Jury 18, 1912] 
NATURE 
499 
INTEGRAL EQUATIONS. 
Introduction a la Théorie des Equations Intégrales. 
By Prof. T. Lalesco. Pp. vii+152. (Paris: 
A. Hermann et Fils, 1912.) Price 4 francs. 
L’ Equation de Fredholm et ses applications a la 
Physique Mathématique. By Prof. H. B. Hey- 
wood and Prof. M. Fréchet. Pp. vi+ 165. 
(Paris: A. Hermann et Fils, 1912.) Price 
5 frances. 
NTEGRAL equations are not a quite modern 
invention, because a particular example was 
solved by Abel so far back as 1826. But an 
immense impetus was given to the subject by the 
papers of Volterra and Fredholm, especially by 
those of the latter; and the reason is not difficult 
to find. In the first place, Fredholm chose a 
standard form of equation obviously suited for a 
process of continued approximation; and what is 
much more important, a happy induction led him 
to the discovery that the solution could be put 
into the form of the quotient of one integral func- 
tion of the parameter (A) by another infegral 
function. In a certain way this is analogous to 
Jacobi’s expression of his elliptic functions as 
ratios of theta-functions; and the simplicity and 
elegance of the formule are due to a similar cause. 
The two works considered here are to a certain 
extent complementary. Prof. Lalesco treats the 
subject from a purely theoretical point of view; 
Profs. Heywood and [I'réchet’ emphasise the 
physical applications. From the latter point of 
view we cannot fail to see that Fredholm’s method 
is really the most “natural” and appropriate one 
hitherto discovered. In the theory of potential, 
for instance, it, so to speak, normalises Poincaré’s 
method of exhaustion, bringing it into the range 
of practical computation: it brings scattered 
results into a closer correlation; and it throws 
additional light on the difficult problem of deter- 
mining Green’s function, although it does not com- 
pletely solve it. 
So far, also, the notations and terminology are 
as simple as could be expected. The so-called 
“kernel” and the derived “resolvent kernel’? have 
received appropriate names; and it would not be 
difficult, if it were convenient, to invent an inverse 
notation, and corresponding names, for the solu- 
tion of Fredholm’s standard equation. It is un- 
likely, however, that this will be done, because 
the equation in question is only one of an in- 
definite series, in relation to which it stands in 
much the same position as the linear differential 
equation of the first order stands to other ordinary 
differential equations. It may be noted that Prof. 
Lalesco shows that every linear differential equation 
may be reduced to an integral equation of Vol- 
terra’s form. We are not surprised, therefore, 
NO. 2229, vot. 89] 
| to find a theory of associated equations analogous 
| to that of a differential equation and its adjoint. 
| The theories of abstract dynamics and Fourier 
series have led to the notion of normal functions, 
and quite naturally a similar theory for Fredholm’s 
equation has been developed by Goursat and others. 
Again, since the nature of the solution is mainly 
conditioned by that of the kernel, we are not sur- 
| prised to find that Hilbert and others have arrived 
at important results by giving special properties 
(such as symmetry) to the kernel, and taking into 
account the distribution of its zeroes and poles. 
It is most interesting to see how the general 
theories of integral functions, and even of trans- 
finite numbers, find their applications in the 
present context: thus illustrating once more the 
organic connection of all analysis. 
Perhaps the pleasantest fact of all is that we 
have in this subject a nascent theory, equally inter- 
| esting to pure and applied mathematicians, which, 
for all that we can tell, may grow into a subject 
as large as that of differential equations; while 
at the same time its rudiments are scarcely more 
difficult than the old problem of the reversion of 
series. A clever schoolboy, fairly proficient in 
ordinary calculus, could appreciate the main points 
of Fredholm’s analysis; and there seems to be no 
reason why some of the theory should not be 
included in, say, Part i. of the mathematical 
tripos. Of course, candidates would not be ex- 
pected to know all the more delicate points of the 
theory; but neither are they supposed to know 
all about the conditions for the convergence of an 
integral, or about the complete theory of the 
invariant factors of a determinant. 
English students may begin the subject with 
Mr. Bécher’s outline in the ‘‘ Cambridge Mathe- 
matical Tracts,” No. 10; they could scarcely do 
| better than go on by reading the two present 
treatises, each of which is clear and elementary 
and has special merits of its own; then they could 
| consult the original papers indicated by these 
authors’ bibliographies. 
There is one very small point to which atten- 
tion may be directed. Messrs. Hermann have, in 
one of these books, adopted a method of emphasis- 
| ing important propositions which is very ugly, 
and we hope will not be imitated. In other 
respects they keep up the best traditions of French 
| printing. In Prof. Lalesco’s book there are a 
rather large number of misprints; of these one of 
the most misleading for a beginner is on p. 10, 
2 
1. 2, where, instead of the second | we should 
£0 
(x . . “fc ~ 
read | , as is obvious, if we draw a figure. 
Jo 
Perhaps it may not be superfluous to point out 
