVENS.—No report.

    SUMNER.—No report.

    THOMAS.—A few foreigners scattered about the
        country; all anglicised.

    TREGO.—No report.

    WABAUNSEE.—Germans and some Swedes in Kaw,
        Newbury, Mill, Farmer, Alma and Washington townships,
        with both preaching and schools in the mother tongue.

    WALLACE.—Swedes, to the number of 300, have
        settled since 1888 in the south west corner of the
        county. They have church and schools in Swedish.

    WASHINGTON.—Germans in Franklin, Charleston,
        Hanover and north part of Sherman townships, have both
        church and schools (6) conducted in German. Bohemians
        are numerous in Little Blue township; French about
        midway in Sherman township; Irish in Barnes, south part
        of Sherman and Koloko townships.

    WICHITA.—No report.

    WILSON.—Swedes have settled since 1870 in Colfax
        township. They have preaching but no schools in Swedish.

    WOODSON.—No report.

    WYANDOTTE.—Germans, 150, in north west corner of
        Prairie township; Swedes, 350, in Kansas City, Kas.;
        both have church service in the native language. Welsh,
        200, in Rosedale, and Irish about midway in Wyandotte
        township.


                              SUMMARIES.

        There are German settlements of thirty or more persons
        in the following counties: Allen, Anderson, Butler,
        Chase, Chautauqua, Cherokee, Cheyenne, Coffey, Comanche,
        Cowley, Crawford, Decatur, Dickinson, Doniphan, Douglas,
        Edwards, Elk, Ellis, Ellsworth, Ford, Garfield, Geary,
        Greenwood, Harper, Harvey, Hodgeman, Jefferson, Kingman,
        Leavenworth, Lincoln, Marion, Marshall, Miami, Mitchell,
        Montgomery, McPherson, Nemaha, Neosho, Norton, Osage,
        Osborne, Phillips, Pottawatomie, Rawlins, Reno, Rice,
        Riley, Rooks, Rush, Saline, Sedgwick, Shawnee, Sherman,
        Smith, Stafford, Stanton, Thomas, Wabaunsee, Washington,
        Wyandotte.                                    Total, 60.

        Scandinavians in settlements of thirty or over are
        found in: Allen, Chautauqua, Cherokee, Cheyenne, Cloud,
        Cowley, Crawford, Decatur, Dickinson, Doniphan, Edwards,
        Elk, Gove, Greeley, Greenwood, Hodgeman, Jackson,
        Jewell, Labette, Lincoln, Logan, Lyon, Marshall, Morris,
        McPherson, Neosho, Osage, Ottawa, Pawnee, Phillips,
        Pottawatomie, Rawlins, Riley, Saline, Sedgwick, Sherman,
        Wabaunsee, Wallace, Wilson, Wyandotte.        Total, 40.

        Settlements of Slavonic peoples, Bohemians, Poles,
        Russians, or Hungarians, in: Decatur, Ellsworth, Harper,
        Lincoln, Marshall, Ottawa, Phillips, Rawlins, Reno,
        Riley, Rooks, Rush, Sedgwick, Washington.     Total, 14.

        Settlements of Irish have been made in: Anderson, Cloud,
        Crawford, Dickinson, Doniphan, Elk, Geary, Jackson,
        Kingman, Marshall, Miami, Nemaha, Osage, Ottawa,
        Pottawatomie, Washington, Wyandotte.          Total, 17.

        French are found in settlements of thirty or more in:
        Cherokee, Cloud, Crawford, Doniphan, Graham, Harper,
        Harvey, Nemaha, Osage, Pottawatomie, Rooks, Sedgwick,
        Washington.                                   Total, 13.

        Italians are in Cherokee, Crawford, Sedgwick.  Total, 3.

        Welsh in Lyon, Osage and Wyandotte.            Total, 3.

        Dutch in Phillips, Reno, Sedgwick.             Total, 3.

        Scotch are reported from Cherokee and Osage.   Total, 2.

        English in Geary and Doniphan.                 Total, 2.

        The following counties report that there are no
        settlements of people of foreign birth within their
        borders: Atchison, Barber, Bourbon, Clarke, Finney,
        Grant, Gray, Hamilton, Haskell, Lane, Linn, Morton,
        Pratt, Seward. Total, 14.

        No reports have been secured from the following
        counties: Barton, Brown, Clay, Franklin, Johnson,
        Kearney, Kiowa, Meade, Ness, Republic, Russell, Scott,
        Sheridan, Stevens, Sumner, Trego, Wichita, Woodson.
        Total, 18.

