ITNCTIONS IN THE THKOKY OF INTEGRAL KQUATIOXS. 147 



We commence with the equation 



= p 





It was shown in ^ 8 that pl\(s t t) is limited, and that, as p tends to oo, its limit is 

 zero for unequal values of * and t. The function pl\ (s, t) a (p, s, t) will therefore 

 satisfy the requirements of the theorem of L, 3, if it can be proved that />r\ (s, t) 

 converges uniformly to zero, for values of s and t such that \ts\ is not less than 

 an arbitrarily assigned positive number T;. That this is so follows at once from the 

 inequality 



which the reader will establish without difficulty. We deduce that, as X tends 

 to oo, 



-\\'[K,(s,t)-l\(,t)]f(t)dt 



Jo 



converges uniformly to zero in (0, ir). 



Let us now suppose that TJ is any positive number less than the least of y, IT 8. 

 Referring to the equation (15), it is evident that the left-hand member is less than 



smh 



pir 



for all values of * lying in (y, 8). It follows that (16) converges uniformly to zero in 

 (y, 8). In the same manner it may be shown that (18) and the difference between 

 the left- and right-hand members of (19) both converge uniformly to zero in this 

 interval, for a fixed value of a.* Hence it appears that the difference between 



p {' T A (s, t) f(t) dt and f" e-* f(s-t) 



Jo 2 Jo 



dt 



converges uniformly to zero in (y, 8), as X tends to oo. Finally, since the same may 

 be proved of the difference between 



p [" I\ (s, t) f(t) dt and P- P e-* f(s + 1) dt, 



It f Jt 



we deduce that the difference between 



-x ("KiM/W* and f -"[/(* 



Jo if Ji 



converges uniformly to zero in (y, 8), as X tends to oo. 



* We assume that a. i >;. 



IT 2 



