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%                                                                         %
% Project Gutenberg's Four Lectures on Mathematics, by Jacques Hadamard   %
%                                                                         %
% This eBook is for the use of anyone anywhere at no cost and with        %
% almost no restrictions whatsoever.  You may copy it, give it away or    %
% re-use it under the terms of the Project Gutenberg License included     %
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%                                                                         %
% Title: Four Lectures on Mathematics                                     %
%        Delivered at Columbia University in 1911                         %
%                                                                         %
% Author: Jacques Hadamard                                                %
%                                                                         %
% Release Date: August 24, 2009 [EBook #29788]                            %
% Most recently updated: June 11, 2021                         %
%                                                                         %
% Language: English                                                       %
%                                                                         %
% Character set encoding: UTF-8                                           %
%                                                                         %
% *** START OF THIS PROJECT GUTENBERG EBOOK FOUR LECTURES ON MATHEMATICS ***
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        ESTABLISHED DECEMBER~17{\footnotesize TH}, 1904}

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      \textbf{\huge FOUR LECTURES \\[18pt]
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Project Gutenberg's Four Lectures on Mathematics, by Jacques Hadamard

This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever.  You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.org


Title: Four Lectures on Mathematics
       Delivered at Columbia University in 1911

Author: Jacques Hadamard

Release Date: August 24, 2009 [EBook #29788]
Most recently updated: June 11, 2021

Language: English

Character set encoding: UTF-8     

*** START OF THIS PROJECT GUTENBERG EBOOK FOUR LECTURES ON MATHEMATICS ***
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Produced by Andrew D. Hwang, Brenda Lewis and the Online
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%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%

\frontmatter

\pagenumbering{roman}
\pagestyle{empty}

\normalsize

%% -----File: 001.png---Folio xx-------

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\TitlePage[1915]
\end{center}
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%[Blank Page]
%% -----File: 003.png---Folio xx-------
\iffalse
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\TitlePage
\end{center}
\fi

%% -----File: 004.png---Folio xx-------
\begin{center}
\textsc{Copyright 1915 by Columbia University Press}
\vfill

{\sffamily\tiny PRESS OF \\
THE NEW ERA PRINTING COMPANY \\[-6pt]
LANCASTER, PA.}\\[8pt]
{\footnotesize 1915}

\end{center}
\clearpage

%% -----File: 005.png---Folio xx-------

\footnotesize%
On the seventeenth day of December, nineteen hundred and four, Edward Dean
Adams, of New York, established in Columbia University ``The Ernest Kempton
Adams Fund for Physical Research'' as a memorial to his son, Ernest Kempton
Adams, who received the degrees of Electrical Engineering in 1897 and Master of
Arts in 1898, and who devoted his life to scientific research. The income of this
fund is, by the terms of the deed of gift, to be devoted to the maintenance of a
research fellowship and to the publication and distribution of the results of scientific
research on the part of the fellow. A generous interpretation of the terms of the
deed on the part of Mr.~Adams and of the Trustees of the University has made it
possible to issue these lectures as a publication of the Ernest Kempton Adams Fund.

\Rules

\normalsize
\begin{center}
\textbf{Publications of the \\
Ernest Kempton Adams Fund for Physical Research}

\rule{1.5in}{0.5pt}
\end{center}

\Advert{Number One.}{Fields of Force.}{Vilhelm Friman Koren Bjerknes}{Professor of Physics
in the University of Stockholm. A course of lectures delivered at Columbia University,
1905-6.}
{Hydrodynamic fields. Electromagnetic fields. Analogies between the two. Supplementary lecture on
application of hydrodynamics to meteorology. 160~pp.}

\Advert{Number Two.}{The Theory of Electrons and its Application to the Phenomena of Light and
Radiant Heat.}{H.~A. Lorentz}{Professor of Physics in the University of Leyden.
A course of lectures delivered at Columbia University, 1906--7. With added notes.
332~pp. Edition exhausted. Published in another edition by Teubner.}{}

\Advert{Number Three.}{Eight Lectures on Theoretical Physics.}{Max Planck}{Professor of
Theoretical Physics in the University of Berlin. A course of lectures delivered at
Columbia University in 1909, translated by \textsc{A.~P. Wills}, Professor of Mathematical
Physics in Colum\-bia University.}
{Introduction: Reversibility and Irreversibility. Thermodynamic equilibrium in dilute solutions.
Atomistic theory of matter. Equation of state of a monatomic gas. Radiation, electrodynamic theory.
Statistical theory. Principle of least work. Principle of relativity. 130~pp.}

\Advert{Number Four.}{Graphical Methods.}{C.~Runge}{Professor of Applied Mathematics in the
University of Göttingen. A course of lectures delivered at Columbia University,
1909--10.}
{Graphical calculation. The graphical representation of functions of one or more independent variables.
The graphical methods of the differential and integral calculus. 148~pp.}

\Advert{Number Five.}{Four Lectures on Mathematics.}{J.~Hadamard}{Member of the Institute,
Professor in the \Typo{Collége}{Collège} de France and in the École Polytechnique. A course of lectures
delivered at Columbia University in 1911.}
{Linear partial differential equations and boundary conditions. Contemporary researches in differential
and integral equations. Analysis situs. Elementary solutions of partial differential equations
and Green's functions. 53~pp.}

\Advert{Number Six.}{Researches in Physical Optics, Part~I, with especial reference to the radiation
of electrons.}{R.~W. Wood}{Adams Research Fellow, 1913, Professor of Experimental
Physics in the Johns Hopkins University. 134~pp. With 10~plates. Edition exhausted.}{}

\Advert{Number Seven.}{Neuere Probleme der theoretischen Physik.}{W.~Wien}{Professor of
Physics in the University of Würzburg. A course of six lectures delivered at Columbia
University in 1913.}
{Introduction: Derivation of the radiation equation. Specific heat theory of Debye. Newer radiation
theory of Planck. Theory of electric conduction in metals, electron theory for metals. The Einstein
fluctuations. Theory of Röntgen rays. Method of determining wave length. Photo-electric effect and
emission of light by canal ray particles. 76~pp.}

\par\footnotesize%
These publications are distributed under the Adams Fund to many libraries
and to a limited number of individuals, but may also be bought at cost from the
Columbia University Press.
\normalsize

%% -----File: 006.png---Folio xx-------
%[Blank Page]
%% -----File: 007.png---Folio xx-------

\Chapter{PREFACE}{}
\label{preface}
\fancyhead{}

The ``Saturday Morning Lectures'' delivered by Professor
Hadamard at Columbia University in the fall of
1911, on subjects that extend into both mathematics and
physics, were taken down by Dr.~A.~N. Goldsmith of the
College of the City of New York, and after revision by the
author in 1914 are now published for the benefit of a wider
audience. The author has requested that his thanks be expressed
in this place to Dr.~Goldsmith for writing out and
revising the lectures, and to Professor Kasner of Columbia
for reading the proofs.
\cleardoublepage

%% -----File: 008.png---Folio xx-------
%[Blank Page]
%% -----File: 009.png---Folio xx-------

\Chapter{CONTENTS}{}
\label{contents}

\setlength{\TmpLen}{1in}

\TocPrep
\TocBox{1}{Lecture I.} The Definition of Solutions of Linear Partial
Differential Equations by Boundary Conditions.
\medskip

\TocPrep
\TocBox{2}{Lecture II.} Contemporary Researches in Differential
Equations, Integral Equations, and Integro-Diff\-er\-en\-tial
Equations.
\medskip

\TocPrep
\TocBox{3}{Lecture III.} Analysis Situs in Connection with Corres\-pond\-ences
and Differential Equations.
\medskip

\TocPrep
\TocBox{4}{Lecture IV.} Elementary Solutions of Partial Differential
Equations and Green's Functions.


%% -----File: 010.png---Folio xx-------
%[Blank Page]
%% -----File: 011.png---Folio 1-------

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\Chapter{LECTURE I}{The Determination of Solutions of Linear Partial Differential
Equations by Boundary Conditions}
\label{chapter:1}
\fancyhead[CE]{\Heading{FIRST LECTURE}}
\fancyhead[CO]{\Heading{LINEAR PARTIAL DIFFERENTIAL EQUATIONS}}

% [** PP: Presumed \Section{1.}]
In this lecture we shall limit ourselves to the consideration of
linear partial differential equations of the second order.

It is natural that general solutions of these equations were
first sought, but such solutions have proven to be capable of
successful employment only in the case of ordinary differential
equations. In the case of partial differential equations employed
in connection with physical problems, their use must be given
up in most circumstances, for two reasons: first, it is in general
impossible to get the general solution or general integral;
and second, it is in general of no use even when it is obtained.

Our problem is to get a function which satisfies not only the
differential equation but also other conditions as well; and for
this the knowledge of the general integral may be and is very
often quite insufficient. For instance, in spite of the fact that
we have the general solution of Laplace's equation, this does
not enable us to solve, without further and rather complicated
calculations, ordinary problems depending on that equation
such as that of electric distribution.

Each partial differential equation gives rise, therefore, not to
one general problem, consisting in the investigation of all solutions
altogether, but to a number of definite problems, each of
them consisting in the research of one peculiar solution, defined,
not by the differential equation alone, but by the system of that
equation and some accessory data.

The question before us now is how these data may be chosen
in order that the problem shall be ``correctly set.'' But what
do we mean by ``correctly set''? Here we have to proceed by
analogy.
%% -----File: 012.png---Folio 2-------

In ordinary algebra, this term would be applied to problems
in which the number of the conditions is equal to that of the
unknowns. To those our present problems must be analogous.
\emph{In general}, correctly set problems in ordinary algebra are characterized
by the fact of having solutions, and in a finite number.
(We can even characterize them as having a unique solution
if the problem is linear, which case corresponds to that of our
present study.) Nevertheless, a difficulty arises on account of
exceptional cases.

Let us consider a system of linear algebraic equations:
\[
\begin{array}{c}
  a_{1}x_{1} + \cdots \cdots + a_{n}x_{n} = b_{1}\\
  \hdotsfor{1}\\
  \hdotsfor{1}
\end{array}
\Tag{(1)}
\]
the number~$n$ of these equations being precisely equal to the
number of unknowns. If the determinant formed by the coefficients
of these equations is not zero, the problem has only
one solution. If the determinant is zero, the problem is in
general impossible. At a first glance, this makes our aforesaid
criterion ineffective, for there seems to be no difference between
that case and that in which the number of equations is greater than
that of the unknowns, where impossibility also generally exists.
(Geometrically speaking, two straight lines in a plane do not
meet if they are parallel, and in that they resemble two straight
lines given arbitrarily in three-dimensional space.) The difference
between the two cases appears if we choose the~$b$'s (second
members of the equation~\Eqno{(1)}) properly; that is, in such manner
that the system becomes again possible. If the number of
equations were greater than~$n$, the solution would (in general)
again be unique; but, if those two numbers are equal, the problem
when ceasing to be impossible, proves to be \emph{indeterminate}.

Things occur in the same way for every problem in algebra.
For instance, the three equations
\begin{align*}
f(x, y, z) &= a\\
g(x, y, z) &= b\\
     f + g &= c
\end{align*}
%% -----File: 013.png---Folio 3-------
between the three unknowns $x$, $y$, $z$, constitute an impossible
system if $c$ is not equal to $a + b$, but if $c$ equals $a + b$, that
system is in general indeterminate.

Moreover, this fact has been both extended and made precise
by a most beautiful theorem due to Schoenflies.

