COLUMBIA UNIVERSITY IN THE CITY OF NEW YORK PUBLICATION NUMBER FIVE OF THE ERNEST KEMPTON ADAMS FUND FOR PHYSICAL RESEARCH ESTABLISHED DECEMBER 17th, 1904 FOUR LECTURES ON MATHEMATICS DELIVERED AT COLUMBIA UNIVERSITY IN 1911 BT J. HADAMARD HEUBER OF THE INSTITUTE, PROFESSOR IN THE COLLEGE DE FRANCE AND IN THE ficOLR POLYTECHNIQUE, LECTURER IN MATHEMATICS AND MATHEMATICAL PHTSICS IN COLUMBIA UNIVEB8ITT FOR 1911 NEW YORK COLUMBIA UNIVERSITY PRESS 1915 ftcus* Copyright 1915 by Columbia University Press 37/ PRESS OF THE NEW ERA PRINTING COMPANY LANCASTER, PA. 1915 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 scien- tific 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. Publications of the Ernest Kempton Adams Fund for Physical Research Number One. Fields of Force. ByViLHELM Friman Koren Bjerkneb, Professor of Physics in the University of Stockholm. A course of lectures delivered at Columbia Univer- sity, 1905-6. Hydrodynamic fields. Electromagnetic fields. Analogies between the two. Supplementary lecture on application of hydrodynamics to meteorology. 160 pp. Number Two. The Theory of Electrons and its Application to the Phenomena of Light and Radiant Heat. By 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. Number Three. Eight Lectures on Theoretical Physics. By Max Planck, Professor of Theoretical Physics in the University of Berlin. A course of lectures delivered at Columbia University in 1909, translated by A. P. Wills, Professor of Mathematical Physics in Columbia 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 1 east work. Principle of relativity. 130 pp. Number Four. Graphical Methods. By C. Rcnge, Professor of Applied Mathematics in the University of Gottingen. 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 ca 'cuius. 148 pp. Number Five. Four Lectures on Mathematics. By J. Hadamard, Member of the Institute, Professor in the College de France and in the Eeole Polytechnique. A course of lectures delivered at Columbia University in 1911. Linear partial differential equations and boundary conditions. Contemporary researches in differen- tial and integral equations. Analysis situs. Elementary solutions of partial differential equations and Green's functions. 53 pp. Nurhber Six. Researches in Physical Optics, Part I, with especial reference to the radiation of electrons. By R. W. Wood, Adams Research Fellow, 1913, Professor of Experimental Physics in the Johns Hopkins University. 131pp. With 10 plates. Edition exhausted. Number Seven. Neuere Probleme der theoretischen Physik. By W. Wien, Professor of Physics in the University of Wurzburg. 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 Rontgen rays. Method of determining wave length. Photo-electric effect and emission of light by canal ray particles. 76 pp. 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. Digitized by the Internet Archive in 2007 with funding from Microsoft Corporation http://www.archive.org/details/fourlecturesonmaOOhadauoft .•• PREFACE The " Saturday Morning Lectures " delivered by Pro- fessor 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 ex- pressed in this place to Dr. Goldsmith for writing out and revising the lectures, and to Professor Kasner of Columbia for reading the proofs. in CONTENTS Lecture I. The Definition of Solutions of Linear Partial Differential Equations by Boundary Con- ditions. Lecture II. Contemporary Researches in Differential Equations, Integral Equations, and In- tegro-Differential Equations. Lecture III. Analysis Situs in Connection with Corres- pondences and Differential Equations. Lecture IV. Elementary Solutions of Partial Differential Equations and Green's Functions. » LECTURE I The Determination of Solutions of Linear Partial Dif- ferential Equations by Boundary Conditions 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 gen- eral 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 solu- tions 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. 2 FIRST LECTURE 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. In general, correctly set problems in ordinary algebra are char- acterized 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: (1) the number n of these equations being precisely equal to the number of unknowns. If the determinant formed by the co- efficients 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 dif- ference between the two cases appears if we choose the b's (second members of the equation (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 indeterminate. Things occur in the same way for every problem of algebra. For instance, the three equations /(•'•> y,z) = a g(.r, y, z) = b f+g = c * » LINEAR PARTIAL DIFFERENTIAL EQUATIONS 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 (2) /(*, y, z) = X, g(x, y, z) = Y, h(x, y, z) = Z be the equations of a space-transformation, the functions /, g, h