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| author | Roger Frank <rfrank@pglaf.org> | 2025-10-15 02:48:13 -0700 |
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| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-15 02:48:13 -0700 |
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diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..6833f05 --- /dev/null +++ b/.gitattributes @@ -0,0 +1,3 @@ +* text=auto +*.txt text +*.md text diff --git a/29788-pdf.pdf b/29788-pdf.pdf Binary files differnew file mode 100644 index 0000000..3d745a3 --- /dev/null +++ b/29788-pdf.pdf diff --git a/29788-pdf.zip b/29788-pdf.zip Binary files differnew file mode 100644 index 0000000..f17fcd5 --- /dev/null +++ b/29788-pdf.zip diff --git a/29788-t.zip b/29788-t.zip Binary files differnew file mode 100644 index 0000000..d8c1f80 --- /dev/null +++ b/29788-t.zip diff --git a/29788-t/29788-t.tex b/29788-t/29788-t.tex new file mode 100644 index 0000000..baf737a --- /dev/null +++ b/29788-t/29788-t.tex @@ -0,0 +1,3862 @@ +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % +% % +% 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 *** +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{29788} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Standard DP encoding. Required. %% +%% %% +%% fix-cm: Variable-sized font scaling. Required. %% +%% %% +%% ifthen: Logical conditionals. 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This ebook may be easily recompiled with + errors and irregularities retained. Please consult the + preamble of the \LaTeX\ source file for instructions.} + {Every effort has been made to reproduce the text of the original + book faithfully, including inconsistencies of spelling, + punctuation, sectioning, and mathematical notation. This ebook may + be easily recompiled with minor corrections effected. Please consult + the preamble of the \LaTeX\ source file for instructions.} + \bigskip + + Figures may have been moved slightly with respect to the surrounding + text. + \bigskip +} + +\newcommand{\TransNoteText} +{ + \TransNoteCommon + + This PDF file is formatted for screen viewing, but may be easily + formatted for printing. 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HADAMARD\\[8pt] + + \makebox[0pt][c]{\Micro% + MEMBER OF THE INSTITUTE, PROFESSOR IN THE COLLÉGE DE FRANCE AND IN THE ÉCOLE POLYTECHNIQUE,}\\[-4pt] + \makebox[0pt][c]{\Micro% + LECTURER IN MATHEMATICS AND MATHEMATICAL PHYSICS IN COLUMBIA UNIVERSITY FOR 1911}\\[48pt] + + \includegraphics[width=1.25in]{./images/pub_device.png}\\[24pt] + + NEW YORK \\[6pt] + \ifthenelse{\equal{#1}{}} + {COLUMBIA UNIVERSITY PRESS} + {COLUMBIA UNIVERSITY PRESS \\[6pt] #1} + \end{minipage}} +} + + + +\DeclareMathOperator{\Div}{div} +\renewcommand{\div}{\Div} + +\newcommand{\Derivative}[4][] +{\ifthenelse{\equal{#1}{}} + {\dfrac{#4 #2}{#4 #3}} + {\dfrac{#4^{#1} #2}{#4 #3^{#1}}}} + +\newcommand{\pderiv}[3][]{\Derivative[#1]{#2}{#3}{\partial}} +\newcommand{\deriv}[3][]{\Derivative[#1]{#2}{#3}{d}} + + +% Mediates corrections; search for \Typo for list of corrections +\newcommand{\Typo}[2]{\ifthenelse{\boolean{FixTypos}}{#2}{#1}} + + +\begin{document}%%%%% BEGIN DOCUMENT %%%%% + +\pagestyle{empty} +\pagenumbering{Alph} +\phantomsection +\pdfbookmark[-1]{Front Matter}{Front Matter} + +%%%% PG BOILERPLATE %%%% +\Pagelabel{PGBoilerplate} +\phantomsection +\pdfbookmark[0]{PG Boilerplate}{Project Gutenberg Boilerplate} + +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +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 *** +\end{PGtext} +\end{minipage} +\end{center} + +\clearpage + + +%%%% Credits and transcriber's note %%%% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang, Brenda Lewis and the Online +Distributed Proofreading Team at http://www.pgdp.net (This +file was produced from images from the Cornell University +Library: Historical Mathematics Monographs collection.) +\end{PGtext} +\end{minipage} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\pdfbookmark[0]{\TransNote}{\TransNote} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% + +\frontmatter + +\pagenumbering{roman} +\pagestyle{empty} + +\normalsize + +%% -----File: 001.png---Folio xx------- + +\pagestyle{empty} +\enlargethispage{0.25in} +\begin{center} +\TitlePage[1915] +\end{center} +\clearpage + +%% -----File: 002.png---Folio xx------- +%[Blank Page] +%% -----File: 003.png---Folio xx------- +\iffalse +\cleardoublepage +\enlargethispage{2in} +\begin{center} +\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------- + +\mainmatter +\pdfbookmark[-1]{Main Matter}{Main Matter} +\pagenumbering{arabic} +\setlength{\headheight}{15pt} +\pagestyle{fancy} +\fancyfoot{} + +\stretchyspace + + +\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------- + +\fancyhead{} +\begin{center} +\makebox[0pt][c]{\textbf{\Huge Columbia University Press}} + +\textbf{Columbia University in the City of New York}\\[12pt] + +\textbf{\large Lemcke \& Buechner, Agents} + +\textbf{30--32 West 27th Street}\hfill \textbf{NEW YORK} + +\Rules + +\textbf{Publications of the \\ +Ernest Kempton Adams Fund for Physical Research} + +\rule{1.5in}{0.5pt} +\end{center} + +\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. +\medskip + +\Advert{Number One.