Let
\[
f(x, y, z) = X, \quad
g(x, y, z) = Y, \quad
h(x, y, z) = Z
\Tag{(2)}
\]
be the equations of a space-transformation, the functions $f$, $g$, $h$
being continuous. Let us suppose that within a given sphere
($x^2 + y^2 + z^2 = 1$, for instance), two points $(x, y, z)$ cannot give
the same single point $(X, Y, Z)$: in other words, that $f(x, y, z)
= f(x', y', z')$, $g(x, y, z) = g(x', y', z')$, $h(x, y, z) = h(x', y', z')$
cannot be verified simultaneously within that sphere unless
$x = x'$, $y = y'$, $z = z'$. Let $S$ denote the surface corresponding
to the surface $s$ of the sphere; that is, the surface described by
the point $(X, Y, Z)$ when $(x, y, z)$ describes $s$. If in equation~\Eqno{(2)}
we consider now $X$, $Y$, $Z$ as given and $x$, $y$, $z$ as unknown, our
hypothesis obviously means that those equations cannot admit
of more than one solution within $s$. Now \textit{Schoenflies' theorem}
says that \textit{those equations will admit of a solution} for any $(X, Y, Z)$
that may be chosen within $S$. Of course the theorem holds
for spaces of any number of dimensions. It is obvious that this
theorem illustrates most clearly the aforesaid relation between
the fact of the solution being \emph{unique} and the fact that that
solution necessarily exists.\footnote
  {We must note nevertheless, that in it the unique solution is opposed not
only to solutions in infinite number (as above), but also to any more than
one. For instance, the fact that $x^{2} = X$ may have no solution in $x$, is, from
the point of view of Schoenflies' theorem, in relation with the fact that for
other values of $X$, it may have two solutions.}

As said above, the theorem is in the first place remarkable for
its great generality, as it implies concerning the functions $f$, $g$, $h$
no other hypothesis but that of continuity. But its significance
is in reality much more extensive and covers also the functional
field. I consider that its generalizations to that field cannot
%% -----File: 014.png---Folio 4-------
fail to appear in great number as a consequence of future discoveries.
\label{page:4}%
This remarkable importance will be my excuse for
digressing, although the theorem in question is only indirectly
related to our main subject. The general fact which it emphasizes
and which we stated in the beginning, finds several applications
in the questions reviewed in this lecture. It may be taken as a
criterion whether a given linear problem is to be considered as
analogous to the algebraic problems in which the number of
equations is equal to the number of unknown. This will be the
case always when the problem is possible and determinate and
sometimes even when it is impossible, if it cannot cease (by
further particularization of the data) to be impossible otherwise
than by becoming indeterminate.

% [** PP: Presumed \Section{2.}]
Let us return to partial differential equations. Cauchy
was the first to determine one solution of a differential equation
from initial conditions. For an ordinary equation such as
$f(x, y, dy/dx, d^{2}y/dx^{2}) = 0$, we are given the values of $y$ and
$dy/dx$ for a particular value of~$x$. Cauchy extended that result
to partial differential equations.

Let $F(u, x, y, z, \partial u/\partial x, \partial u/\partial y, \partial u/\partial z, \partial^{2}u/\partial x^{2}, \cdots) = 0$ be a given
equation of the second order and let it be granted that we can
solve it with respect to~$\partial^{2}u/\partial x^{2}$. Thus we obtain $(\partial^{2}u/\partial x^{2}) + F_{1}
= 0$ where $F_{1}$ is a function of all the above quantities, except
$\partial^{2}u/\partial x^{2}$. Then Cauchy's problem arises by giving the values
\[
u = \varphi(y, z), \quad \pderiv{u}{x} = \psi(y, z)
\Tag{(3)}
\]
of $u$ and $\partial u/\partial x$ for $x = 0$. (These data must be replaced by
analogous data if, instead of the plane $x = 0$, we introduce
another surface.) Indeed, under the above hypothesis concerning
the possibility of solving the equation with respect to $\partial^{2}u/\partial x^{2}$,
and on the supposition that the functions $F_{1}$,~$\phi$ and~$\psi$ are holomorphic,
Cauchy, and after him, Sophie Kowalevska, showed
that in this case there is indeed one and only one solution.
This solution can be expanded by Taylor's series in the form
$u = u_{0} + xu_{1} + x^{2}u_{2} + \cdots$ where $u_{0}$, $u_{1}$, $\cdots$ can be calculated.
%% -----File: 015.png---Folio 5-------

The above theorems are true for most equations arising in
connection with physical problems, for example
\[
\nabla^{2}u = \pderiv[2]{u}{t}.
\Tag{(E)}
\]

\emph{But in general these theorems may be false.} This we shall
realize if we consider Dirichlet's problem: to determine the
solution of Laplace's equation
\[
\nabla^{2}u = \pderiv[2]{u}{x}+\pderiv[2]{u}{y}+\pderiv[2]{u}{z} = 0
\Tag{(e)}
\]
for points within a given volume when given its values at every
point of the boundary surface~$S$ of that volume.

It is a known fact that this problem is a correctly set one: it
has one, and only one, solution. Therefore, this cannot be the
case with Cauchy's problem, in which \emph{both}~$u$ and one of its
derivatives are given at every point of~$S$. If the first of these
data is by itself (in conjunction with the differential equation)
sufficient to determine the unknown function, we have no right
to introduce any \emph{other} supplementary condition. How is it
therefore that, by the demonstration of Sophie Kowalevska, the
same problem with both data proves to be possible?

Two discrepancies appear between the sense of the question
in one case and in the other: (\textit{a})~In the theorem of Sophie
Kowalevska, $u$~has only to exist in the immediate neighborhood
of the initial surface~$S$. In Dirichlet's problem, it has to exist
and to be well determined in the whole volume limited by~$S$.
We therefore require more in the latter case than in the former,
and it might be thought that this is sufficient to resolve the
apparent contradiction met with above.

In fact, however, this is not the case and we must also take
account of the second discrepancy. (\textit{b})~The data, in the case of
the Cauchy-Kowalevska demonstration, are, as we said, supposed
to be analytic: the functions $\varphi$,~$\psi$ (second members of~\Eqno{(3)})
considered as functions of $y$,~$z$, are taken as given by convergent
Taylor's expansions in the neighborhood of every point
%% -----File: 016.png---Folio 6-------
of the plane~$x = 0$ in the region where the question is to be solved.
Nothing of the kind is supposed in the study of Dirichlet's
problem. Not even the existence of the first derivatives of~$u$,
corresponding to displacements on~$S$, is postulated, and in some
researches, certain discontinuities of these values are admitted.
Both these circumstances play their rôle in the explanation of
the difference between the two results discussed above.

That (\textit{a}) is one reason for that difference is evident, for of
course, if a function is required to be harmonic (i.\,e.\ to admit
everywhere derivatives and to verify Laplace's equation) within
a sphere, its values and those of its normal derivative, may not
together be chosen arbitrarily on the surface even if analytic.

To show that (\textit{a}) is not sufficient for the required explanation,
let us take the geometric terms of the problem in the same way
as Cauchy. We therefore suppose that, $u$ being defined by
Laplace's equation, the accessory data given to determine it
are the values of~$u$\Typo{,}{} and~$\partial u/\partial x$ on the plane $x = 0$, or, more
exactly, on a certain portion~$\Omega$ of that plane; $u$ will also not be
required, now, to exist in the whole space; its domain of existence
may be limited, for instance, to a certain distance, however small,
from our plane $x = 0$ (in the environs of~$\Omega$) provided that
distance be finite and not infinitesimal.

Now under these conditions, in general such a function~$u$
does \emph{not} exist, if the data are not analytic and are chosen arbitrarily.
One sees then a fact which never appeared as long as
ordinary differential equations were alone concerned, namely,
that the results are utterly different according as the analytic
character of the data is postulated or not.

%[**PP: No section 1. or 2.; presumed locations marked above.]
\Typo{\Section{3.}{}}{}

Of these two opposite results which is to be considered as
giving us a more correct and adequate idea of the nature of
things? I do not say as the true one, for of course each one is so
under proper specifications.

Some mathematicians still incline to prefer the old point
%% -----File: 017.png---Folio 7-------
of view of Cauchy, one of their reasons being that, as known
since Weierstrass, any function, analytic or not, can be replaced
with any given approximation by an analytic one, (more precisely
by a polynomial). Therefore the fact that a function
belongs to one or the other of those two categories seems to them
to be immaterial. I cannot agree with this point of view.
That the thing is \textit{not} immaterial, seems to me to follow directly
from what we have just stated. And it cannot fail to be put in
evidence if we think not only of the mere existence of the solution,
but of its properties and the means of calculating it. If
Cauchy's problem, for equation~\Eqno{(e)}, ceases to be possible, as a
rule, when the functions designated by $\varphi$, $\psi$ are not analytic,
then every expression for the solution must depend essentially
on that analyticity and especially upon the radii of convergence
of the developments of $\varphi$, $\psi$. In other words, let us imagine
that the functions $\varphi$, $\psi$ be replaced by other functions $\varphi_{1}$, $\psi_{1}$,
the differences $\varphi_{1} - \varphi$, $\psi_{1} - \psi$ being very small for every
system of real values of $y$, $x$ within~$\Omega$ (and perhaps also the
differences of some derivatives being small). However slight
the alteration may be it rigorously follows from the aforesaid
theorem of Weierstrass, that the radii of convergence of
the developments in power series (if existing at all) may and
will be, in general, completely changed; so the calculations leading
to the solution will necessarily be changed also.

If that solution itself should undergo but a slight change, this
would at once show us that these methods of calculation ought
to be of quite an artificial nature, masking completely the qualitative
properties of the required result.\footnote
  {The solution by development in Taylor's series is, in general, for problems
  of that kind, the only one which can be given. I know but one exception,
  which is Schwarz's method for minimal surfaces, when a curve of the surface
  and the corresponding succession of tangent planes are given. This method
  rests on the favorable and exceptional circumstance that complex variables
  can be employed for the study of real points of such a surface.}
But in fact, it is clear
that matters are not as just assumed above. The alteration
$u_{1} - u$ produced on the values of~$u$ by our slight modification
%% -----File: 018.png---Folio 8-------
of $\varphi$,~$\psi$ will be generally important and often complete, as is
evident\footnote
  {If $u_1 - u$ should be uniformly very small at the same time as $\varphi_1 - \varphi$,
  $\psi_1 - \psi$, it follows from the well-known convergence theorem of Cauchy that,
  letting the analytic functions $\varphi_1$,~$\psi_1$, converge towards certain (non-analytic)
  limiting functions $\varphi$,~$\psi$, the corresponding solution~$u_1$ ought to converge
  uniformly towards a certain limit~$u$, which would be \Typo{}{a} solution of the problem
  with the data $\varphi$,~$\psi$.}
by the fact that $u$ will cease completely to exist when
$\varphi$,~$\psi$ become non-analytical. This proves, first of all, that the
application of Weierstrass' theorem in that case is illegitimate,
since it gives an approximation for the data but nothing of the
kind for the unknown.

Then we see also that such a problem and calculation, the
results of which are utterly changed by an infinitesimal error in
starting, can have no meaning in their applications.

This leads to my second and chief reason for considering
only the results which correspond to non-analytic data, namely,
the remarkable accordance between them and the results to
which physical applications bring us.

This accordance is the more interesting from the fact of its
results being unexpected. Our former point of view---i.\,e.\ that
of the Cauchy-Kowalevska theorem---evidently constitutes a
complete analogy to the case of ordinary differential equations.
But from our latter point of view---which is also the point of
view in problems set by physical applications---every analogy
seems to be upset. The results often seem almost incoherent\Typo{,}{;}
they will give opposite conclusions in apparently similar
questions.

A first instance of this was given above. We know that
Cauchy's problem is now impossible for Laplace's equation
\[
\Typo{\Delta}{\nabla}^2u
 = \pderiv[2]{u}{x}+\pderiv[2]{u}{y}+\pderiv[2]{u}{z} = 0;
\Tag{(e)}
\]
but, on the contrary, in the equation of spherical waves
\[
\pderiv[2]{u}{x}+\pderiv[2]{u}{y}+\pderiv[2]{u}{z}-\pderiv[2]{u}{t} = 0,
\Tag{(E)}
\]
%% -----File: 019.png---Folio 9-------
or of the cylindrical waves
\[
\pderiv[2]{u}{x} + \pderiv[2]{u}{y} - \pderiv[2]{u}{t} = 0,
\Tag{(E')}
\]
we may assign arbitrarily the values (whether analytical or not)
of~$u$ and~$\delta u/\delta t$ for $t = 0$, and Cauchy's problem set in that way
has a solution (which is unique). In this latter case it is like
a problem in algebra in which the number of equations is equal
to the number of unknowns; in the former, like a problem in
which the number of equations is superior\footnote
  {We could be tempted to apply in that case the remark made in the beginning
  (\Pageref{4}) concerning such impossible problems, which, notwithstanding
  that circumstance, must be considered as resembling ``correctly set'' ones.
  This, however, is not really applicable; for we have seen that the category
  alluded to is recognized by the fact that the problem may, under more special
  circumstances, become indeterminate. Now, this can never be the case in
  the present question: it follows from a theorem of Holmgren (``Archiv für
  Mathematik'') that the solution of Cauchy's problem, if existent, is in every
  possible case unique.}
to the number of
unknowns.

