Davis: Present Status of Integral Equations 19 
Generalizations of the integral equation problem were made in 
another direction by extending the definition of integration to in- 
clude the integrals of Stieltjes and Lebesgue. With this work a,re 
associated the names of G. C. Evans, E. W. Hobson, E. E. Levi, 
W. H. and Grace C. Young and more recently C. A. Fischer.* 
The postulational foundations of the theory of integral equations 
was examined by E. H. Moore 75 in 1910 and the methods of general 
analysis used to describe the basic principles which underlie the 
theory. To quote Professor Moore: 
The existence of analogies between central features of various theories 
implies the existence of a general theory which underlies the particular theories 
and unifies them with respect to those central features. 
In this characterization the property of linearity plays an essential 
role. 
Another somewhat similar generalization of the problem of 
integral equations is found in the calculus of permutable functions and 
functions of composition. 
Two continuous functions K x (x,y), K 2 (x,y) are called permutable 
if they satisfy the relation: 
Ki (x,t) K 2 (t,y) dt = r K 2 (x,t) Ki (t,y) dt. 
X X 
The preceding operation is called the composition of the two 
functions, and the resulting function a function of composition. 
Such functions were first studied by Volterra in connection with 
the solution of the Volterra equation of second kind. Thus he showed 
that the solution of equation (2) can be written in the form 
u(x) = f(x) + X I f(t) k(x,t) dt, 
where we have made the abbreviations, 
k(x,y) = Ki (x,y) + X K 2 (x,y) + X 2 K 3 (x,y) + 
and 
The fundamental theorem which states that if K(x,t) is inte- 
grable over the triangle a < t < x < b, we shall have 
for all values of i less than n, shows that Ki(x,y) and K n -i(x,y) are 
permutable. The function k(x,y) belongs to the class of functions 
of composition. 
*See ref. XIX of the Bibliography. 