Seventy-four of our Kansas counties report settlements of citizens of
foreign birth in numbers above 30. In so many cases there is no report
or estimate of numbers that it is not worth while to give summaries.
Probably there are not actually ten counties that have not such
settlements.

        Church services in a foreign tongue are held as follows:
        Allen S.,[4] Anderson G., Butler G., Chase G., Cheyenne
        G., Cherokee G., Cloud F. S., Coffey G., Decatur G.,
        Dickinson G. S., Doniphan G., Douglas G., Edwards G. S.,
        Ellis G., Ellsworth G., Ford G., Geary G., Graham F.,
        Greeley S., Greenwood G. S., Harper G. Hung., Harvey
        G., Hodgeman G., Jefferson G., Leavenworth G., Lincoln
        G. Du., Logan S., Lyon W. G., Marion G. Boh., Marshall
        G., Miami G., Mitchell G., Montgomery G., Morris S.,
        McPherson S. G., Nemaha G., Neosho G. S., Norton G.,
        Osage S. Welsh, Osborne G., Pawnee S., Phillips G.
        Du., Pottawatomie G. S., Rawlins G., Reno G. Du. Rus.,
        Rice G., Riley S., Rooks F. G., Rush G., Saline G. S.,
        Sedgwick G., Sherman G. S., Smith G. Du., Stafford G.,
        Wabaunsee G., Wallace S., Washington G. Wilson S.,
        Wyandotte G. S.                              Total, 58.

This total of fifty-eight counties in which church service is held in
a foreign tongue does not at all indicate the number of such churches.
In many of the reports received the number is not given, or merely
in the plural. These very incomplete reports indicate one hundred
thirty-eight such churches; it is safe to say that the number is nearly
double this.

        More interesting is the number of schools conducted in a
        foreign tongue. The counties having them are: Allen S.,
        Anderson G., Chase G., Cheyenne G., Cherokee, G., Cloud
        F., Dickinson G. S., Douglas G., Ellis G., Ellsworth G.,
        Ford G., Geary G., Greeley S., Harvey G., Leavenworth
        G., Lincoln G. S., Logan S., Marion G., Marshall G.,
        Mitchell G., Morris S., McPherson S. G., Nemaha G.,
        Osborne G., Phillips G., Pottawatomie G. S., Rawlins G.,
        Reno G. Du. Rus., Riley S., Rush G., Saline S., Sedgwick
        G., Sherman G. S., Smith G. Du., Wabaunsee G., Wallace
        S., Washington G.
                                                     Total, 37.

The number of separate schools in a foreign language so far as reported
is seventy-four, and here, too, it is safe to say that the actual
number is much larger.


EXPLANATION.

The spaces indicating settlements are in many cases too small to admit
a complete description of the inhabitants, and accordingly they have
been marked by races rather than by nationalities and tribes. “German”
is made to do duty for all inhabitants of Germany whether Low or High,
as well as for Austrians, German Swiss, and Russo-German Mennonites.
The last are reported simply as Mennonites, but are, I believe, in
all cases of German origin. “Scandinavian” is used instead of Swede,
Norwegian and Dane, because in some cases the distinction was not made
in the reports, and in order to limit the number of colors on the map
which is to come. In the case of Scotch I have been unable to secure
information whether they are Highlanders or Lowlanders, and in case of
Irish, to what extent, if at all, they speak the old Irish language.

                                            W. H. CARRUTH.

[Illustration: A PRELIMINARY MAP OF FOREIGN SETTLEMENTS IN KANSAS.

B - Bohemians (in a few cases other Slavs)

G - Germans (including Dutch and Russian Mennonites)

S - Scandinavians (Danes, Swedes and Norwegians)

I - Irish   W - Welsh   It - Italians   F - French

H - Hungarians]

Footnote:

[4] G = German, S = Scandinavian, F = French, W = Welsh, Du = Dutch.




The Great Spirit Spring Mound.


BY E. H. S. BAILEY.

The “Waconda” or Great Spirit Spring, which is situated in Mitchell
County, Kansas, about two miles east of Cawker City, has been described
in detail by G. E. Patrick in vol. vii, p. 22, Transactions of the
Kansas Academy of Science. An analysis of the water, and of the rock
forming the mound on which the spring is located, is also given.

The spring is upon a conical, limestone mound 42 feet in height, and
150 feet in diameter at the top. The pool itself is a nearly circular
lake about 50 feet in diameter, 35 feet deep, and the water rises to
within a few inches of the top of the basin. There is a level space on
all sides of the spring so wide that a carriage can be readily driven
around it.