It never could have been imagined \textit{a~priori} that such a difference
could depend on the mere changing of sign of a coefficient in
the equation. But it is entirely conformable to the physical
meaning of the equations. Equation~\Eqno{(E')}, for instance\Typo{}{,} governs
the small motions of a homogeneous and isotropic medium, like a
homogeneous gas; and the corresponding Cauchy's problem,
enunciated above, represents the definition of the motion by
giving the state of positions and speeds at the origin of times.
On the contrary, equation~\Eqno{(e)}, which also governs many physical
phenomena, never leads to problems of that kind but exclusively
to problems of the Dirichlet type. The analytical criterion by
which those two kinds of partial differential equations are to be
distinguished, is known: it is given by what are called the
\textit{characteristics of an equation}. The characteristics of an equation
correspond analytically with what the physicist calls the \textit{waves}
compatible with this equation, and are calculated in the following
way. Let a wave be represented by the equation $P(x, y, z, t) = 0$.
%% -----File: 020.png---Folio 10-------
In the given equation, for instance, if $\Typo{\Delta}{\nabla}^2u - 1/a^2 · \partial^2u/\partial t^2 = 0$
and $\Typo{\Delta}{\nabla}^2u$ be replaced by $(\partial P/\partial x)^2 + (\partial P/\partial y)^2 + (\partial P/\partial z)^2$ and
$- (1/a^2)(\partial^2u/\partial t^2)$ by $- (1/a^2)(\partial P/\partial t)^2$ the condition thus obtained
is
\[
\left(\pderiv{P}{x}\right)^2
  + \left(\pderiv{P}{y}\right)^2
  + \left(\pderiv{P}{z}\right)^2
  - \frac{1}{a^2}\left(\pderiv{P}{t}\right)^2 = 0
\]
%{\stretchyspace
(which is a partial differential equation of the first order).
It must be verified by the function~$P$. When this holds,
$P(x, y, z, t) = 0$ is said to be a characteristic of the given equation.

For equation~\Eqno{(E)}, such characteristics exist (that is, are real);
this case is called the \textit{hyperbolic one}.

Laplace's equation, $\Typo{\Delta}{\nabla}^2u = 0$, on making the above substitution,
leads to the equation
\[
\left(\pderiv{P}{x}\right)^2
  + \left(\pderiv{P}{y}\right)^2
  + \left(\pderiv{P}{z}\right)^2 = 0
\]
which has no real solution. Therefore, in this case there are no
waves and we have the so-called elliptic case.\footnote
  {An intermediate case exists $\Typo{\Delta}{\nabla}^2u - k(\partial u/\partial t) = 0$. This is semi-definite
  and is termed the parabolic one (example: the equation of heat).}
Cauchy's problem
can be set for a hyperbolic equation, but not for an elliptic one.
Does this mean that for a hyperbolic equation Cauchy's problem
will always arise? No, the matter is not quite so simple. For
instance, in equation~\Eqno{(E)} or~\Eqno{(E')}, we could not choose arbitrarily
$u$~and~$\partial u/\partial y$ for $x = 0$; this would lead us again to an
impossible problem (in the non-analytic case, of course).

The physical explanation of this lies in the fact that there are,
besides the partial differential equation, two kinds of conditions
determining the course of a phenomenon, viz., the initial and the
boundary conditions. The former are of the type of Cauchy
and they alone intervene in Cauchy's problem quoted above
for the equation of sound.

But the boundary conditions are always of the type of Dirichlet.
They are the only ones which can occur in an elliptic
equation, but even in a hyperbolic one they generally present
%% -----File: 021.png---Folio 11-------
themselves together with initial ones. This gives place to so-called
\emph{mixed problems} where the two kinds of data (belonging
respectively to the Cauchy and to the Dirichlet type) intervene
simultaneously for the determination of the unknown.

In equation~\Eqno{(E)}, $t = 0$ represents the origin of time and can
give place to initial conditions, having the form of Cauchy.
But no such conditions can correspond to $x = 0$, which represents
a geometric boundary.

More or less complicated cases can arise for various dispositions
of the configurations, giving place to other paradoxical
and apparently contradictory results, which can however all be
explained in the same way. Moreover, there are other types
of linear partial differential equations,\footnote
  {The so-called \emph{non-normal} hyperbolic equations, such as
    \[
      \pderiv[2]{u}{x_1} + \cdots \pderiv[2]{u}{x_m}
    - \pderiv[2]{u}{y_1}   \cdots \pderiv[2]{u}{y_{\Typo{m}{n}}} = 0
    \quad \text{($m > 1$, $n > 1$)\Typo{}{.}}
    \]}
which do not govern any
physical phenomena. The determination of solutions has been
studied\footnote
  {By Hamel (Inaugural Dissertation, Göttingen) and Coulon (thesis, Paris)\Typo{}{.}}
in the analytic case but no sort of determination of
that kind for non-analytic data has been discovered hitherto.

We see that from this non-analytic point of view the accordance
between mathematical results and the suggestions of
physics holds perfectly. This accordance must not surprise us,
for, as we saw above, it corresponds to the fact that a problem
which is possible only with analytic data can have no physical
meaning. But it remains worth all our attention. No other
example better illustrates Poincaré's views\footnote
  {Lectures delivered at the first International Mathematical Congress,
  Zurich, 1897; reproduced in ``La Valeur de la Sciences.''}
on the help which
physics brings to analysis as expressed by him in such statements
as the following: ``It is physics which gives us many important
problems, which we would not have thought of without it,''
and ``It is by the aid of physics that we can foresee the solutions.''
%% -----File: 022.png---Folio 12-------


\Chapter{LECTURE II}
{Contemporary Researches in Differential Equations,
Integral Equations, and Integro-Differential
Equations}
\label{chapter:2}
\fancyhead[CE]{\Heading{SECOND LECTURE}}
\fancyhead[CO]{\Heading{CONTEMPORARY RESEARCHES IN EQUATIONS}}

\Section{1.}{Partial Differential Equations and Integral Equations}

I reminded you at the end of the last lecture what indispensable
help the physicist renders to the mathematician in furnishing
him with problems. But that help is not always free from
inconveniences, and the task of the mathematician is often a
thankless one. Two cases generally occur: it may happen that
the physical problem is easily soluble by a mere ``rule of three''
method, but if not, it is so extremely difficult that the mathematician
despairs of solving it at all; and he will strive after
that solution for two centuries and, when he obtains it, our
interest in the particular physical problem may have been lost.
Such seems to be the case with some problems concerning partial
differential equations. Just after the discovery of infinitesimal
calculus, physicists began by needing only very simple methods
of integration, the problems in general reducing to elementary
differential equations. But when higher partial differential
equations were introduced, the corresponding problems almost
immediately proved to be far above the level of those which
contemporary mathematics could treat.

Indeed, those problems (such as Dirichlet's) exercised the
sagacity of geometricians and were the object of a great deal of
important and well-known work through the whole of the
nineteenth century. The very variety of ingenious methods
applied showed that the question did not cease to preserve its
rather mysterious character. Only in the last years of the
century were we able to treat it with some clearness and understand
%% -----File: 023.png---Folio 13-------
its true nature. This clearness seemed to come too late,
for at that time, physics began its present evolution in which it
seems to disregard partial differential equations and to come
back to ordinary differential equations, but of course in problems
profoundly different from the simple cases which were
familiar to \Typo{Bernouilli}{Bernoulli} or Euler.

Happily, for it would have been a humiliating thing to work so
uselessly, this disregard was only in appearance, and the ancient
problems have not lost their importance by the fact that other
ones have been superposed on and not substituted for them.
In fact, the solution now obtained for Dirichlet's problem has
proved useful in several recent researches of physics.

Let us therefore inquire by what device this new view of
Dirichlet's problem and similar problems was obtained. Its
peculiar and most remarkable feature consists in the fact that
the partial differential equation is put aside and replaced by a
new sort of equation, namely, the integral equation. This new
method makes the matter as clear as it was formerly obscure.

In many circumstances in modern analysis, contrary to the
usual point of view, the operation of integration proves a much
simpler one than the operation of derivation. An example of
this is given by integral equations where the unknown function
is written under such signs of integration and not of differentiation.
The type of equation which is thus obtained is much
easier to treat than the partial differential equation.

The type of integral equations corresponding to the plane
Dirichlet problem is
\[
\phi(x) - \lambda \int_A^B \phi(y)K(x, y)\,dy = f(x)
\Tag{(1)}
\]
where $\phi$ is the unknown function of~$x$ in the interval $(A, B)$,
$f$~and~$K$ are known functions, and $\lambda$ is a known parameter. The
equations of the elliptic type in many-dimensional space give
similar integral equations, containing however multiple integrals
and several independent variables. Before the introduction of
%% -----File: 024.png---Folio 14-------
equations of the above type, each step in the study of elliptic
partial differential equations seemed to bring with it new difficulties;
not only did the various methods imagined for Dirichlet's
problem not cast more than a partial light on the question,
but the principles of most of them were peculiar to that special
problem: they seemed to disappear if Laplace's equation was
replaced by any other equation of the same type, or even (except
for Neumann's method, which, as we shall soon see, is directly
related to integral equations) if for the same Laplace's equation
Dirichlet's problem was replaced by any analogous one
such as presented by hydrodynamics or theory of heat. Each
of them, besides, was rather a proof of existence than a method
of calculation.

Then they seemed again quite insufficient for another series
of questions which mathematical physics had to solve, viz., the
study of harmonics. The existence of those harmonics (such as
the different kinds of resonance of a room filled with air) was
physically evident, but for the mathematician it offers an immense
difficulty. Schwarz, Picard and Poincaré gave a first
solution which was rather complicated as each harmonic requires
for its definition a new infinite process of calculation after the
preceding one has been determined. Nevertheless it has demonstrated
rigorously the chief properties of the quantities in question
(namely, certain special values of the parameter in equation~\Eqno{(1)}),
i.\,e.\ that they exist and form a discrete infinity, only a finite
number of them lying within any finite interval.

But at the same time a discovery even more important, in a
certain sense, was made by Poincaré, namely the near relation
between that question of harmonics and the method which had
been indicated by Neumann for Dirichlet's problem. This
discovery of Poincaré paved the way for Fredholm's work. The
latter treats every one of the aforesaid questions, and any
which can be assimilated to them, by one and the same method,
which consists in the reduction to an equation such as~\Eqno{(1)}.
This gives all the required results at once and for all the possible
types of such problems.
%% -----File: 025.png---Folio 15-------

In all this, the mathematician seems to play again the
unfortunate \Typo{role}{rôle} we alluded to in the beginning; for those
results are nothing but the mathematical demonstration of facts
each of which was familiar to every physicist long before the
beginning of all those researches. But of course their interest
is not in fact limited in demonstration; they can and do serve
as starting points for the discovery of new facts. They are
useful as giving the proper method of calculation. Previously,
in the calculation of the resonance of a room filled with air,
the shape of the resonator had to be quite simple, which requirement
is not a necessary one for the case where integral equations
are employed. We need only make the elementary calculation
of the function~$K$ and apply to the function so calculated the
general method of resolution of integral equations.

There are two chief methods for the solution of the equations.
It is not always easy to get numerical results.