being continuous. Let us suppose that within a given sphere (x2 + y2 + z2 — 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) = /0'> 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 = x1 ', 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 (A", Y, Z) when {x, y, z) describes s. If in equation (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 Schoenflies' theorem says that 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 unique and the fact that that solution necessarily exists.1 As said above, the theorem is in the first place remarkable for its great generality, as it implies concerning the functions /, 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 1 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 x2 = 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. 4 FIRST LECTURE fail to appear in great number as a consequence of future dis- coveries. 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. Let us return to partial differential equations. Cauchy was the first to determine one solution of a differential equa- tion from initial conditions. For an ordinary equation such as f(x, y, dy/dx, d?y/dx2) = 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, du/dx, du/dy, du/dz, d2u/dx2, • • • ) = 0 be a given equation of the second order and let it be granted that we can solve it with respect to d2u/dx2. Thus we obtain (d2u/dx2) + Fi = 0 where Fi is a function of all of the above quantities, except dhi/dx2. Then Cauchy's problem arises by giving the values OIL (3) u =
and \p are holo-
morphic, 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 — u0 + xu\ + xhii + • • • where u0, Wi, • • • can be calculated.
»
LINEAR PARTIAL DIFFERENTIAL EQUATIONS 5
The above theorems are true for most equations arising in
connection with physical problems, for example
(E) v>M=f|.
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
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 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 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: (a) In the theorem of Sophie
Kowalevska, u has only to exist in the immediate neighborhood
of the initial surface ) — + —
^ ; dx* + dp?
we may assign arbitrarily the values (whether analytical or not)
of u and hujU 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 superior1 to the number of
unknowns.
It never could have been imagined 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 (E'), for instance 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 (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
characteristics of an equation. The characteristics of an equa tion
jrrespond analytically with what the physicist calls the 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)
1 We could be tempted to apply in that case the remark made in the be-
ginning (p. 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 fiir
Mathematik") that the solution of Cauchy's problem, if existent, is in every
possible case unique.
10 FIRST LECTURE
= 0. In the given equation, for instance, if A2u — l/a?-d2u/d1? = 0
and A2u be replaced by (dP/dx)2 + (dP/dy)2 + (dP/dz)2 and
- (l/a2)(d2u/dP) by - (l/a2)(dP/dt)2 the condition thus obtained
(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 (E), such characteristics exist (that is, are real);
this case is called the hyperbolic one.
Laplace's equation, A2u = 0, on making the above substitu-
tion, leads to the equation
which has no real solution. Therefore, in this case there are no
waves and we have the so-called elliptic case.1 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 (E) or (E')t we could not choose arbi-
trarily u and du/dy 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 Dirich-
let. They are the only ones which can occur in an elliptic
equation, but even in a hyperbolic one they generally present
1 An intermediate case exists A*u — k(du/dt) = 0. This is semi-definite
and is termed the parabolic one (example: the equation of heat).
LINEAR PARTIAL DIFFERENTIAL EQUATIONS 11
themselves together with initial ones. This gives place to so-
called 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 (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 disposi-
tions 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,1 which do not govern any
physical phenomena. The determination of solutions has been
studied2 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 accord-
ance 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 Poincare's views3 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."
1 The so-called non-normal hyperbolic equations, such as
*» + ...** _** A2!L = 0(m>l,n>l)
dx?^ dxj dyi> dyj
* By Hamel (Inaugural Dissertation, Gottingen) and Coulon (thesis, Paris)
* Lectures delivered at the first International Mathematical Congress,
Zurich, 1897; reproduced in "La Valeur de la Sciences."
