Davis: Present Status of Integral Equations 
7 
j* K(x,t) h(t) dt — 0, 
then K(x,y) is called a closed kernel .* 
By a Schmidt kernel is meant one of the form A-(x) B(y) S(x,y) 
where S(x,y) is a symmetric function and A(x) and B(y) are of one 
sign in the rectangle A < x < B, a < y < b. 
Kernels of the form 
K(x,y) = SX i (x)Y i (y) 
i = l 
are called bilinear kernels . 
4. The Origin of Integra! Equations-! The first phase in the 
historical development of integral equations centered around the 
question of the inversion of integrals. As early as the end of the 
eighteenth century and with increasing frequency in the early part 
of the next, problems presented themselves in differential equations, 
mechanics, and other applied subjects in the form of equations of 
the following type, 
f(x) = J* K(x,t) u(t) dt, 
where f(x) was a known function, u(t) a function to be determined 
and S a path in the complex plane, usually the axis of reals. 
The credit for the first example in this new calculus apparently 
goes to Laplace,! who in 1782 studied the equation 
f (x) = J* e Xt u(t) dt 
in connection with linear difference and differential equations. 
Abel 198 later employed this equation in the more general form 
f(x,y,z,. . . 
xu+yv-j-zp. . 
e 
<p(u,v,p, . . . ) du dv dp . . . , 
and developed a number of the fundamental properties of the Laplace 
transformation, altho no general method for finding its solution is 
indicated. 
The work of Abel and Laplace has been the inspiration of a 
number of memoirs on this type of equation. E. Borel§, for example, 
has made it of fundamental importance in his theory of divergent 
*We here exclude all null-functions, i.e. discontinuous functions which are zero except at the points 
of a set of measure zero. Complete generality is not obtained in theorems on closed kernels. wit.hout 
the, use of Lebesgue integrals; 
fFor a comprehensive historical outline to 1910 consult H. Bateman, ref. 4. The author is greatly 
indebted to this paper. In the pages that follow reference numbers refer to the titles listed in the 
bibliography. 
{Oeuvres, vol. 10, p. 235. 
§Lecons sur les series divergentes, Paris (1901). 