Liouville and Neumann (in solving Dirichlet's problem)
really worked out a method of solving integral equations. A
second method is due to Fredholm. The first method leads to
series which may converge slowly but they are easy to calculate.
The method of Fredholm gives a quotient of two series (entire
functions of~$\lambda$) the terms of which have to be calculated independently,
while in the first method each is obtained from the
one immediately preceding it. While we must add that Erhard
Schmidt has shown how the first method can be made to supply
a more rapidly convergent series, Fredholm's method is of
greater value to physics because of the theoretical point of view.
It gives easily (what was impossible before its appearance) not
only the existence of harmonics, but their properties. For
instance, older methods could not have succeeded, at least not
without great difficulties and a large amount of calculation, in
obtaining the order of magnitude of the successive upper harmonics
(i.\,e.\ the corresponding great values of~$\lambda$). They would
probably have been quite unable to predict the order or magnitude,
as is done in the recent works of Hermann Weyl, so as to
%% -----File: 026.png---Folio 16-------
show its relation the volume of the room to which they
correspond. But it has even proved of great importance for
physics to know mathematically, and not only empirically, that
the harmonics corresponding to equations of the form~\Eqno{(1)} are a
discrete infinity. For in the case of the spectral frequencies we
get series which tend to accumulate towards definite positions.
Since Fredholm's theory we can assert that such series are not
compatible with the form of integral equation given at the
beginning of this lecture.

Fredholm himself investigated new forms (as also did Walther
Ritz). The introduction of the integral equation has made even
the above problem accessible. The older method would not have
been able to decide whether the distribution in question was possible
or not. The hypothesis proposed by Fredholm leads to an
integral equation such as
\[
\phi(x) - \frac{1}{k-\lambda^{2}}\int_{a}^{b} \phi(y)K(x, y)\,dy = f(x)
\Tag{(2)}
\]
Here the frequencies will accumulate in the neighborhood of
$\lambda = \sqrt{K}$.

I must immediately add that, as Ritz showed, Fredholm's type
is not sufficient to give a correct explanation of the phenomena.
But this does not change the essential fact that by the aid of the
new method we are immediately able to decide what the asymptotic
distribution of harmonics can or cannot be, so that comparison
with observation becomes possible; and this we owe
entirely to Fredholm's method.

\Section{2.}{Coming Back to Ordinary Differential Equations}

As we said in the beginning, the subject of partial differential
equations which was the main and almost the only occupation
of mathematical physics, ceases nowadays to be so. As a consequence
of the general admission of the discrete structure of
matter, physical problems tend now to lead to ordinary differential
equations. These differential equations are to be studied
%% -----File: 027.png---Folio 17-------
under the most difficult circumstances because we must follow
the form of the solutions for very long periods of time, that is,
of the independent variable~$t$. One can say that such a study
did not exist before Poincaré, and even his researches on the
subject, I mean especially his four chief memoirs in the ``Journal
de Mathematiques,'' 1887 (\textit{On the shape of Curves Defined by
Differential Equations}), lead us, like Socrates, to begin to feel
that we know nothing.

We cannot, in this place, lay stress on the extraordinary complications
and paradoxes which he discovered. We shall mention
only one of them, because it helps to correct an error frequently
committed in hydrodynamical and electrical problems, concerning
the lines of force and the lines of flow. These lines are all
defined by ordinary differential equations. The general form
is $dx/X = dy/Y = dz/Z$. In a very general category of cases
the vector~$XYZ$ has the property that
\[
\div (XYZ) = \left(\pderiv{X}{x} + \pderiv{Y}{y} + \pderiv{Z}{z}\right) = 0
\]
Now, whenever such conditions existed, physicists used to say
that the tubes of force---or tubes of flow, or tubes of vortices---were
closed (if they did not go to infinity or come to the
boundaries of the domain of existence of the vector $X$,~$Y$,~$Z$).

They were, I think, led to say so by the examples given by
some simple peculiar cases in which the differential equations
could be integrated, for one could not suspect before Poincaré's
work that such cases are exceptional, generally giving
a quite inadequate and deformed view of things. In fact, the
assertion in question is an utterly false one.\footnote
  {A demonstration is frequently given to justify it, the error of which
  consists in an incomplete enumeration of possible cases.}
If you allow me
such a crude comparison, it is not true that the tube of force
must get back home and put its key in the lock. Rather does
it put its key above and below and on either side, and never
succeeds in getting it in exactly. It will, it is true, nearly get
%% -----File: 028.png---Folio 18-------
back an infinite number of times. The only consequence which
can be correctly drawn from the equation $\div(XYZ) = 0$ is
that the area of the cross section of the tube cannot have changed.
But its shape may, and generally will, have done so. If it were,
let us say, circular in starting, it will have become elliptic when
coming back and its ellipticity will increase at each return.
Finally it will become a long flat strip and only a part of it will
come back to the neighborhood of its original position. In \Figref{1},
the successive appearances of the same tube of force are shown.
The tube of force may have been originally circular, but on its
first recurrence or return, it may have become elliptic in cross
section and thus it has only partly returned to its original
position. Still more is this the case in the second recurrence of
the tube of force, which may be assumed by this time to have
become very flat in cross section.

\Graphic{1}{3.5in}{028}% [Illustration: \textsc{Fig.~1} ]
As Mr. Birkhoff kindly pointed out to me, it is interesting
to remark that in most cases, the deformed and flattened tube
will even pass \textit{simultaneously} indefinitely near to any point of
the considered medium.

A rather curious fact must nevertheless be stated. Although
the principle that the tube is closed is completely false, the
%% -----File: 029.png---Folio 19-------
conclusions drawn from it by physicists are most often true.
Why is this so? Perhaps the explanation lies in the fact that
under that same hypothesis, $\div (X, Y, Z) = 0$, a line defined
by our differential equations generally returns indefinitely near
and an infinite number of times to its starting point. (This is
called ``Stabilité a la Poisson.'') Poincaré has shown that though
not every line in question necessarily does this, the fact occurs
for an infinitely greater number of cases than those in which it
does not occur.

\Section{3.}{Application to Molecular Physics}

We see by this single example how complicated and unexpected
the shapes of curves defined by differential equations may be,
and how far we are from understanding them when considered
for great values of the independent variable.

But could we be satisfied with our work if we succeeded in
doing so? This even is doubtful. I cannot help thinking of
a bequest left to the French Academy of Science for a prize to
the first person who should be able to communicate with a
planet other than Mars! The case of molecular physics reminds
me of that rather difficult requirement. The discussion of the
molar effects (i.\,e.\ the effects on quantities of matter accessible
to observation) of molecular movements is a mathematical
problem, which, logically speaking, would presuppose a rather
advanced knowledge of curves defined by differential equations,
and take this as a starting point, in order to discuss the questions
of probability connected with such curves.

That probability plays its \Typo{role}{rôle} in the movements of almost any
dynamical system, follows from the statements we just quoted.
If the initial positions and the initial speeds of the moving points
are exactly given, so will be the final positions and speeds after
any (however long) given period of time. But if this period is
long, and if we make a very small error in the initial conditions,
the small error will have a much magnified effect and even cause
a total change in the results at the end of the long period of
time, and this is precisely Poincaré's conception of hazard.
%% -----File: 030.png---Folio 20-------
It is like a roulette game at Monte Carlo where we do not know
all the conditions of launching the ball which induces the hazard.
And so we know nothing more about the conditions than the
gamblers. In other words, molecules are finally mixed just as
cards after much shuffling. It is this fundamental hazard which
plays the main part in Gibbs's method. A sort of mixing function
ought to be introduced. Let us start on one of the lines of
force. If we know exactly the point of departure~$A$ we should
know accurately the point of arrival. If $A$ is but approximately
known, that point of arrival may occupy all sorts of positions;
and indeed, in many differential problems, it may coincide
(approximately) with any point~$B$ within the domain where the
differential system is considered (though this is not exactly so
for dynamical problems on account of the energy integral or
other uniform integrals which the equations may admit).

Therefore, the starting point being approximately~$A$, there
will be a certain probability that the point of arrival will be in a
certain neighborhood of another given point~$B$; and that probability
will be a certain function of the positions of the two
points~$A$,~$B$.

Now, logically speaking, in order to solve the question set
for us by kinetic theories, we ought to take such a ``mixing
function,'' assuming it to be known, as a base for further and
perhaps complicated reasoning. In fact, the main present
theories in statistical mechanics rest on certain assumptions
concerning that function, which are very plausible. But, rigorously
speaking, we are not able to consider them as theorems.

Happily, things are greatly simplified by the fact that in such
mixings the aforesaid function, characteristic of the law of
mixing, only intervenes by some of its properties and may be
changed to a large extent without changing the final result.
This is what Poincaré showed for the ordinary shuffling of cards
in his ``Calcul des \Typo{Probabilitès}{Probabilités}'' (second edition). In one
shuffling the peculiar habits of the player certainly intervene
and so do they more or less after only a few shufflings. But
%% -----File: 031.png---Folio 21-------
after many shufflings the results become totally independent of
those habits. Poincaré also shows (though with some exceptions
which do not however seem to play a great practical rôle),
that such is likewise the case in the kind of mixing introduced by
molecular theories.

Some known facts in the history of these theories give a
striking instance of this. Such is the work of Boltzmann and
Gibbs in the treatment of the kinetic theory of gases and
statistical mechanics. They both obtained the result that if
we consider the probability of the average number of molecules
in $6$-dimensional space and call it~$P$, and integrate~$\log P$
over the whole mass, the conclusion drawn will be that the
integral obtained is constantly increasing. Critics, and among
them my colleague and friend Brillouin, say: ``We have not
to congratulate ourselves on the result, because the two speak
of quite different things and yet they agree. Gibbs does not
mention the collision of molecules, while Boltzmann's analysis
is founded on the collisions of molecules. The primitive order
of the molecules is disturbed by such collisions and a mixing is
produced. Gibbs gets a similar mixing by the mere consideration
of differential equations existing over long periods of time.''
In both cases, if we consider systems which are ``molecularly
organized,'' after a certain time the molecules will be so much
less organized and more mixed up.

We are surprised to find this coincidence of the results of
Gibbs and of Boltzmann in such circumstances. We shall, however,
cease to consider it as fortuitous and perceive its true
signification by precisely what we just remarked on the shuffling
of cards, which makes us understand that such final results may
and do depend on properties which are, in general, common to
utterly various laws of mixing.

But the difficulties met with in partial or ordinary differential
equations are not the only ones which we had to consider at the
present time. The mathematicians have contrived to introduce
a new sort of equation, more difficult than the previous ones, the
integro-differential equation.
%% -----File: 032.png---Folio 22-------

\Section{4.}{Integro-\Typo{differential}{Differential} Equations}

We are now forced to consider this new form. Here the unknown
function simultaneously appears in integrals and in differentials.
We have at least two completely different cases of such equations
to consider. Their difference corresponds to the two sorts of
variables which intervene in all physical problems, the space
variables $x$,~$y$,~$z$, and the time variable~$t$. (There may be more
than three variables in the first group.)

Type~1: Differentiation with respect to $x$,~$y$,~$z$; integration
relative to~$t$. Type~2: Differentiation with respect to~$t$; integration
relative to $x$,~$y$,~$z$. And even though this type dates only
from 1907, we have already found cases of both kinds.

Volterra was led to consider the first one in connection with
``The Mechanics of Heredity.'' This is the case where the
properties of the system depend on all the previous facts of its
existence (such as magnetic hysteresis, strains of glass, and
permanent deformations in general).

Volterra considers elastic hysteresis. Let $T$ be any component
of strains; $E$~the component of deformation. (There are six $T$'s
and six~$E$'s.) Then formerly we considered $T_{hk} = \sum a_{hk}E_{hk}$. There
are $6$~equations of this type. There are $21$,~$36$, $6$~or $2$~$a$'s depending
on the theories. If we consider heredity, we must introduce
new terms. Suppose that at the time~$0$ there were no strains; then
$T_{hk} = \sum aE_{hk} + {\displaystyle\int_0^t} (\sum aE)_{t}\, d\tau$ where $\tau$ is the variable time. This
is an equation in which we have derivatives with respect to $x$,~$y$,~$z$,
and an integral with respect to the time; and the same
character subsists if, from those values of the~$T$'s, we deduce
the equations of movement. Water waves furnish us with an
instance of the opposite type. One knows that waves on the
surface of water are the most common examples of an undulatory
phenomenon and that, for this reason, they are most frequently
used to give to the beginner a first idea of what such phenomena
are.