There is but little indication of organic matter in the water of the
large spring, though there is a slimy white deposit adhering to the
bottom and sides, but the water is colorless, clear, and transparent.
The excess of water, instead of overflowing the bank, escapes by
numerous small fissures, from 10 to 20 feet down on the sides,
especially on the side away from the bluff. In these lateral springs
there is an abundance of green algæ, and a whitish scum, which seems
to be detached from the bottom and to float to the surface. This has
a slimy, granular feeling suggesting in a very marked manner hydrated
silica.

The mound is situated within about 200 feet of a limestone bluff,
which rises perhaps 20 feet above the level of the spring. The natural
inference would be that the harder material of the mound protected
it from the erosion which carried away the rock in the valley of the
Solomon on the south, and the rock between the spring and the bluff.

Is it not possible however that the mound has been really made by the
successive deposits from the spring? Although the mound is plainly
stratified, this need not interfere with the theory, for the water
may have been intermittent in its flow. The rock is very porous, and
on being ground to a thin section is shown to be concretionary in
structure.

An analysis of the water of the spring (loc. cit.) showed that it
contained over 1120 grains of mineral matter per gallon, of which 775
grains were sodium chloride and 206 grains sodium sulphate, with 66
grains of magnesium sulphate, 41 grains of magnesium carbonate, and 31
grains of calcium carbonate. An analysis by the author shows that there
are 0.874 grains of silica.

Samples of the rock composing the mound, and of the adjoining bluff
were secured, and comparative analyses made, with the following results:

                                                              GREAT
                                               COUNTRY       SPIRIT
                                                 ROCK.        MOUND.

    Silica and insoluble residue                  2.14         4.10
    Oxides of Iron and Alumina                    3.22      [5]2.66
    Sulphuric Anhydride                            .00         0.34
    Carbon Dioxide                               40.90        39.10
    Calcium Oxide                                51.90        49.28
    Magnesium Oxide                               0.63         1.15
    Water and organic matter, undetermined        1.21      [6]3.37
                                                ——————       ——————
                                                100.00       100.00

    Specific gravity                              2.52         2.79

The rocks are entirely different in appearance and structure, that of
the mound being twice as hard as that of the bluff. The former contains
much organic matter as is shown by blackening when it is heated in a
tube and giving off the characteristic odor. The iron is practically
of the ferrous variety, probably combined with carbonic acid, and the
rock contains traces of chlorides. The particular sample taken was at
some distance from the spring, and had been thoroughly exposed to the
weather.

The rock of the mound is of just such a character as might have been
built up by deposition from the water, as it contains the least soluble
constituents of the water. The process of solidification would have
been assisted by the silica in the water, forming insoluble cementing
silicates, as noticed by Prof. Patrick. The analysis given above shows
that there is abundant silica in the water for this purpose.

Mention has been made of the organic growth in the adjacent springs.
The mixed scum on being heated changes from a dull green to a vivid
grass-green, and if ignited it swells up and emits an ill-smelling
vapor, which is evidently nitrogenous in its character. A grayish white
ash is left, which contains much carbonate of lime. This is evidently
freshly deposited, as it is entangled in the algæ in granular lumps.

A specimen of the white scum, noticed above, only slightly mixed with
the green algæ, was analyzed. The acid solution of the ash contains
1.26 per cent of soluble silica. This was of course a combined silica,
probably calcium silicate, which becomes the cementing material in
the rock. In another sample of ash, after removing all the substances
soluble in hot water, the residue was found to contain 76.46 per cent
of silica.

The siliceous residue from the scum was examined by Dr. S. W.
Williston. It consists mostly of diatoms. He recognized

    Navicula—    2 species
    Nitzschia—   2 species
    Asteronella— 1 species.

All three genera are found both in fresh and salt or brackish water.

The green material consists essentially of Oscillaria and Confervæ.
If the scum is allowed to stand for a short time a very strong
sulphuretted odor is developed, strangely suggestive of salt water
marshes or mud flats; and indeed the same odor is noticed in the
vicinity of the spring. No characteristic salt water organisms, that
should occasion this peculiar odor have, however, yet been observed
here. A more extended and special study of the organic life of these
interior salt water marshes and springs would be of great interest.

Footnotes:

[5] Mostly FeO, and so calculated.

[6] With alkalies.




On Pascal’s Limaçon and the Cardioid.


BY H. C. RIGGS.

The inverse of a conic with respect to a focus is a curve called
Pascal’s Limaçon. From the polar equation of a conic, the focus
being the pole, it is evident that the polar equation of the limaçon
may be written in the form:

                           e              1
                      r = ——— cos_x_  +  ——— ;
                           p              p

where e and p are constants, being respectively the eccentricity and
semi-latus rectum of the conic.