LECTURE II
Contemporary Researches in Differential Equations,
Integral Equations, and Integro-Differential
Equations
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 mathe-
matician 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 under-
12
CONTEMPORARY RESEARCHES IN EQUATIONS 13
stand its true Dature. 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 prob-
lems profoundly different from the simple cases which were
familiar to 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 differentia-
tion. 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
(1) *(*) — Xjf 2), so that, in that respect, the case of two dimen-
sions proves more complicated than that of three or more
dimensional spaces.
These peculiar distinctions are closely connected with the fun-
damental distinctions of analysis situs. They are due to the fact
that there are many ways essentially distinct from each other, of
1 It is interesting to add that as far as ordinary (closed) surfaces are con-
cerned, 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 2, of genus g > 1,
corresponds to one (and only one) point of a second closed surface 2' 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 2 corresponding to same
point on 2 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.
*,
ANALYSIS SITUS 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
polybedra would give us the most simple and elementary in-
stances in which the results are profoundly modified by con-
siderations 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 infini-
tesimal 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 neces-
sarily turn our attention towards the mode of synthesis of those
details which introduce the point of view of analysis situs.
LECTURE IV
Elementary Solutions of Partial Differential Equations
and Green's Functions
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: V2m = 0; namely the elementary Newtonian
potential 1/r, where
,2+^2=0
f- V(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
dx2+ dy2
Here we must consider the logarithmic potential, log 1/r, where
r = V (x — a)2 -f- (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 coordinates of two points
(x, y, z) and (a, b, c) [the corresponding element log 1/r in the
plane being similarly a function of (.r, y; a, b)]. If considered
as a function of x, y, z, alone (a, b, c, being supposed to be con-
stant) in the real domain, 1/r is singular for r = 0; and r = 0
40
1
ELEMENTARY SOLUTIONS 41
only when x = a, y = b and z = c simultaneously. But for
complex points, 1/r is singular when the line that joins (.r, 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 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 vibra-
tions 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.
42 FOURTH LECTURE
The gap is filled by a theorem of Delassus. Let us consider any
linear partial differential equation of the second order, and sup-
pose that u is a solution which would be singular along all points
of a certain surface, irix, 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 V2u = 0,
the (non-linear) partial differential equation of the first order
{%)H%>&)'-°
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 x (the other terms of the given equation being considered
as cancelled). This is the 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 uir since u = F sin fiir, where F is the parameter corresponding
to the frequency, and ir is a function of x, y, z. This form of u
seems to show no singularity, for the sine is a holomorphic function
It is nevertheless what one may call "practically singular." If
we suppose that the absolute magnitude of n is large, the function
varies very rapidly from + 1 to — 1, it has derivatives which
contain u 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 "approxima-
tive" analysis, it must be considered as analogous to certain
discontinuous functions. From that point of view the three
notions of waves are closely connected.
ELEMENTARY SOLUTIONS 43
This view of Delassus is the one which will interest us now
because in the case of the elementary solution 1/r the char-
acteristic 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 char-
acteristic cone is replaced by a 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
d2u d2u
(1) W+W=CU
Not every equation of the general type
. d2u , _ d2u , „ d2u , A„fc , „„3ti , „
AM+B3^+CW+2Dl>:r+2Eay+Fu=0
can be reduced to that form. But in the elliptic case (B2 —AC
< 0) it can, by a proper change of independent variables, be
reduced to the form
,,. d2u , d2u du du ,
(1) d?+W+adx+bdy+CU=°
(in which the characteristic lines are the isotropic lines of the
plane). Sommerfeld and Hedrick treated this more general
form and showed for equation (1), as Picard had done for the
equation (1'), that there exists an elementary solution, possessing
all the essential properties of log 1/r. It is
P log 1/r + Q
P and Q being regular functions of x and y. P has the value 1,
4
44 FOXTRTH LECTURE
x = a, y = b. In the hyperbolic case (real characteristics),
the form to which the equation can be reduced is Laplace's form
, . d2u . du , , 3m ,
if the change of variables is real; and the corresponding ele-
mentary solution is of the type
PlogV(*-o)(y-4) + Q
P and Q having the same significations as above (P is nothing
else than the function which plays the chief role in Riemann's
method for equation (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 (2) or as well, (1). The only
case in which that reduction is not at all possible, is when
B2 — 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 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
du , , du , du , „
dx dy dz
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 singu-
larity which is the characteristic conoid. Suppose first that we
ELEMENTARY SOLUTIONS 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 = xpF (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 consider-
ations 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
F(x, y, z) i v , s
— h Fi(x, y, z)
(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
F + x"Ft
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 conoid, which
has (a, b, c) for a conic point, things behave differently; p cannot
have an arbitrary value. If the number of independent vari-
ables is n, we must have
n- 2
p= 5—, or
(-_ii+1), -(!=!+,).
