4 
Indiana University Studies 
In order to indicate the progress of this new branch of analysis 
and to show the broad scope of its problems, it would seem that a 
short historical summary might be of value. The Bibliography at the 
end will show that there are now plenty of technical treatises avail- 
able for workers in this field, but because of the rapid development 
of the subject and the numerous topics which it embraces, these must 
all be limited in both content and point of view. 
The admirable treatment of the history of integral equations 
prepared by Professor H. Bateman for the British Association for the 
Advancement of Science gives an exact and complete account of the 
development of the subject up to 1910. In the present essay, there- 
fore, the emphasis has been placed upon the extension and progress 
of the theory since that time. Each year in the past quarter of a 
century has seen important new developments, and the subject seems 
to be an almost inexhaustible mine for research workers in mathe- 
matics. 
2. Classification of Equations. By an integral equation in its 
general sense is meant any functional equation in which the unknown 
function appears under one or more signs of integration, and by its 
order the highest number of dimensions of any integral which occurs 
in it. For example 
f(x) 
= J* cos xt u(t) dt, u(x) = f (x) + X j* (x -t) 
u(t) 
dt, 
/ *y x u(t) 
dt, 
° l/Z=t 
are integral equations of first order in a single variable and 
u(x,y) = f (x,y) + xf f log[(x - s ) 2 + (y - t) 2 ] u(s,t) ds dt, 
** s\. a. 
is an integral equation of second order in two variables. 
It is clear from the examples just given that further classification 
is desirable; hence any equation in which the unknown function 
enters to the first degree only has been called a linear integral 
equation and all others non-linear integral equations. 
Linear equations may be divided further into four classes for 
convenience of reference altho it will be seen that the general 
theory of the third class will include the first two and the general 
theory of the second class will include the preceding one. Thus we 
mean by a Volterra equation of first kind the following integral 
equation, 