But it is a general, though astonishing fact, that the most
%% -----File: 033.png---Folio 23-------
simple of daily phenomena are the most difficult to understand.
While the theory of aërial or even elastic waves is rather simple,
at least as long as viscosity is left aside,\footnote
  {In a viscous gas, waves cannot exist, strictly speaking. They are replaced
  by quasi-waves which were first considered by Duhem, and more profoundly
  studied in an important memoir presented by Roy to the French Academy
  of Sciences.}
and now classically
reduced to analytical principles (related to notion of characteristics
as we saw in the preceding lecture), the properties of surface
waves in liquids are much more hidden. The few results classically
known on that subject are even of a contradictory nature.
One of them is the differential equation given by Lagrange in
the case of small (and constant) depth, which has served as a
model for the dynamical theory of tides, the equation obtained
as governing the phenomenon being in both cases a partial
differential equation of the \emph{second} order. But, for the same
phenomenon on a liquid of indefinite depth, Cauchy gets a
partial equation of the \emph{fourth} order. The truth is that the
problem does not lead to a differential equation at all, but to
an integro-differential equation. For an originally plane surface
with small displacements, where $z$ is the vertical displacement
at~$(x, y)$, then
\[
\deriv[2]{z}{t} = \iint Z_{Q} \phi (P, Q)\, dS_{Q}\Typo{}{.}
\]
Thus, for any determinate point~$P$ of the surface defined by its
coördinates, $(x, y)$, the vertical acceleration depends on the
values of~$z$ in every other point~$Q(x', y')$. Here $S_{Q}$ is~$dx'\, dy'$
and $\phi$ is a known function of $(x, y, x', y')$. The above equation
is of the second form of integro-differential equations.

Volterra succeeded in the case of isotropic bodies in reducing
the problem to the solution of a partial differential equation and
an ordinary integral equation. But things are not so simple
for crystalline media.\footnote
  {Since these lectures were delivered, Professor Volterra has given a comprehensive
  view of his methods and solutions in a course of lectures at the
  University of Paris. See the issue of those lectures by J.~Peres (Paris, Gauthier
  Villars).}
%% -----File: 034.png---Folio 24-------

The two types of integro-differential equations, which we
just enumerated, are completely different in their treatment.
Volterra's type resembles the partial differential equations (of
the elliptic or sometimes parabolic genus in the examples hitherto
given). The equation must be completed by accessory conditions
which are nothing else than boundary conditions (cf.\
Lecture~I). The methods given by Volterra run exactly parallel
to those which are applied for Dirichlet's problem (such as the
formation of Green's functions).

In the second type described above, the accessory conditions
are initial ones; and are to be treated in the manner, not
of partial, but of ordinary differential equations---such methods
as Picard's successive approximations being of great use in that
case.
%% -----File: 035.png---Folio 25-------


\Chapter{LECTURE III}{Analysis Situs in Connection with Correspondences and
Differential Equations}
\label{chapter:3}
\fancyhead[CE]{\Heading{THIRD LECTURE}}
\fancyhead[CO]{\Heading{ANALYSIS SITUS}}

\Section{1.}{}

We are going to speak of the rôle of analysis situs in our
modern mathematics. This theory is also called the geometry of
situation. It is the study of connections between different parts
of geometrical configurations which are not altered by any continuous
deformation. We suppose that we can let a system
undergo any deformation whatever, however arbitrary it may be,
only that it preserves continuity. For instance, a sphere and a
cube are considered as one and the same thing from the point
of view of the geometry of situation, because one can be transformed
into the other without separating parts, or uniting parts
which formerly were separated. The circle and the rectangle
are identical from the same point of view. But the lateral
surface of a cylinder and the surface of a rectangle are not
identical, because, for the transformation of one into the other,
we must make a cut along a generatrix. Also one is limited by
two lines (the base circles) while the other is limited by one.
The total surface of a cylinder is entirely closed; it is identical
with the surface of a sphere. There is no difficulty in the
transformation.

If we consider the ``anchor ring,'' the case is different.
This is a closed surface but it has a hole which is not found
in the surface of the sphere, and the surface of the sphere cannot
be transformed continuously in it. It would have to be
transformed by several cuts, the first of them (\Figref{2}) giving a
broken ring, which for us is identical with the lateral surface of
a cylinder. This may be cut into a rectangle and then transformed
%% -----File: 036.png---Folio 26-------
into a sphere. But the transformation of an anchor
ring into a sphere cannot be done without cutting and piecing.
The principles of analysis situs, for surfaces in ordinary space,
\Figure{2}{2.5in}{036a}% [Illustration: \textsc{Fig.~2.}]
are well known and I do not intend to go over them at this moment.
We shall take them for granted. According to them,
a surface of two dimensions is defined from our present point of
view by the number of boundaries and another number, namely
%% -----File: 037.png---Folio 27-------
the \textit{genus}. The genus is zero for the sphere and one for the
anchor ring. For a pot with two ``ears'' (\Figref{3}) we have the
genus two.
\Graphic{3}{2.5in}{036b}% [Illustration: \textsc{Fig.~3.}]

Analysis situs started with trifling problems, such as that
treated by Euler of the bridges of Königsberg over the Pregel
river. There are seven bridges; the problem is to go over all
of them without passing twice over any one (\Figref{4}). The great
\Figure{4}{3.5in}{037}% [Illustration: \textsc{Fig.~4.}]
Euler did not disdain to occupy himself with this and many
other apparently childish problems. But what interests us in
this one especially is that it involves the geometry of situation,
in the sense in which we have used the term. For even if the
islands in the river had other shapes and the bridges had the
queerest forms, the reasoning would be exactly the same, provided
the numbers of islands and bridges should not change, and
each bridge should join the same islands in both cases.

We have here an example of an important theory which
develops from a childish exercise. Some would think that it was
a disadvantage to mathematics that we should occupy ourselves
with such problems. The fact is, as we see, that they may,
though exceptionally, lead to valuable results.

That this notion of analysis situs was really an important one,
appears first from the researches of Riemann. You know that
Riemann was the fellow founder with Cauchy of the modern
theory of analytic functions. These two schools applied their
%% -----File: 038.png---Folio 28-------
theories to the study of algebraic functions. Cauchy's methods,
in the hands of their author and of Puiseux, were capable of
casting light on some important parts of the problem, but did
not however completely elucidate it, and (in particular) Riemann
alone could discover the fundamental notion of the \emph{genus} of an
algebraic curve.

What were the elements of Riemann's success and superiority
over Cauchy? A remark must first be made which perhaps,
strictly speaking, would not be within our subject, but which
is nevertheless, as we shall see, most closely and necessarily
connected with it.

Let us consider the real domain. Suppose that we have to
study the algebraic function~$y$ defined by $x^{2} + y^{2} = 1$ (or any
quadratic equation defining~$y$ as a function of~$x$ corresponding to
an ellipse). This function is real only for values of~$x$ which are
\Figure{5}{4in}{038}% [Illustration: \textsc{Fig.~5.}]
comprised between $-1$~and~$+1$ (in the second case, for values
between $x_{0}$~and~$x_{1}$). Riemann considered the function in the segment
comprised between these values. He remarked that this
is an incomplete view of the equation, for~$y$ is not well defined,
%% -----File: 039.png---Folio 29-------
because it has two different values. But if we change our straight
line into two slightly different straight lines, then we may admit
that the superior segment corresponds to the $+$~value of~$y$,
and the inferior one to the $-$~value, the two segments being
supposed to join each other at their common ends. To each
point of the drawing, after that modification, one and only one
system of values of $x$ and~$y$ verifying the given equation will
correspond. Besides, in that case, we obtain a figure which
from the point of view of analysis situs, is identical with the
ellipse represented by the given equation itself.

But Riemann applied that same method in the complex
domain, and was led to the celebrated kind of representing surfaces
which bear his name.

This principle is a very general one. It must be applied, in
any case, before using the geometry of situation. We must
inquire whether the domain used is adequate to represent the
states of variation to be studied. I shall give an instance which
I think is due to Sophus Lie. It is concerned with the singular
solution of differential equations of the first order. Given the
differential equation
\[
f(x, y, y') = 0
\Tag{(1)}
\]
the question, as well known, is whether some solution exists which
is not represented in the general integral. In that case such a
solution must verify not only the original equation, but also
\[
\pderiv{f}{y'} = 0
\Tag{(2)}
\]
Darboux showed that this was not sufficient, and that, in general,
the system of equations \Eqno{(1)}~and~\Eqno{(2)} does not represent an actual
solution, but that the curve which it defines is the locus of the
cusps of the solutions of equation~\Eqno{(1)} (\Figref{5}). We now shall
see that this result, the analytical proof of which requires some
complicated calculations, appears of itself by the above geometric
considerations.

Equation~\Eqno{(1)} defines $y'$ as a function of $x$ and~$y$, but this function
%% -----File: 040.png---Folio 30-------
has several determinations or branches. This state of things
is not satisfactory from our point of view above. In order to
avoid this, let us consider the surface $f(x, y, z) = 0$ in space. For
each point of that surface, we have
\[
dy/dx = z
\Tag{(3)}
\]
\Graphic{6}{4in}{040}% [Illustration: \textsc{Fig.~6.}]
So that the problem becomes to trace on the surface, those curves
which have $dy/dx$ equal to~$z$. Geometrically speaking, such
curves must, in each point, be tangent to a certain direction, viz.\Typo{}{,}
the intersection of the tangent plane to the surface with a certain
vertical plane (represented by~\Eqno{(3)}). The system~\Eqno{(1)} and~\Eqno{(2)}
%% -----File: 041.png---Folio 31-------
represents the ``horizontal boundary'' of the surface. At each
point~$m$ on it, the tangent plane is vertical (\Figref{6}). What
happens there? We see that in~$m$, the two planes which define
the tangent to our curve are vertical (the plane corresponding to~\Eqno{(3)}
being so in any case). Therefore, this tangent itself is also
vertical. This gives immediately the desired result; for it is
well known that by projecting a space curve on a plane perpendicular
to one of its tangents, we obtain a projection curve which
has a cusp. The only exception would be when our two planes
would coincide and this indeed gives the supplementary condition
for the existence of a singular solution.

A difficult question in differential equations is thus reconducted
to an elementary result of analytical geometry; and this
is obtained by the mere fact of depicting correctly (in the sense
of Riemann) $y'$ as a function of $x$ and~$y$. Only when this adequate
representation of the domain of variation is obtained,
analysis situs is to be applied.

Before seeing it in operation, let us notice that Cauchy had an
opportunity of discovering its importance. This is a curious
historical fact in his work; for it was one of his few errors.
It was done in his youthful period, when dealing with the theorem
of Euler on polyhedrons. This theorem connects the number of
faces, summits and edges. It expresses that $F + V = E + 2$,
where $F$ is the number of faces, $V$ is the number of vertices, and
$E$ the number of edges. Cauchy's demonstration was false,
and so is even the theorem itself. This theorem holds effectively
(and this is the reason why Euler and Cauchy believed it to be
true) for a very large category of polyhedra, among which every
convex one occurs. But others had been overlooked, such as
those which have the general shape of an anchor ring, and these
do not verify the above relation. If Cauchy had perceived
that error; if he had noticed that exception to Euler's theorem,
it may be presumed with some probability that he would not
have left to Riemann the glory of founding a complete theory
of algebraic functions.
%% -----File: 042.png---Folio 32-------

Let me remind you of the difference between the method of
Cauchy (and of Puiseux) and that of Riemann. If we consider
the algebraic function defined by $F(x, y) = 0$, then $y$, in general,
in the environs of $x_{0}$ and~$y_{0}$, is a regular analytic function of~$x$
and is given by a Taylor's series within a certain circle around~$x_{0}$.
Inside this circle, the principles of Cauchy and Weierstrass
permit us to study the function. At critical points~$x_{1}$, where
$y$ is not a holomorphic function of~$x$, Puiseux studied this.
He took $X = (x - x_{1})^{1/p}$, $p$~being properly chosen. Then $y$ can
be developed in powers of~$X$ instead of in terms of~$x - x_{1}$.
Everything seems at first to be settled then. But really we still
ignore some fundamental properties. The reason of this is that
we do not get the direct idea of the total domain, but only an
indirect idea of it by a series of smaller regions.