The first of these values is, however, the only essential one,
because, if we have formed the (unique) solution corresponding
to p = — (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
46 FOURTH LECTURE
therefore, on account of what we just said, there is, in general,
no solution of the above form
We have to replace this by
« = ^ + p1iogr
in which T would again be equal to r2, r meaning a distance in
n-dimensional space, if the higher terms (of the second order)
of the given equation are of the form V2w- However, if these
terms are arbitrary, T should be replaced by the first member
of the equation of the characteristic conoid of summit (a, b, c).
The functions P, Q, Pi 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 (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 equa-
tions 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. 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.
ELEMENTARY SOLUTIONS 47
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 ellip-
tic case, this latter is not to be considered, and we have to
face others, such as Dirichlet's problem. For Dirichlet's prob-
lem (i. e. to find u taking given values all over the surface of
the volume S, and satisfying y~u = 0), 1/r is 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 Greens 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 main-
tained 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 interipr 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 V2 V2u = 0 (in two variables x and y only) and the role of
elementary solution being played by r2 log r.
Like 1/r and like the elementary solution itself, any Green's
function depends on the coordinates 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
48 FOURTH LECTURE
remember that each of those two problems depends on three
kinds of elements:
1. A given differential equation;
2. A given surface (or hyper-surface in higher spaces) S;
3. A certain distribution of given quantities at the different
points of S.
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 role of
the first element may be compared to that of the coefficients of
the unknowns (by the help of which such complicated expres-
sions as the determinant and its minor determinants must be
formed) while the role 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 role 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 role as the third element. For instance, in Rie-
mann's method for Cauchy's problem concerning equation (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 V2" = 0, one Dirichlet's
problem for the sphere, one for the ellipsoid, one for the paral-
ELEMENTARY SOLUTIONS 49
lelepipedon; 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 elemen-
tary solution depends on nothing else than the given equation
and the coordinates 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 coordinates, but also on the form of
the boundary S.1
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 8n (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 gBA for the surface S,
and if we change from S to S', gA changes. The change is
«n * * f fd9nA d9nB* jo
danA
—j — is the rate of change of gA relative to the change of n.
Here 8ndS is an element of volume comprised between the
surfaces S, S'. Similar formulas hold for Green's functio.ns 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
S0a = S{y)K{x, y)dy = /(*)
where <£ is the unknown function of x in the interval {A, B), f
and K are known functions, and X 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
14 SECOND LECTURE
equations of the above type, each step in the study of elliptic
partial differential equations seemed to bring with it new diffi-
culties; 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 equa-
tion 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 im-
mense difficulty. Schwarz, Picard and Poincare 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 demon-
strated rigorously the chief properties of the quantities in ques-
tion (namely, certain special values of the parameter in equation
(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 Poincare, namely the near relation
between that question of harmonics and the method which had
been indicated by Neumann for Dirichlet's problem. This
discove.y of Poincare 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 tire reduction to an equation such as (1).
This gives all the required results at once and for all the possible
types of such problems.
CONTEMPORARY RESEARCHES IN EQUATIONS 15
In all this, the mathematician seems to play again the
unfortunate role 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 require-
ment 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 equa-
tions. 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 X) the terms of which have to be calculated inde-
pendently, 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 har-
monics (i. e. the corresponding great values of A). They would
probably have been quite unable to predict the order of magni-
tude, as is done in the recent works of Hermann Weyl, so as to
16 SECOND LECTURE
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 (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 pos-
sible or not. The hypothesis proposed by Fredholm leads to an
integral equation such as
*<*> ~ kZTtf£