}{Fields of Force.}{Vilhelm Friman Koren Bjerknes}{Professor of Physics +in the University of Stockholm. 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Con- + [] + +[57 + +] [58] +Overfull \hbox (27.59485pt too wide) in paragraph at lines 2680--2680 +[]\OT1/cmtt/m/n/9.2 End of Project Gutenberg's Four Lectures on Mathematics, by + Jacques Hadamard[] + [] + + +Overfull \hbox (5.94514pt too wide) in paragraph at lines 2682--2682 +[]\OT1/cmtt/m/n/9.2 *** END OF THIS PROJECT GUTENBERG EBOOK FOUR LECTURES ON MA +THEMATICS ***[] + [] + + +Overfull \hbox (8.27509pt too wide) in paragraph at lines 2729--2729 +[]\OT1/cmtt/m/n/9.2 Section 1. 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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] % +% % +% Language: English % +% % +% Character set encoding: ISO-8859-1 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK FOUR LECTURES ON MATHEMATICS *** +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{29788} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Standard DP encoding. Required. %% +%% %% +%% fix-cm: Variable-sized font scaling. Required. %% +%% %% +%% ifthen: Logical conditionals. Required. %% +%% %% +%% amsmath: AMS mathematics enhancements. 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HADAMARD\\[8pt] + + \makebox[0pt][c]{\Micro% + MEMBER OF THE INSTITUTE, PROFESSOR IN THE COLLÉGE DE FRANCE AND IN THE ÉCOLE POLYTECHNIQUE,}\\[-4pt] + \makebox[0pt][c]{\Micro% + LECTURER IN MATHEMATICS AND MATHEMATICAL PHYSICS IN COLUMBIA UNIVERSITY FOR 1911}\\[48pt] + + \includegraphics[width=1.25in]{./images/pub_device.png}\\[24pt] + + NEW YORK \\[6pt] + \ifthenelse{\equal{#1}{}} + {COLUMBIA UNIVERSITY PRESS} + {COLUMBIA UNIVERSITY PRESS \\[6pt] #1} + \end{minipage}} +} + + + +\DeclareMathOperator{\Div}{div} +\renewcommand{\div}{\Div} + +\newcommand{\Derivative}[4][] +{\ifthenelse{\equal{#1}{}} + {\dfrac{#4 #2}{#4 #3}} + {\dfrac{#4^{#1} #2}{#4 #3^{#1}}}} + +\newcommand{\pderiv}[3][]{\Derivative[#1]{#2}{#3}{\partial}} +\newcommand{\deriv}[3][]{\Derivative[#1]{#2}{#3}{d}} + + +% Mediates corrections; search for \Typo for list of corrections +\newcommand{\Typo}[2]{\ifthenelse{\boolean{FixTypos}}{#2}{#1}} + + +\begin{document}%%%%% BEGIN DOCUMENT %%%%% + +\pagestyle{empty} +\pagenumbering{Alph} +\phantomsection +\pdfbookmark[-1]{Front Matter}{Front Matter} + +%%%% PG BOILERPLATE %%%% +\Pagelabel{PGBoilerplate} +\phantomsection +\pdfbookmark[0]{PG Boilerplate}{Project Gutenberg Boilerplate} + +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +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] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK FOUR LECTURES ON MATHEMATICS *** +\end{PGtext} +\end{minipage} +\end{center} + +\clearpage + + +%%%% Credits and transcriber's note %%%% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang, Brenda Lewis and the Online +Distributed Proofreading Team at http://www.pgdp.net (This +file was produced from images from the Cornell University +Library: Historical Mathematics Monographs collection.) +\end{PGtext} +\end{minipage} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\pdfbookmark[0]{\TransNote}{\TransNote} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% + +\frontmatter + +\pagenumbering{roman} +\pagestyle{empty} + +\normalsize + +%% -----File: 001.png---Folio xx------- + +\pagestyle{empty} +\enlargethispage{0.25in} +\begin{center} +\TitlePage[1915] +\end{center} +\clearpage + +%% -----File: 002.png---Folio xx------- +%[Blank Page] +%% -----File: 003.png---Folio xx------- +\iffalse +\cleardoublepage +\enlargethispage{2in} +\begin{center} +\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------- + +\mainmatter +\pdfbookmark[-1]{Main Matter}{Main Matter} +\pagenumbering{arabic} +\setlength{\headheight}{15pt} +\pagestyle{fancy} +\fancyfoot{} + +\stretchyspace + + +\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------- + +\fancyhead{} +\begin{center} +\makebox[0pt][c]{\textbf{\Huge Columbia University Press}} + +\textbf{Columbia University in the City of New York}\\[12pt] + +\textbf{\large Lemcke \& Buechner, Agents} + +\textbf{30--32 West 27th Street}\hfill \textbf{NEW YORK} + +\Rules + +\textbf{Publications of the \\ +Ernest Kempton Adams Fund for Physical Research} + +\rule{1.5in}{0.5pt} +\end{center} + +\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. +\medskip + +\Advert{Number One.}{Fields of Force.}{Vilhelm Friman Koren Bjerknes}{Professor of Physics +in the University of Stockholm. 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