6 
Indiana University Studies 
A fourth type of linear equation, first introduced by A. Kneser, 
wlio derived it from certain problems in mechanics, has been generally 
called the mixed integral equation. It is of the form 
u(x) = 2 <Pi(x) u(y T f K(x, t) u(t) dt, 
i = l J a 
where Si , & , S n are any distinct points in the interval (a,b). 
Another class of functional equations that have considerable 
importance in applied problems are those to which Volterra has given 
the name of integro-differential equations. These in the general case 
include the unknown function under one or more signs of integration 
and as a derivative. 
By a solution of an integral equation is meant a function which, 
when substituted in the equation, reduces it to an identity. 
3. Types of Kernels. To the function K(x,t) has been given 
the name kernel (French, noyau; Italian, nucleo; German, Kern). 
The quantity X is called the parameter of the equation. 
Since the characteristics of solutions of integral equations 
are largely determined from the characteristics of their kernels, it 
has been found desirable to classify linear equations by means of 
some of these important properties. 
Thus by a symmetric kernel is meant one which has the property 
K(x,t) : K(t,x) 
and by a skew-symmetric kernel one in which we have 
K(x,t) EE - K(t,x). 
Two kernels are called orthogonal if they satisfy the two con- 
ditions 
/: 
Ki(x,t) K 2 (t,y) dt = 0, 
r 
K 2 (x,t) Ki(t,y) dt = 0, 
and semi-orthogonal if only one of these conditions is satisfied. 
If a kernel is symmetric and possesses the additional property 
that I > 0, where 
K(x,y) h(x) li(y) dx dy, 
no matter what function h(x) may be, provided only that it has an 
integral square, then the kernel is called positive , and if I ^ 0, 
it is called negative. If I > 0 or if I < 0, then the kernel is called 
definite. 
If K(x,y) is a symmetric kernel and if there exists no function 
h(x) such that 