From the above equation it is readily seen that the curve may be traced
by drawing from a fixed point O on a circle any number of chords
and laying off a constant length on each of these lines, measured
from the circumference of the circle. The point O is the node of the
limaçon; and the fixed circle, which I shall call the base circle, is
the inverse of the directrix of the conic. This is readily shown as
follows:—the polar equation of the directrix is r = p/(e cos_x_). Hence
the equation of its inverse is r = (e cos_x_)/p, which is the equation of
the base circle of the limaçon.

If the conic which we invert be an ellipse, the point O will be an
acnode on the Limaçon; if the conic be a hyperbola, the point O is a
crunode. If the conic be a parabola, O is then a cusp and the inverse
curve is called the Cardioid.

The limaçon may also be traced as a roulette.

Let the circle A C have a diameter just twice that of the circle A B.
Then a given diameter of A C will always pass through a fixed point Q
on the circle A B, (Williamson’s Diff. Cal. Art. 286) and will have its
middle point on the circle A B. Now any point P on the diameter of A C
will always be at a fixed distance from C and will therefore describe a
limaçon of which A B will be the base circle.

The pedal of a circle with respect to any point is a limaçon. This may
be inferred from the general theorem that the pedal of a curve is the
inverse of its polar reciprocal, (Salmon’s H. P. C. Art. 122). For the
polar reciprocal of a conic from its focus is a circle and hence its
pedal is a limaçon.

The base circle is the locus of the instantaneous centre for all points
on the limaçon. Let B O P be a line cutting a circle in B and Q. Let
the line revolve about B, Q following the circle; the point P will
trace a limaçon.

Now, for any instant, the instantaneous center will be the same whether
Q be following the circle or the tangent at the point where the line
cuts the circle. Therefore the instantaneous center for the point P
is found by erecting a perpendicular to the line P B, through B, and
a normal to the circle at Q. (Williamson’s Diff. Cal. Art. 294). The
intersection (C) of these two lines is the instantaneous center for the
curve at the point P. But by elementary geometry C is on the circle.
Now as the line P B revolves through 360° around B, the line B C which
is always perpendicular to it also makes a complete revolution and the
instantaneous center C moves once round the base circle.

Below we give a list of theorems obtained by inverting the
corresponding theorems respecting a conic. In these theorems any circle
through the pole is called a nodal circle, any chord through the pole
is called a nodal chord, and the line through the pole perpendicular to
the axis of the curve is called the latus rectum. The letters _e_ and
_p_ signify respectively the eccentricity and half the latus rectum of
the inverted conic.

    The locus of the point of intersection
    of two tangents to a parabola which
    cut one another at a constant angle is
    a hyperbola having the same focus and
    directrix as the original parabola.

                    The locus of the point of intersection
                    of two nodal tangent circles to a
                    cardioid which cut each other at a
                    constant angle is a limaçon having the
                    same double point and director circle.

    The sum of the reciprocals of two focal
    chords of a conic at right angles to
    each other is constant.

                    The sum of any two nodal chords of a
                    limaçon at right angles to each other
                    is constant.

    P Q is a chord of a conic which
    subtends a right angle at the focus.
    The locus of the pole of P Q and the
    locus enveloped by P Q are each conics
    whose latera recta are to that of the
    original conic as √2 : 1 and 1 : √2
    respectively.

                    If P and Q be two points on a limaçon
                    such that they intercept a right
                    angle at the node, then the locus
                    of the point of intersection of the
                    two nodal circles tangent at P and Q
                    respectively, is a limaçon whose latus
                    rectum is to that of the original
                    limaçon as ½√2 : 1. And the envelope
                    of the circle described on P Q as a
                    diameter is a limaçon, whose latus
                    rectum is to that of the original
                    limaçon as 1 : ½√2.

    If two conics have a common focus,
    two of their common chords will pass
    through the point of intersection of
    their directrices.

                    If two limaçons have a common node,
                    two nodal circles passing each through
                    two points of intersection of the
                    limaçons, will pass through the point
                    of intersection of their base circles.

    Two conics have a common focus about
    which one of them is turned; two of
    their common chords will touch conics
    having the fixed focus for focus.

                    Two limaçons have a common node about
                    which one of them is turned; two of
                    the nodal circles through two of their
                    points of intersection will envelope
                    limaçons having fixed node for node.

    Two conics are described having the
    same focus, and the distance of
    this focus from the corresponding
    directrix of each is the same; if the
    conics touch one another, then twice
    the sine of half the angle between
    the transverse axes is equal to the
    difference of the reciprocals of the
    eccentricities.

                    If two limaçons are described having
                    the same node and base circles of the
                    same diameter, and if the limaçons
                    touch each other, then twice the sine
                    of half the angle between the axes of
                    the limaçons is equal to the difference
                    of the eccentricities.

