Davis: Present Status of Integral Equations 11 
which is essentially a system of linear equations, no elaboration of 
the method appears anywhere in his published work. 
In Volterra’s words the following principle seems to underlie 
the theory of both linear functional and linear algebraic equations :* 
To any problem, algebraic or differential, whose solution leads to a 
function expressed as the quotient of entire functions of a certain number of 
variables, there corresponds an integral or integro-differential problem whose 
solution is wholly expressed by means of the quotient of entire functions of the 
same variables. Two such problems are called correlative and it is possible to 
pass from the solution of one to that of the other. 
Thus the integral equation 
u(x) = f(x) + r K(x,t) u(t) dt, 
o 
can be thought of as the limiting form of the equation 
u (x) =• f(x) + S K(x, qj u(q) Ail (5) 
i = i 
as n — > oo ; where the ti are n equally spaced points in the interval 
(0,x), t = 0, t = x, and A- = t. - t. . 
In a similar way equation (5) may be thought of as the limiting 
form of a system of algebraic equations 
I - 1 
u(tj) = f(tj) + 2 K(tj, q) u.(tj) Mb (j = 1 , 2, . . .n) 
i = l 
for which a solution exists of the form 
u(tj) = f(tj) + 2 k(q, q) f(q) Ai, (j = 1, 2, . . , n). 
i = l 
By considering the value of this solution as n — *- °o f Volterra was 
able to determine a limiting form for k (tj,ti) and thru it arrived 
heuristically at a function u(x). That this was a solution of the 
original integral equation he was able to show by direct substitution. 
R. D. Carmichael f has recently examined in detail this method 
of arriving at solutions of transcendental problems thru the limiting- 
form of algebraic systems and has indicated its scope and power 
by deriving oscillation, comparison, and expansion theorems for 
various types of functional equations. 
It was I. Fredholm who first used this new tool in the integral 
equation which bears his name. { The first of these papers appeared 
in 1900 under the title : “Sur un nou velle methode pour la resolution 
*Ref. 292, p. 84. 
fAlgebraic Guides to Transcendental Problems. Bull. Amer. Math. Soc., vol. 28 (1922), pp. 
179-210. 
tit is interesting to note that Fredholm himself proposed the name “equation fonctionnelle 
abelienne’’ for the equation of first kind. Ref. 137, p. 365. 