It is true that these smaller regions are such that, taken altogether,
they cover the totality of the domain in question, and
for that reason, they finally may enable us to master it completely.
But the error was to believe that this could be without
a special study of the manner in which those partial regions
are united.

I should compare this (though the comparison is very incomplete)
to the map of a large country, which is given by a
series of partial leaves. We must take account, not only of
each separate leaf, but of the ``assembling table'' showing their
general disposition, so as to pass from the detail to the whole.
The capital and unexpected fact, the discovery of which belongs
to Riemann, is that such ``assembling tables'' are not at all
like each other; that there are several quite different kinds of
them: therefore, the synthesis of the details of the solution cannot
be well understood without noticing these differences.

\Section{2.}{}

It is now evident that the importance of these considerations
is not limited to algebraic functions. They are connected with
every synthesis of the above mentioned kind, that is to say,
%% -----File: 043.png---Folio 33-------
theoretically speaking, with every employment of integral
calculus.

They constitute a sort of revenge of geometry on analysis.
Since Descartes, we have been accustomed to replace each geometric
relation by a corresponding relation between numbers,
and this has created a sort of predominance of analysis. Many
mathematicians fancy they escape that predominance and consider
themselves as pure geometers in opposition to analysis; but most
of them do so in a sense I cannot approve: they simply restrict
themselves to treating exclusively by geometry questions which
other geometers would treat, in general quite easily, by analytical
means; they are of course, very frequently forced to choose
their questions not according to their true scientific interest,
but on account of the possibility of such a treatment without
intervention of analysis. I am even obliged to add that some
of them have dealt with problems totally lacking any interest
whatever, this total lack of interest being the sole reason
why such problems have been left aside by analysts. Of course,
I not only admit geometrical treatment, but use it every time
I find it possible, for, if applicable at all, it gives us, in general, a
much better view of the subject than an analytical one. But
very important problems may be inaccessible to it. We must
use all means at our disposal and choose, not this or that one
\Typo{a~priori}{\emph{a~priori}}, but the one best adapted to our question.

But here geometry has over analysis a more certain advantage.
I consider that analysis could not, or could only
with great difficulty, and probably after a long series of sterile
efforts, have replaced the geometrical views we have just alluded
to for resolving the corresponding part of the problem. I mean
that passage from the solution in small regions to the solution
over the whole domain.\footnote
  {Logically speaking, even the results of analysis situs can be rigorously
  stated in numerical language; but such statements have been made only
  after the results have been found, and some parts of this analytic treatment
  are of extreme difficulty (such as Jordan's theorem).}
%% -----File: 044.png---Folio 34-------

Let us, for instance, admit that that domain is a two-dimensional
one. Then according to analytical methods, we ought to
individualize any point of it by giving the values of two parameters,
$x$~and~$y$. But the representation of a geometrical
problem by means of functions of $x$ and $y$ often makes us lose
some element of the problem: functions in a domain in two
dimensions may be something else than the functions of $x$
and~$y$. The simultaneous variation of $x$ and $y$ represents a
plane. Now a plane has not the same general shape as a sphere
or anchor ring, and those differences are lost in Descartes's
method. We can have, for instance, as many examples of this
difference in rational dynamics as we please. One knows that
when a dynamical problem has two degrees of freedom the corresponding
differential equations, i.\,e.\ the equations of Lagrange,
are defined, the parameters which define the position of the
system being designated by $x$ and~$y$, if one gives the expression
$2T = E(x, y)x'^{2} + 2F(x, y)x'y' + G(x, y)y'^{2}$ for the vis viva
and the expression $U = \varphi (x, y)$ for the force function. Therefore,
if two problems of dynamics correspond to the same expression
of~$T$ and the same expression of~$U$, their studies ought
to be exactly identical and reducible to each other. That matters
may really be quite different is to be immediately seen
by the following example:

(1)~Consider the material particle acted on by no forces.
The trajectories will be straight lines. (2)~Let us have a vertical
standard. The arms $AA'$~and~$BB'$ are solidly attached and
$A$ and $B$ are fixed (\Figref{7}). The only motion of the system is
a rotation about~$AB$. $A'B'$ is a second axis about which a rigid
body homogeneous and of revolution can rotate. The system
has two degrees of freedom. We have to study the motion of the
system. There will be no force function. Only rotations are
possible (two independent ones around $AB$ and one around~$A'B'$).

Analytically, the two problems are one and the same, for in
both cases, $U = 0$ and the coefficients $E$,~$F$,~$G$ in~$2T$ are constants
(which can always, by a linear transformation in $x$,~$y$, be reduced
%% -----File: 045.png---Folio 35-------
to $E = G = 1$, $F = 0$). Nevertheless, there is evidently no
comparison between the motions in case~(1) and case~(2), so
that to a certain extent, we are deceived by analytic methods.
The assemblage of all possible positions of system~\Eqno{(2)} can be
represented not on a plane, but on the surface of an anchor ring.
\Graphic{7}{3in}{045}% [Illustration: \textsc{Fig.~7.}]

We know since the researches of Poincaré that the study of
trajectories represented by differential equations must be founded
on analysis situs. For instance, $f(x, y, y') = 0$ is geometrically
represented by a certain surface, and on this surface defines a
geometrical correspondence as follows: for each point of the
surface it defines a certain direction (with its sense) in the
tangent plane. We have then to draw at each point of the surface
a curve which is tangent to the direction thus defined.
%% -----File: 046.png---Folio 36-------
Poincaré showed that such a problem cannot be handled unless
we know what the genus of the surface is. This already appears
in a simple preliminary question which arises in that study. We
have said that we have a certain direction at each point of our
surface. Can we \emph{in general} do this without exception? In
general we cannot. In each point, in general, we shall have a
certain tangent direction defined, but there will be certain
singular points in the correspondence. The only case in which
the correspondence can be complete is when the surface is of
genus one. For instance, there \emph{must} be singular points for the
genus zero. In that case, Poincaré stated that every trajectory
is either a closed one, or finishes in a singular point, or is asymptotic
to a closed curve. For genus one, singular points may be
absent, but the shapes of curves verifying the equation may
yet be much more complicated.

Differential equations of higher order will also of course (and
did indeed in some parts of Poincaré's work) require the intervention
of analysis situs. But the difficulty will be much greater,
as in hyper-spaces this theory becomes as complicated as it was
simple in Riemann's hands when applied to ordinary surfaces.
These higher chapters of analysis situs begin, however, to be well
known, and though they could not hitherto be applied to differential
equations, their rôle is already clear, owing to the works
of Picard and Poincaré, in the natural generalization of Riemann's
original theory. I mean the difficult theory of algebraic surfaces
and algebraic functions of two or more independent variables.

In the line of partial differential equations, we must point out
a very remarkable analogous example due to Volterra and concerning
the problem of elasticity. Generally speaking, if the
external forces and also the peripheric efforts acting on a homogeneous
solid body are zero, so will be the stress at every point
of its substance. More precisely in such a body of simply connected
shape, stress could only appear under those conditions if
singular points would exist where they would cease to obey the
general laws known for their distribution. But the contrary can
%% -----File: 047.png---Folio 37-------
take place if the body has an annular form, and in fact Volterra
practically constructed such annular bodies in which stress exists
and can be experimentally perceived, without any external action
and without any singular point.

\Section{3.}{}

But examples of a much more elementary character, belonging
to the very beginning of the differential calculus, can be given.
Let us consider a point-to-point correspondence, defined by such
equations as
\[
X = f(x, y),\qquad Y = g(x, y).
\]
When does that system of equations admit one and only one
solution in $x$,~$y$ if $X$,~$Y$ are supposed to be given?

It is classical that this, above all, depends on the functional
determinant
%[F1: the vertical spacing in the matrix below could use some work]
\[
\frac{D(X,Y)}{D(x,y)} =
\begin{vmatrix}
  \pderiv{f}{x}&\pderiv{f}{y} \\[2ex]
  \pderiv{g}{x}&\pderiv{g}{y}
\end{vmatrix}.
\]
Suppose that this is not zero in a certain point $x_{0}$,~$y_{0}$. We are
taught that in the \emph{neighborhood} of $(X_{0}, Y_{0})$ the system will have
one and only one solution. The tempting conclusion is to
suppose that if everywhere this determinant is not zero, then
everywhere we will have a one-to-one correspondence. This is
not true, and indeed errors have been committed on that subject.
Even in the simplest case, in which the representation of the
\emph{whole} plane of~$XY$ on the \emph{whole} plane of~$xy$ is considered, a supplementary
condition at infinity must be added in order to
ascertain that the transformation is one-to-one.

But now let us replace our planes by two spheres, a correspondence
being considered between a point $(x, y, z)$ of the surface
of the first sphere, and a point $(X, Y, Z)$ of the surface of the
second. In this case we find that if a condition analogous to
that above holds at every point of the first surface it will actually
insure a regular one-to-one correspondence.
%% -----File: 048.png---Folio 38-------

But if we replace our spheres by two anchor rings, the results
will again be completely and utterly changed. Several points
on the surface of one anchor ring may correspond to one and the
same point on the surface of a second one, although in the
neighborhood of each point everything seems to take place just
as in a one-to-one correspondence. To see this, one has only
to note that a point on the torus depends on two angles, $\Theta$,~$\varphi$.
If we call $\Theta'$, $\varphi'$ the two similar angles for the second surface,
we have only to define the correspondence by $\Theta' = p \Theta$, $\varphi' = q \varphi$,
$p$ and~$q$ being two arbitrary integers.\footnote
  {It is interesting to add that as far as ordinary (closed) surfaces are concerned,
  the genus~$1$ is the only one for which such a paradoxical circumstance
  can occur, in the sense that, if each point of a closed surface~$\Sigma$, of genus $g > 1$,
  corresponds to one (and only one) point of a second closed surface~$\Sigma'$ \emph{of the
  same genus}, and if, in the neighborhood of each point, the relation thus defined
  takes the character of a one-to-one regular correspondence, it is such on the
  whole surfaces.

  This is easily seen in noting that, more generally, if we place ourselves
  under the same conditions except that we do not suppose the two genera,
  $g$,~$g'$ to be equal, and if $h$ be the number of points of~$\Sigma$ corresponding to \Typo{}{the} same
  point on \Typo{$\Sigma$}{$\Sigma'$,} this number~$h$ (which must be the same everywhere, on account of
  the absence of singular points) is connected with $g$,~$g'$ by the equation
  $g-1 = h(g'-1)$: a fact which results from the generalized Euler's theorem.}

A curious fact is that the same thing occurs with respect to
two circles. It is evident that if two points respectively move
on the two circumferences with uniform speed, one turning
exactly $p$~times ($p$ being an integer) while the other turns once,
each position of the former will correspond to $p$~distinct positions
of the latter, although the ratio of speeds never changes signs,
nor even becomes zero or infinite.

Nothing of the kind could, as we saw, occur on the surfaces
of our two spheres (nor of two hyperspheres in $n$-dim\-en\-sional
space, if $n > 2$), so that, in that respect, the case of two dimensions
proves more complicated than that of three or more
dimensional spaces.

These peculiar distinctions are closely connected with the fundamental
distinctions of analysis situs. They are due to the fact
that there are many ways essentially distinct from each other, of
%% -----File: 049.png---Folio 39-------
passing from one point to another of a circumference (according
to the number of revolutions performed around the curve) whilst
any line joining two points of the surface of a sphere can be
changed into any other one by continuous deformation.

This question of correspondences and Euler's theorem on
polyhedra would give us the most simple and elementary instances
in which the results are profoundly modified by considerations
of analysis situs, if another one did not exist which
concerns the principles of geometry themselves. I mean the
Klein-Clifford conception of space. But since this conception
has been fully and definitively developed in Klein's Evanston
Colloquium, there is no use insisting on it. We want only to
remember that this question bears to a high degree the general
character of those which were spoken of in the present lecture.
Klein-Clifford's space and Euclid's ordinary space are not only
approximately, but fully and rigorously identical as long as
the figures dealt with do not exceed certain dimensions. Nothing
therefore can distinguish them from each other in their infinitesimal
properties. Yet they prove quite different if sufficiently
great distances are considered.