    If a circle of a given radius pass
    through the focus (S) of a given conic
    and cut the conic in the points A,
    B, C, and D; then SA. SB. SC. SD
    is constant.

                    If a circle of a given radius pass
                    through the node (S) of a given limaçon
                    and cut it in A, B, C, and D; then

                            1
                    ———————————————  is constant.
                    (SA. SB. SC. SD)

    A circle passes through the focus of
    a conic whose latus rectum is 2l and
    meets the conic in four points whose
    distance from the focus are r₁, r₂,
    r₃, r₄, then

       1       1       1       1       2
      ———  +  ———  +  ———  +  ———  =  ——— .
       r₁      r₂      r₃      r₄      l

                    A circle passes through the node of
                    a limaçon whose latus rectum is 2l,
                    meeting the curve in four points whose
                    distances from the node are r₁, r₂, r₃,
                    r₄, then

                    r₁ + r₂ + r₃ + r₄ = 2l.

    Two points P and Q are taken, one on
    each of two conics which have a common
    focus and their axes in the same
    direction, such that PS and QS are at
    right angles, S being the common focus.
    Then the tangents at P and Q meet on a
    conic the square of whose eccentricity
    is equal to the sum of the squares of
    the eccentricities of the original
    conics.

                    Two points P and Q are taken one on
                    each of two limaçons which have a
                    common node and their axes in the same
                    direction, such that PS and QS are at
                    right angles, S being the common node.
                    Then the nodal tangent circles at P and
                    Q intersect on a limaçon the square of
                    whose eccentricity is equal to the sum
                    of the squares of the eccentricities of
                    the original limaçons.

    A series of conics are described with
    a common latus rectum; the locus
    of points upon them at which the
    perpendicular from the focus on the
    tangent is equal to the semi-latus
    rectum is given by the equation

    p = -r cos 2_x_

                    If a series of limaçons are described
                    with the same latus rectum, the locus
                    of points upon them at which the
                    diameter of the nodal tangent circle
                    is equal to the semi-latus rectum, is
                    given by the equation

                    pr = -cos 2_x_

    If POP₁ be a chord of a conic through a
    fixed point O, then will tan ½P₁SO tan ½PSO
    be a constant, S being the focus of the conic.

                    If POP₁ be a nodal circle of a limaçon
                    passing through a fixed point O, then
                    will tan ½ P₁SO tan ½ PSO be a constant,
                    S being the node.

    Conics are described with equal latera
    recta and a common focus. Also the
    corresponding directrices envelop
    a fixed confocal conic. Then these
    conics all touch two fixed conics, the
    reciprocals of whose latera recta are
    the sum and difference respectively of
    those of the variable conic and their
    fixed confocal, and which have the same
    directrix as the fixed confocal.

                    Limaçons are described with equal
                    latera recta and a common node. Also
                    the director circles envelop a fixed
                    limaçon having a common node. Then
                    these limaçons all touch two fixed
                    limaçons whose latera recta are the
                    sum and difference respectively of the
                    reciprocals of the variable limaçon
                    and of the fixed limaçon, and which
                    have the same base circle as the fixed
                    limaçon.

    Every focal chord of a conic is cut
    harmonically by the curve, the focus,
    and the directrix.

                    Every nodal chord of a limaçon is
                    bisected by the base circle.

    The envelope of circles on the focal
    radii of a conic as diameters is the
    auxiliary circle.

                    The envelope of the perpendiculars at
                    the extremities of the nodal radii of
                    a limaçon is a circle having for the
                    diameter the axis of the limaçon.

Below we give a number of theorems respecting the cardioid obtained by
inverting the corresponding theorems concerning the parabola.

    The straight line which bisects the
    angle contained by two lines drawn
    from the same point in a parabola,
    the one to the focus, the other
    perpendicular to the directrix, is a
    tangent to the parabola at that point.

                    The nodal circle which bisects the
                    angle between the line drawn from any
                    point on a cardioid to the cusp and the
                    nodal circle through the point which
                    cuts the director circle orthogonally,
                    is a tangent circle at that point.

    The latus rectum of a parabola is equal
    to four times the distance from the
    focus to the vertex.

                    The latus rectum of a cardioid is equal
                    to its length on the axis.

    If a tangent to a parabola cut the axis
    produced, the points of contact and of
    intersection are equally distant from
    the focus.

                    If a nodal tangent circle cut the
                    axis of a cardioid, the points of
                    intersection and of tangency are
                    equally distant from the cusp.

    If a perpendicular be drawn from the
    focus to any tangent to a parabola, the
    point of intersection will be on the
    vertical tangent.

