ON THE THEORY OF INTEGRAL EQUATIONS. 357 



CO 



where li(x,t) = k-(x,t) + 2V~ K x (x,t) 



i 



x 



and K,(x,t) = U m (x,y)^ r - m (y>t)dy 



K\{x,t) = />(#,/). 



The series for K(.r,/) is convergent for all values of X if f(x) and i,{x,t) 

 are bounded integrable functions, and there is only one bounded 

 integrable solution of the equation. 



The integral equation 



X 



f(x) -f(a) = ^li(x,y)^{y)dy 



a 



may be reduced to the previous one by differentiation, for we have 



X 



fix) = U(x,x)p(x) + &&&mdy. 



a 



It is convenient now to assume that /H(x,x)/ has a lower limit which 

 is different from zero in an interval a < x <a + A, and that the function 



H(y,y) dx 



is a bounded integrable function in this interval. The previous method 

 may then be applied to determine the function 



4*(x) — H(x,x)(l>(x). 

 Yolterra's extension of Abel's theorem is as follows : — 



n Ay) -/(«) = j" §^$*W dx - (a < i) 



a 



where f(y) and /'(?/) are finite and continuous for a < y < « + A, and if 

 G(x,y) and ^— =G 2 {x,y) are finite and continuous for all values of x and y 



contained in the interval (a, a 4- A) and the absolute value of the lower 

 limit of g(y) = G(y,y) is different from zero, then here is only one finite 

 continuous function (\>(x) which satisfies the equation for« < x < a -f A, 

 and this solution is given by the formulae 



X 



m= ^^ 1 [f'{x)%%{*,z)dx 



tv g(z) J 



a 



* g(z) J v* — U z - y 



T (x,z) = _* 



\z — xy A 



X 



T 1 («,f)«fs (£ I *)T < : i (* > £)« 



