FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 117 



As a corollary to this theorem it should l>e noticed that, with the same hypotheses, 



?g(*,t t n)f(t)dl 



Ja 



converges uniformly to 



for values of s in (a,, &,), as n tends to oo. 



4. The theorem we have just proved, like that of 1, admits of generalisation. 

 On account of its importance in what follows, we state the theorem : 



Let g(s, t, u, n) be a function defined at all points of the rectangular parallelepiped, 

 P, which consists of the points a 2 ^s^b 2 , a 1 ^t^b l , a^u^b, and for a set of 



values of n which have + oo for an improper limiting point. Let this function be 



(b 

 g (s, t, u, n) du 

 . a ' 



exists as a Lebesgue integral for all values of s, t, and n. Let g (s, t, u, n) be related 

 to a limited function g(s, t, u) defined in the whole ofP in such a way that, as n tends 

 to CD, g (s, t, u, n) converges uniformly to g (s, t, u), either in the whole of P or, for each 

 positive number 17, in those parts of it which correspond to 



\U-a m s-p m t-y m \ >r), (m = I, 2, ..., M), 



where the numbers a m , /J m are all finite. Then, iff(n] is any function which has a 



Lebesgue integral in (a, b), 



ft 

 g(s,t,u,n)f(u)du 



.a 



converges uniformly to 



for a^cz-b, a l S / rS &,, a. 2 S s ^ b 2 , as n tends to oo. 

 lu particular, we see that 



f g(s, t,u,n)f(u)du 



Ja 



converges \iniformly to 



in the rectangle a, t ^ 6,, a 3 i .s < b.^ 



Again, let us suppose that a = a,, b = t> 2 ; then, writing c = s, we see that, 

 as n tends to oo, 



\'g(s,t,u,n)f(u)du 



Ja 



converges uniformly to 



\' . ff(*,t, n)/(ti)d 



Ja 