                    If a nodal circle be drawn tangent to a
                    cardioid, the diameter of such circle
                    passing through the cusp will be a
                    common chord of this circle and another
                    described on the axis of the cardioid
                    as diameter.

    The directrix of a parabola is the
    locus of the intersection of tangents
    that cut at right angles.

                    The base circle is the locus of the
                    intersection of nodal circles tangent
                    to a cardioid, which cut orthogonally.

    The circle described on any focal chord
    of a parabola as diameter will touch
    the directrix.

                    The circle described an any nodal chord
                    of a cardioid as diameter will be
                    tangent to the base circle.

    The locus of a point from which two
    normals to a parabola can be drawn
    making complementary angles with the
    axis, is a parabola.

                    The locus of the point through which
                    two nodal circles, cutting a cardioid
                    orthogonally, and making complementary
                    angles with the axis, can be drawn is a
                    cardioid.

    Two tangents to a parabola which
    make equal angles with the axis and
    directrix respectively, but are not at
    right angles, meet on the latus rectum.

                    Two nodal circles tangent to a cardioid
                    which make equal angles with the axis
                    and latus rectum, respectively but do
                    not cut orthogonally intersect on the
                    latus rectum.

    The circle which circumscribes the
    triangle formed by three tangents to a
    parabola passes through the focus.

                    If three nodal circles be drawn tangent
                    to a cardioid, the three points of
                    intersection of these three circles are
                    on a straight line.

    If the two normals drawn to a parabola
    from a point P make equal angles with
    a straight line, the focus of P is a
    parabola.

                    If the two nodal circles cutting a
                    cardioid orthogonally and pass through
                    the point P, make equal angles with a
                    fixed nodal circle, the locus of P is a
                    cardioid.

    Any two parabolas which have a common
    focus and their axes in opposite
    directions intersect at right angles.

                    Any two cardioids which have a common
                    cusp and their axes in opposite
                    directions intersect at right angles.

A number of other theorems on the limaçon and cardioid are given in
Professor Newson’s article in this number of the QUARTERLY,
and these need not be repeated here.




Dialect Word-List.


BY W. H. CARRUTH.

The following are some of the dialect words that have come to one
observer’s ears within the past triennium. They are all from Kansas,
unless otherwise noted. They are printed here to interest others, and
to secure a basis for observation. The writer will be under obligations
to any one who will note his familiarity with any of these words,
insert others, or other meanings, and send them, with a statement of
his place of birth and childhood, to him at Lawrence:

=among:= all of, as, Where are you going among you?

=all:= all gone, as, The corn is all. (Indiana, Penn.) Comp. German.

=bat:= a ‘hard case.’

=bid:= in, to bid the time of day. (Indiana.)

=beeslings:= preparation of artificially curdled milk. (Indiana and
Kansas.)

=become:= to look well in, as, He becomes that coat.

=bad:= desperate, as in, A bad citizen = a desperate fellow.

=behave:= to behave well, as in, Do behave now!

=bump on a log:= something lifeless, as, He sat there like a bump on a
log.

=bier:= sham, as in pillow-bier. (Vermont.)

=branch:= a small stream.

=breeze:= a torrent of talk, as in, He gave me a breeze.

=boo:= dried mucous.

=buckle down:= to work persistently.

=conniptions:= a fit, also ‘conniption fit.’

=caba:= an old valise. (Penn.)

=craps:= a game with dice; playing, it is called, ‘shooting craps.’

=crawl:= to try to escape from an embarrassing situation without
admitting one’s mistake.

=crawfish:= same as ‘crawl.’

=crock:= an earthenware vessel, a large bowl.

=chuck:= lunch.

=chuck-a-luck:= loaded (of dice).

=coddy:= odd, out of fashion.

=chug:= to strike a blow, as in, Chug him one.

=could:= to be able, as in, He used to could.

=cod:= a bit of deceit, as in, He gave the teacher a cod.

=Chenuk:= a Canadian. (Note the pronunciation.)

=dast:= to dare, as in, He don’t dast to do it.

=dew-claws:= hands and knees (?), as in, Get down on your d., = apply
yourself intensely.

=Dick’s hatband:= in the phrase, As contrary as Dick’s hatband.
What is the origin of this?

=dick-nailer:= anything quite satisfactory, as in, He (it) is a
dick-nailer.

=drop:= advantage, as in, to get the drop on a person (allusion to
dexterity in drawing a revolver). Comp. also: bulge, inside-track, whip
handle, dead-wood, all used in the same way and with same sense.

=diven=, past participle of _dive_.

=drug:= pret. of drag.

=east:= yeast.

=emptings:= bread dough set to ferment. Note the expression “It
will come out all right in the emptings,” i. e. after it has had a
chance to stand.

