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Indiana University Studies 
theory for kernels expansible in a power series and also of the 
form K(x)= 1 a n e PnX . 
n = 1 
7. Applications. Side by side with the purely theoretical 
studies of integral equations, we find an ever growing number of 
memoirs devoted to the application of these equations to applied 
problems. We have already indicated how integral equations pre- 
sented themselves early and naturally in connection with differential 
equations of mathematical physics. Consequently, after the solutions 
of Volterra and Fredholm appeared, the dilemma of Du Bois- 
Reymond no longer existed, and to change a problem from a differ- 
ential equation to an integral equation was to bring it under a theory 
rich in new methods and results. 
The general tendency in the study of an applied problem has 
been to continue to use the methods of the pioneers. The physical 
situation is first formulated as a differential equation, the problem then 
converted by means of a Green’s function to one in integral equations, 
and the resulting equation used as a bridge to the solution. But in 
his study of the kinetic theory of gases, D. Hilbert 44 has taken a 
more fundamental point of view. He goes directly from the physical 
.problem to an integral equation and asserts, further, that this direct 
use of integral equations is essential in the foundation of the theory. 
He says, in part:* 
In all of the applications of the theory of integral equations which we have 
discussed up to the present time — wdiether they were analytic or geometric 
problems or problems in the field of theoretical physics — there was always an 
ordinary or partial differential equation or a system of such differential 
equations which served us by intervention in setting up the integral equation. 
In the following I make a new, direct application of the theory of integral 
equations, for I show that there is a certain linear integral equation of second 
kind with symmetric kernel which forms the mathematical foundation of the 
kinetic theory of gases and without whose investigation according to the 
modern methods of the theory of integral equations a systematic foundation 
of the theory of gases is impossible. 
Further extensions of this idea should be looked for in the 
future application of integral equations. The concept of ‘‘rate of 
change” is fundamental both in differential equations and in the 
relations between the physical properties of matter, so that it has 
been natural to express physical laws by means of derivatives. But 
the notion of “summation” seems to be just as fundamental' in a 
study of the phenomena of nature, so that we would naturally 
expect to be able to pass from a physical problem to one in integral 
*Ref. 9, p. 267. 