This example, as you see, exactly like the previous ones,
teaches us that some fundamental features of mathematical
solutions may remain hidden as long as we confine ourselves
to the details; so that in order to discover them we must necessarily
turn our attention towards the mode of synthesis of those
details which introduce the point of view of analysis situs.
%% -----File: 050.png---Folio 40-------


\Chapter{LECTURE IV}{Elementary Solutions of Partial Differential Equations
and Green's Functions}
\label{chapter:4}
\fancyhead[CE]{\Heading{FOURTH LECTURE}}
\fancyhead[CO]{\Heading{ELEMENTARY SOLUTIONS}}

\Section{1.} {Elementary Solutions}

The expressions we are going to speak of are a necessary base
of the treatment of every linear partial differential equation,
such as those which arise in physical problems. The simplest
of them is the quantity employed in all theories of the classical
equation of Laplace: $\nabla^{2}u = 0$; namely the elementary Newtonian
potential~$1/r$, where
\[
r = \sqrt{(x-a)^{2} + (y-b)^{2} + (z-c)^{2}}
\]
and $(a, b, c)$ is a fixed point.

The potential was really introduced first and gave rise to the
study of the equation. All known theories of this equation
rest on this foundation. The analogous equation for the plane is
\[
\pderiv[2]{u}{x} + \pderiv[2]{u}{y} = 0\Typo{}{.}
\]
Here we must consider the \emph{logarithmic potential}, $\log 1/r$, where
$r = \sqrt{(x-a)^{2} + (y-b)^{2}}$. By this we see that if we wish
to treat any other equation of the aforesaid type, we must try
to construct again a similar solution which possesses the same
properties as $1/r$ possesses in the case of the equation of Laplace.
How is such a solution to be found? To understand it, we must
examine certain properties of~$1/r$. First let us note that that
quantity~$1/r$ is a function of the coördinates of two points
$(x, y, z)$ and $(a, b, c)$ [the corresponding element $\log 1/r$ in the
plane being similarly a function of $(x, y; a, b)$]. If considered
as a function of $x$,~$y$,~$z$, alone ($a$,~$b$,~$c$, being supposed to be constant)
in the real domain, $1/r$ is singular for $r = 0$; and $r = 0$
%% -----File: 051.png---Folio 41-------
only when $x = a$, $y = b$ and $z = c$ simultaneously. But for
complex points, $1/r$ is singular when the line that joins $(x, y, z)$
and $(a, b, c)$ is part of the isotropic cone of summit~$(a, b, c)$.

This isotropic cone is not introduced by chance, and not any
surface could be such a surface of singularity. It is what we
shall call the \emph{characteristic cone} of the equation. We already
met with the notion of characteristics in our first lecture, and
saw that it is nothing else than the analytic translation of
the physical expression ``waves.'' I must nevertheless come
back to it this time in order to remind you that the word
``waves'' has two different senses. The most obvious one is the
following: Let a perturbation be produced anywhere, like sound;
it is not immediately perceived at every other point. There are
then points in space which the action has not reached in any
given time. Therefore the wave, in that sense a surface,
separates the medium into two portions (regions): the part
which is at rest, and the other which is in motion due to the
initial vibration. These two portions of space are contiguous.
It was only in 1887 that Hugoniot, a French mathematician,
who died prematurely, showed what the surface of the wave can
be; and even his work was not well known until Duhem pointed
out its importance in his work on mathematical physics.

A second way of considering the wave is more in use among
physicists. We have not in the first definition implied vibrations.
If we now suppose that we have to deal with sinusoidal vibrations
of the classical form, the motion is general and embraces
all the space occupied by the air. Tracing the locus of all
points of space in which the phase of the vibration is the
same, we determine a certain wave surface (or surfaces).

It is clear that these two senses of the word ``waves'' are
utterly different. In the first case, we have space divided into
two regions where different things take place, which is not so
in the second case. Certainly, physically speaking, we feel a
certain analogy between them. But for the analyst, there seems
to be a gap between the two points of view.
%% -----File: 052.png---Folio 42-------

The gap is filled by a theorem of Delassus. Let us consider any
linear partial differential equation of the second order, and suppose
that $u$~is a solution which would be singular along all points
of a certain surface, $\pi(x, y, z) = 0$. By making some very simple
hypotheses as to the nature of the singularity, Delassus found
that this surface must be a characteristic as defined in our first
lecture; that is, it must verify, if the given equation is $\nabla^{2}u = 0$,
the (non-linear) partial differential equation of the first order
\[
\left(\pderiv{\pi}{x}\right)^{2}
  + \left(\pderiv{\pi}{y}\right)^{2}
  + \left(\pderiv{\pi}{z}\right)^{2} = 0
\]
obtained by substituting for the partial derivatives of the second
order of the unknown function~$u$ in the given equation, the
corresponding squares or products of derivatives of the first
order of~$\pi$ (the other terms of the given equation being considered
as cancelled). This is the \emph{characteristic equation} corresponding
to our problem. It is the same as the one found by Hugoniot
in studying the problem from the first point of view. This third
definition will show us the connection between the first two. In
the first case, the wave corresponds to discontinuity, for the
speeds and accelerations change suddenly at the wave surface:
such a discontinuity is evidently a kind of singularity. In the
vibratory motion the general equation contains the factor
$\sin \mu\pi$ since $u = F \sin \mu\pi$, where $F$ is the parameter corresponding
to the frequency, and $\pi$ is a function of $x$,~$y$,~$z$. This form of~$u$
seems to show no singularity, for the sine is a holomorphic function\Typo{}{.}
It is nevertheless what one may call ``practically singular.'' If
we suppose that the absolute magnitude of~$\mu$ is large, the function
varies very rapidly from $+1$~to~$-1$, it has derivatives which
contain~$\mu$ in factor, and these derivatives are therefore very
large. It has a resemblance to discontinuous function because
of the large slope. So that, in what may be called ``approximative''
analysis, it must be considered as analogous to certain
discontinuous functions. From that point of view the three
notions of waves are closely connected.
%% -----File: 053.png---Folio 43-------

This view of Delassus is the one which will interest us now
because in the case of the elementary solution~$1/r$
the characteristic
cone is a surface of singularity. We see now in what
direction we may look for the solution of the problem. We
have to find what will be the characteristic cone or surface
corresponding to it. Then we must construct a solution having
this as a singularity. The first question is answered by the
general theory of partial differential equations of the first order.
We must have a conic point at~$(a, b, c)$. In general the characteristic
cone is replaced by a \emph{characteristic conoid} which has
curvilinear generatrices which correspond to the physical ``rays.''
Secondly, we must build a solution which will have this for a
surface of singularity. The first work of general character in this
direction was that of Picard in 1891. He considered the case
of two variables and treated more especially the equation
\[
\pderiv[2]{u}{x} + \pderiv[2]{u}{y} = cu\Typo{}{.}
\Tag{(1')}
\]
Not every equation of the general type
\[
A \pderiv[2]{u}{x}
  + B \frac{\partial^2 u}{\partial x\, \partial y}
  + C \pderiv[2]{u}{y}
  + 2D \pderiv{u}{x} + 2E \pderiv{u}{y} + Fu = 0
\]
can be reduced to that form. But in the elliptic case $(B^{2} - AC
< 0)$ it can, by a proper change of independent variables, be
reduced to the form
\[
\pderiv[2]{u}{x} + \pderiv[2]{u}{y} + a \pderiv{u}{x} + b \pderiv{u}{y} + cu = 0
\Tag{(1)}
\]
(in which the characteristic lines are the isotropic lines of the
plane). Sommerfeld and Hedrick treated this more general
form and showed for equation~\Eqno{(1)}, as Picard had done for the
equation~\Eqno{(1')}, that there exists an elementary solution, possessing
all the essential properties of~$\log 1/r$. It is
\[
P \log 1/r + Q\Typo{}{,}
\]
$P$ and $Q$ being regular functions of $x$ and~$y$. $P$~has the value~$1$,
%% -----File: 054.png---Folio 44-------
$x = a$, $y = b$. In the hyperbolic case (real characteristics),
the form to which the equation can be reduced is Laplace's form
\Pagelabel{44}%
\[
\frac{\partial^{2} u}{\partial x \,\partial y}
  +   \pderiv{u}{\Typo{u}{x}}%[** PP: N.B. Not ``a \pderiv{u}{x}'']
  + b \pderiv{u}{y} + cu = 0
\Tag{(2)}
\]
if the change of variables is real; and the corresponding elementary
solution is of the type
\[
P \log \sqrt{(x-a)(y-b)} + Q\Typo{}{,}
\]
$P$ and $Q$ having the same significations as above ($P$~is nothing
else than the function which plays the chief rôle in Riemann's
method for equation~\Eqno{(2)}). Of course, if imaginary changes were
admitted (which is possible only if the coefficients are supposed
to be analytic) elliptic equations, as well as hyperbolic ones,
could be reduced to the type~\Eqno{(2)} or as well,~\Eqno{(1)}. The only
case in which that reduction is not at all possible, is when
$B^{2} - AC = 0$, the parabolic case. This is a much more difficult
case. It has been treated only recently. There is a new type
of elementary solution which was given in 1911 by Hadamard in
the \textit{Comptes Rendus}, and for the equation of heat with more than
two variables by Georey that same year (in the same periodical).

Even if we leave the parabolic case aside, the question has a
new difficulty arising because it is not possible to simplify by
changing variables as before when there are more than two of
them, so that we must then treat the general case. The problem
was, however, first treated in the case of
\[
\nabla^{2} u + a \pderiv{u}{x} + b \pderiv{u}{y} + c \pderiv{u}{z} + 1u = 0\Typo{}{.}
\]
But not every partial differential equation of the second order in
three variables can be reduced to this form. It is important
nevertheless. Holmgren obtained a solution in form analogous
to $1/r$, namely $P/r$, where $P=1$ for $r=0$.

If we wish to treat the general case where the coefficients are
quite arbitrary, we must try first to form the surface of singularity
which is the characteristic conoid. Suppose first that we
%% -----File: 055.png---Folio 45-------
have any regular characteristic surface of our equation and
suppose that by a change of variables, $x = 0$ is the surface.
Let us write $u = x^{p} F (x, y, z)$. One can show that, giving $p$
any positive value, solutions of this form can be found, $F$ being
regular. Such is not the case when $p$ is a negative integer; and
this gives us again an interesting illustration of the considerations
explained in our first lecture in connection with Schoenflies'
theorem. Let $p$ be a negative integer and suppose that there is
a solution. Then we have also other values of~$u$ of the form
\[
\frac{F(x, y, x)}{x^{p}} + F_{1}(x, y, z)\Typo{}{.}
\]
(We can form an infinity of these solutions because the differential
equation possesses an infinity of regular solutions.) But those
values of~$u$ can be written
\[
\frac{F + x^{p} F_{1}}{x^{p}}\Typo{}{,}
\]
\Typo{So}{so}
that, if our question is possible, it has an infinity of solutions.
By the same reasoning as in the first lecture, we must not wonder
at its being in general not possible. There is again this balancing
between infinity of solutions and their existence.

But we have supposed our characteristic surface to be a
regular one. If we deal with our characteristic \emph{conoid}, which
has $(a, b, c)$ for a conic point, things behave differently; $p$~\emph{cannot}
have an arbitrary value. If the number of independent variables
is~$n$, we must have
\[
p = - \frac{n-2}{2}, \quad \text{or} \quad -\left(\frac{n-2}{2} + 1\right), \quad -\left(\frac{n-2}{2} + 2\right), \quad \dots\Typo{}{.}
\]
The first of these values is, however, the only essential one,
because, if we have formed the (unique) solution corresponding
to $p = \Typo{-(n-2)2}{-(n-2)/2}$,
which depends on $x$,~$y$,~$z$, $a$,~$b$,~$c$, we can
deduce all others from it: we need merely to differentiate with
respect to $a$,~$b$,~$c$.