=fat up:= to increase a stake at cards.

=find:= to supply with board, as in, Pay five dollars a week and
find him; I get five dollars and found.

=fresh:= impudent, (due to greenness).

=fog:= to filch.

=fluke:= to steal. (Indiana).

=flat:= plug tobacco. (Arkansas).

=gallery:= church, as in, He’s in the gallery.

=gag:= an improbable story intended to deceive, as in, He tried to give
me a gag.

=go with:= to become of, as in, What has gone with my hat? Ohio; also in
Pall Mall Gazette.

=grub-stake:= to give board.

=girling:= a ‘girl-boy,’ in contempt.

=gaumy:= not neat. (Arkansas.)

=gob, or gaub:= a shapeless mass, as, a gob of mud, then sportively,
gaubs of wisdom.

=gray:= an awkward fellow.

=get to:= to get an opportunity to, as in, He didn’t get to do it.

=go to:= to intend to, as in, I didn’t go (for) to do it.

=gumbo:= a peculiar, putty-like dark soil. (Kansas.)

=hen:= feminine, as in, hen-party; comp. stag-party, a gathering of men
only.

=honey:= a fine fellow, generally ironically.

=hump:= to bestir, as, Hump yourself.

=hole:= bad condition financially; as, He is in the hole, i. e. he has
lost.

=huckleberry:= indifferent, in, a huckleberry Christian.

=huckleberry:= the right person, as in, You’re my huckleberry.

=hornswoggle:= to discomfit, as in, I’ll be hornswoggled if I’ll do it.

=in it:= on the successful side, as, He is not in it, i. e., He has no
prospect of success. This phrase is universal in 1891.

=infare:= the reception after a wedding.

=in:= on the credit side, as, I was in five dollars.

=jay:= a green, conceited fellow.

=jag:= a bit of anything; a spree, a brief drunk.

=jack mosquito:= a large insect of the mosquito family, three times the
size of the pestiferous kind; this one does not bite.

=jimmy:= to meddle, as, to jimmy with a thing or person, to ‘fool with.’
Comp., to ‘monkey with.’

=jump:= to leave without notice, as to jump the town, to jump bail; to
jump a board-bill is to leave it unpaid.

=joint:= an illegal saloon. What is the origin of it?

=jigger= or =chigger:= a minute red mite, which frequents weeds and
lawns, burrows beneath the human skin and causes excruciating itching.

=keep:= board and lodgings, as, He works for his keep.

=lay over:= to surpass, as, That lays over anything I know.

=larofamedlers:= a phrase used generally as equivalent to, It’s none
of your business. (Maryland, Penn., Ohio, Arkansas.) The word is a
corruption of Lay-over for medlers, a lay-over being a bear-trap
consisting of a pit covered with boughs.

=light out:= to start on the run, as, He ‘lit’ out for home.

=lagniap:= the extra in a bargain, as, Five dollars, and a hat for
lagniap. (Louisiana.)

=lush:= to drink heavily, to ‘swill.’

=mog:= to move, as, Mog along with you.

=mogle:= the same.

=main:= very, as, It’s main strange. (Worcester county, Mass.)

=mosey.:= to move along with a strut.

=move:= motion, as in, Get a move on you.

=mealer:= one who takes only meals at a boarding-house.

=mind off:= to ward off (flies, etc.).

=meet up with:= to meet. (Tennessee.)

=peter out:= to dwindle.

=pail:= to milk, as, to pail the cow. (Penn.)

=possessed:= anything, as, He acted like ‘all possessed.’

=quill:= to write. (_The Writer._)

=quill-wheel:= a ‘rattle-trap’ wagon.

=ruther:= choice, as, If I had my ruther; also, druther.

=ride and tie:= verbal phrase, describing a mode of travel in which
one vehicle is used by two sets of people, one riding ahead a given
distance and tying the team where the others who have walked will come
up to it, the first walking on ahead until overtaken and passed by the
second, and so on. (Colorado.)

=red up:= to make tidy.

=ring off:= to desist or cease talking, technical phrase from the
telephone, but passed into common usage. Comp., “saw off.”

=rucus:= quarrel, rumpus.

=saddy:= thanks, thank you. (Penn.)

=saw off:= ‘ring off;’ a short person is said to look ‘sawed off.’

=shet, shut, shed:= rid, as, to get shut of anything.

=shear off:= to pour off (water from settlings). (Ohio.)

=shapin’s:= young peas and beans—the unfilled pod. (Arkansas.)

=should have said:= said, as, He should have said yes, i. e.,
indeed he said yes.

=shin:= to climb, as, to shin up a tree.

