Davis: Present Status of Integral Equations 
5 
f(x) = aJ* K(%t) u(t) dt, 
a 
(U 
and by a Volterra equation of second kind , 
u(x) = f(x) + XI K(x,t) u(t) dt. 
(2) 
a 
Similarly we may classify the equation 
►b 
f(x) = 4 K(x,t) u(t) dt 
(3) 
a 
as a Fredholm equation of first kind and 
•b 
u(x) = f(x) + 4 K(x,t) u(t) dt 
(4) 
a 
as a Fredholm equation of second kind. 
We should note that the general theory of the Fredholm equation 
will include that of the Volterra equation as well, since it is only 
necessary in the Fredholm equation to choose a function which will 
equal K(x,t) for t < x and which will be identically zero for t > x. 
In the general case this introduces a line of singularities along t = x, 
but functions of this type are not excluded in the theory. 
D. Hilbert has suggested that the equation 
h(x) u(x) = f (x) + > K(x,t) u(t) dt 
should be called a Fredholm equation of third kind. But when h(x) 
is different from zero for x taken between the limits a and b this 
equation is no more general than (4). On the other hand, the case 
where h(x) vanishes in the interval (a,b) corresponds to the case 
where f(x) and K(x,t) of (4) have singularities in x and it seems 
preferable to call it the singular Fredholm equation of second kind. 
Under this name also equations are included in which at least one of 
the limits of the integral is infinite. Similar remarks apply to 
the Volterra case. 
The third type of equation, characterized by the fact that the 
limits of the integral are functions of x, might be properly named 
after G. Andreoli, who first gave a general discussion of its properties. 
It is obvious that the Andreoli equation 
u(x) = f(x) + X 
will include both the Fredholm and Volterra equations as special 
cases. 