If $n$ is even, those values of~$p$ become negative integers and
%% -----File: 056.png---Folio 46-------
therefore, on account of what we just said, there is, in general,
no solution of the above form
\[
u = \frac{P}{\Gamma^{p}} + Q\Typo{}{.}
\]
We have to replace this by
\[
u = \frac{P}{\Gamma^{p}} + P_{1} \log \Gamma\Typo{}{,}
\]
in which $\Gamma$ would again be equal to~$r^{2}$, $r$~meaning a distance in
$n$-dimensional space, if the higher terms (of the second order)
of the given equation are of the form~$\nabla^{2} u$. However, if these
terms are arbitrary, $\Gamma$~should be replaced by the first member
of the equation of the characteristic conoid of summit~$(a, b, c)$.

The functions $P$, $Q$, $P_{1}$ can easily be developed in convergent
Taylor's series if the coefficients of the equation are analytic.
If not, they still exist but are much more difficult to find. The
first result of Picard, concerning the special equation~\Eqno{(1')}, was
however, obtained (by successive approximations) without any
assumption on the analyticity of~$c$: Later, E.~E.~Levi solved the
problem in the same sense for the general elliptic equation.

The principle of these methods of Picard and Levi in reality
is the same. Both may be considered as peculiar cases of one
indicated by Hilbert and consisting in the introduction of the
first approximation, which presents a singularity of the required
form, but does not need to verify the given equation. The
investigation of the necessary complementary term leads
again to an integral equation. I must add that, for equations
of a higher order, the extension of this seems to offer
difficulties of an entirely new kind, owing to the fact that the
characteristic conoid generally admits other singularities than its
summit (viz.\Typo{}{,} cuspidal lines). For the very special case in which
there are no other terms than those of the highest order, the
coefficients of those terms being constant, it has however been
reduced to Abelian integrals by a beautiful analysis of Fredholm's.
%% -----File: 057.png---Folio 47-------


\Section{2.}{Green's Functions}

Elementary solutions are a necessary instrument for the
treatment of the partial differential equations of mathematical
physics. They are not always sufficient. They are sufficient
for the simplest of the problems alluded to in our first lecture,
namely Cauchy's problem. But we know that for the elliptic
case, this latter is not to be considered, and we have to
face others, such as Dirichlet's problem. For Dirichlet's problem
(i.\,e.\ to find~$u$ taking given values all over the surface of
the volume~$S$, and satisfying $\nabla^{2} u = 0$), $1/r$ is \emph{not} a sufficient
function. We must introduce a new function of the form $1/r + h$
where $h$ is a regular function; and $h$ must be such that $1/r + h$
must be zero at every point of the boundary surface. This is
called \emph{Green's Function}. It is the potential produced on the
surface~$S$ by a quantity of electricity placed at~$(a, b, c)$ interior
to the surface, this surface being hollow, conducting, and maintained
at the potential zero. This is its physical interpretation.

For any other linear partial differential equation of the elliptic
type, one has to consider such Green's functions in which the
term~$1/r$ is to be replaced by the elementary solution (so that,
at any rate, the formation of this latter is presupposed), $h$~still
being a regular function (at least as long as $(a, b, c)$ remains fixed
and interior to~$S$).

Similar sorts of Green's functions are also known for higher
differential equations, e.\,g.\ for the problem of an elastic plate
rigidly fastened at its outline, the differential equation being
then $\nabla^{2} \nabla^{2} u = 0$ (in two variables $x$ and~$y$ only) and the rôle of
elementary solution being played by $r^{2} \log r$.

Like $1/r$ and like the elementary solution itself, any Green's
function depends on the coördinates of two points, $A(x, y, z)$
and $B(a, b, c)$. But the chief interest in the study of those
Green's functions, the important difference between them and
the above mentioned fundamental solutions, corresponds to a
similar difference between Cauchy's and Dirichlet's problems,
such as defined in our first lecture. To understand this, let us
%% -----File: 058.png---Folio 48-------
remember that each of those two problems depends on three
kinds of elements:
\begin{itemize}
\item[1.]{A given differential equation;}
\item[2.]{A given surface (or hyper-surface in higher spaces)~$S$;}
\item[3.]{A certain distribution of given quantities at the different
points of~$S$.}
\end{itemize}

Each of those elements has of course its influence on the
solution but not to the same degree. The influence of the form
of the equation cannot but be a profound one. On the contrary,
the influence of the quantities mentioned in~3 is comparatively
superficial, in the sense that the calculations can be carried pretty
far before introducing them. In other terms, if we compare this
to a system of ordinary linear algebraic equations, the rôle of
the first element may be compared to that of the coefficients of
the unknowns (by the help of which such complicated expressions
as the determinant and its minor determinants must be
formed) while the rôle of the third element resembles that of the
second members which have only to be multiplied respectively
by the minor determinants before being substituted in the
numerator.

But as to the rôle of our second element, the shape of our
surface~$S$, the answers are quite different according to cases.

If we deal with Cauchy's problem, that shape plays just as
superficial a rôle as the third element. For instance, in Riemann's
method for Cauchy's problem concerning equation~\Eqno{(2)},
every element of the solution can be calculated without knowing
the shape of~$S$ (which in that case is replaced by a curve, the
problem being two-dimensional) till the moment when they have
to be substituted in a certain curvilinear integral which is to be
taken along~$S$.

But matters are completely different in that respect in the
case of Dirichlet's problem. While one can practically say that
there is only one Cauchy's problem for each equation, there is,
for the same and unique equation $\nabla^{2} u = 0$, one Dirichlet's
problem for the sphere, one for the ellipsoid, one for the parallelepipedon;
%% -----File: 059.png---Folio 49-------
and these different problems present very unequal
difficulties.

It is clear that the same differences will appear in the mode
of treatment corresponding to the two problems. The elementary
solution depends on nothing else than the given equation
and the coördinates $x$,~$y$,~$z$, $a$,~$b$,~$c$, of the two points $A$,~$B$.

The Green's function on the contrary depends, not only on
this equation and these coördinates, but also on the form of
the boundary~$S$.\footnote
  {All these observations quite similarly hold for the``mixed problems''
  alluded to in our first lecture, and for the expressions introduced in their
  treatment corresponding to Green's functions.}

The interesting question arising therefrom is to find how the
properties of Green's functions are modified by the change of
the shape of the surface. Let us replace $S$ by~$S'$, defined by its
normal distance~$\delta n$ (which may be variable from one point of~$S$
to another). Take two given points $A$~and~$B$ within~$S$. Then
there is a certain form of Green's function~$g^{B}_{A}$ for the surface~$S$,
and if we change from~$S$ to~$S'$, $g^{B}_{A}$~changes. The change is
\Pagelabel{49}%[** PP: Notation??]
\[
\delta g^{B}_{A} = \iint \deriv{g^{n}A}{n}\, \deriv{g^{n}B}{n}\, \delta n\, dS\Typo{}{;}
\Tag{(3)}
\]
$\deriv{g^{n}A}{n}$ is the rate of change of~$g_{A}$ relative to the change of~$n$.

Here $\delta n\, dS$ is an element of volume comprised between the
surfaces $S$,~$S'$. Similar formulas hold for Green's functions for a
plane area. They are like those given by the calculus of variations
of integrals, though its methods are not directly applicable.

A curious consequence is that from all the Green functions
for all the elliptic partial differential equations, we can deduce
by proper differentiations expressions verifying one and the
same integro-differential equation, namely
\[
S \phi^{B}_{A} = S \phi^{n}_{A} \phi^{B}_{n}\, \delta n\, dS
\]

The fact that in the second member of the equation~\Eqno{(3)}, the
coefficient of $\delta n\, dS$ is quadratic and symmetric with respect to
%% -----File: 060.png---Folio 50-------
expressions depending on the points $A$~and~$B$ respectively, is also
an important one. Useful inequalities, which could not easily
be obtained otherwise, can be deduced therefrom.

Besides that study of the variation of the numerical values
of Green's functions, the influence of the shape of~$S$ can be
studied from another point of view, I mean its influence on their
analytical properties, and this has been the occasion for important
recent results. The complementary term~$h$ in a Green's function
remains regular as long as one of the points remains fixed and
interior to the considered domain; but it offers a peculiar
singularity when the two points $A$,~$B$ simultaneously approach
the same point~$P$ of the boundary; and that singularity looks
at first like a very difficult one. Its study is nevertheless
simplified by the fact that it only depends on the shape of~$S$
\emph{in the immediate neighborhood} of~$P$.
\Figure{8}{3in}{060}% [Illustration: \textsc{Fig.~8.}]

In the case of the plane, for instance, if two closed contours
$S$,~$S'$, limiting two different areas have a certain arc~$MN$ in
common\footnote
  {The two contours are understood to be one and the same side of that
  arc~$MN$.}
(\Figref{8}), if $P$ is a point of this arc, and if $G$,~$G'$ be the
two Green's functions corresponding respectively to those contours,
the difference $G - G'$ will be a completely regular function
(admitting a development in a convergent Taylor's series) when
$A$ and $B$ are both very near to~$P$.
%% -----File: 061.png---Folio 51-------

We have now to inquire what the singularity of~$G$, for instance,
will be. After having received a first partial answer in interesting
papers by several Italian geometers, this question has been
completely solved by E.~E. Levi for a function analogous to the
ordinary Green's function, and more recently by P.~Levy for
this latter itself.

The answer thus obtained is remarkably simple in the case
of two dimensions. P.~Levy also works out the three-dimensional
problem, but there the results are much more complicated.

As to Green's function as a whole (and not only the singular
part of it) it must be well understood that its value for any two
given points of the area or even such elements as its normal
derivative in one point of the contour, profoundly depends on
the form of every part of this latter, however distant from the
point or points in question.

By paying attention to this fact, we must expect, on account
of what was seen in the preceding lecture, that considerations of
analysis situs will be important in that question. At first this
does not seem to be the case, and the most important methods
for the resolution of Dirichlet's problem are common to areas
of any genus (although with some modifications of detail, as
will be seen for Fredholm's method in Kellogg's Dissertation).
But other views of the problem will show that the influence of
analysis situs does exist here and is perhaps even more astonishingly
profound than in any of the questions examined in our last
lecture.

If we consider again Dirichlet's problem for an area in the
plane, we shall see that the analytical properties of the corresponding
Green's function are very different if that area has one
or several boundaries.

Let us take the first case. In this case, the plane area can
be represented conformally on a circle of unit radius with the
origin as center. It is easily seen that, in such a conformal
representation, Green's function keeps its values, and this brings
to light a remarkable consequence concerning the six Green's
%% -----File: 062.png---Folio 52-------
functions generated by four points taken two by two. The
six quantities have a relation between them and give rise to a
peculiar sort of geometry, which not only resembles the ordinary
non-Euclidean geometry, but can be reduced to it by a simple
transformation.

In an area with two boundaries (annular area) matters are
quite different. Schottky has shown that if we take two such
areas, $S$,~$S'$, having each two boundaries, they are \emph{not} in general
conformally representable on one another. Each one of them
will be represented on the area between two concentric circles.
But the ratio of the radii of these circles must, in each case, be
chosen properly, and, therefore, will not, in general, be the same
for~$\Sigma$ and for~$\Sigma'$.

In this last case, the relation between the six Green functions
will \emph{not} hold, and the properties of our Green's functions will be
far less simple. They will become still more complicated for
more than two boundaries. We again have here an important
instance of the rôle played by analysis situs in analytical properties,
and as we have stated that Green's functions are related
to all the chief topics treated in our preceding lectures, this is
perhaps the best conclusion to be given to the ensemble of them.
\clearpage
%% -----File: 063.png---Folio 53-------

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[]\OT1/cmtt/m/n/9.2 electronic works in formats readable by the widest variety 
of computers[] 
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[]\OT1/cmtt/m/n/9.2 http://pglaf.org/fundraising.  Contributions to the Project
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[]\OT1/cmtt/m/n/9.2 array of equipment including outdated equipment.  Many smal
l donations[] 
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[]\OT1/cmtt/m/n/9.2 against accepting unsolicited donations from donors in such
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[]\OT1/cmtt/m/n/9.2 Professor Michael S. Hart is the originator of the Project 
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[]\OT1/cmtt/m/n/9.2 Most people start at our Web site which has the main PG sea
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