=shut off:= to make to stop talking, as, Do shut him off.

=shebang:= anything run-down, as house, carriage, affairs.

=scrooch or scrooge:= to cringe.

=skin-away:= a small boy. (Civilized Sac Indians.)

=skin:= to run, as, Skin out, i. e., run away.

=skid:= to sneak through examinations. (Yale.)

=skid:= a sharp-pointed instrument.

=skit:= a mild lie.

=skads:= great quantities, as, Skads of money, of books, etc.; also =
money, as, He hasn’t the skads.

=singed cat:= a shrewd ‘rustler,’ of unpretentious appearance.

=skulduggery:= knavery.

=skip:= to run away, as, Now skip, i. e., Go away from here.

=skip:= to leave hastily, as, He skipped the town.

=slouch:= a gawky fellow; then anything imperfect, as in the phrase,
He’s no slouch, i. e., He is an expert; no slouch of a horse, i. e., a
first-rate horse.

=sloomiky:= not neat.

=slander:= to saunter.

=slump:= to fail to meet requirements, as, in examinations.

=slumps:= great quantities. (Clark’s Second Hand Catalogue, N. Y.)

=sleep:= to give lodgings. I have heard, We can eat and sleep him.

=smoodle:= a sycophant. (Kansas University.) Comp., ‘swipe.’

=smokewood:= dried water-soaked wood used by small boys as
substitute for cigars.

=smearcase:= a preparation of clabber, often called ‘Dutch cheese.’

=snake:= to snatch stealthily.

=snum:= to vow, as in, Well, I snum. Reported as common among girls.

=snouge:= unfair, as, a snouge game.

=snide:= inferior, unfair, as, a snide game, a snide watch, etc.

=so fashion:= thus, as, Do it so fashion.

=soap:= bribe money in elections.

=sugar:= same as soap.

=sugar=, (explet): pshaw!

=split:= anything, as, He ran like split, also, lickety split.

=spunky:= pouting, incensed.

=sprinkle=   ┐
             ├ =:= a small number; also a considerable number.
=sprinkling= ┘

=stag:= masculine, as, A stag-party.

=stag:= to go to an entertainment without a lady companion, as, to
stag it.

=Stoughton-bottle:= an unimpressionable fellow. (From Stoughton’s
Bitters, common in the 50’s.)

=streak:= rapid rate, as, He talked a streak, or more commonly, a blue
streak.

=streak:= to run, as, He streaked it for home.

=steer:= to manage (votes), as, A steering committee, the same as
‘whips’ in Parliament.

=striffin=, or =strifning:= the membrane surrounding the abdominal
viscera. (Missouri.)

=swan:= to vow, as, in exclamation, I swan!

=swat:= to slap or strike, as, Swat him in the eye.

=suz:= (excl.)=:= me, as in, Dear suz, and Law suz.

=swipe:= a sycophant. (Harvard.)

=tacky:= not fashionably dressed.

=tewed:= harrassed, as, I’m tewed and fretted.

=that:= so, as in, Not that far.

=throw over:= to ‘cut’ (an acquaintance).

=throw over:= to stop, as, I threw her over, i. e., stopped talking.
Common among railroad men; derived from the use of the reverse lever.

=tear-down:= to thrash, as, He gave the boy a good tearing down.

=toad on a tussock:= anything dull or lifeless. He sat there like a toad
on a tussock.

=tousle:= to disarrange (hair).

=tousey:= frowsy.

=topside:= on top of, as in, The best man topside o’ God’s green earth.

=trade-lash:= an exchange of compliments. (Wellesley.)

=trottin’-riggin’s:= best suit of clothes.

=two sticks:= anything, as, He’s as cross as two sticks.

=up above:= up, as in, Up above stairs.

=whootle-dasher:= a ‘rustler.’

=want:= for was not, were not, etc.

=wamus=, =wampus=, =warmus:= a close, generally knit jacket. (Illinois,
Pennsylvania, Ohio, Wisconsin, New England.)




PROSPECTUS.


The KANSAS UNIVERSITY QUARTERLY is maintained by the University of
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research by members of the University. Papers will be published only
upon recommendation by the Committee of Publication. Contributed
articles should be in the hands of the Committee at least one month
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_separata_ will be furnished free to contributors.

The QUARTERLY will be issued regularly, as indicated by its
title. Each number will contain fifty or more pages of reading matter,
with necessary illustrations. The four numbers of each year will
constitute a volume. The price of subscription is two dollars a volume,
single numbers varying in price with cost of publication. Exchanges are
solicited.

Communications should be addressed to

                             V. L. KELLOGG,
                                       University of Kansas,
                                                     Lawrence.