128 DR. .1. MERCER, STURM-LIOUVILLE SERIES OF NORMAL 



and, turning an arbitrarily small positive, <, to be assigned, let N be chosen great 



enough to ensure that 



Then we have 



s) I dt ' 



I f *, (Ox (-, 

 U% 



which is certainly less than ?* for all values of * in K 6.). Again, since /, (t + s) 

 is limited, we can choose a positive integer m such that 



|f *. 



for 



n S m and a, s s 6* It follows that, for these values of n and s, we have 



and hence that 



|f *b 



converges uniformly to zero in (a 2 , 6 a ). 



It may be proved in the same way that each of the other integrals 



has this property. Hence the theorem : 



Let the system of normal functions |i(s), ^ 2 (s), ..., $(s), be such that (1) the 

 upper limit of \ \ji n (s) \ in (a, b) is less than a fixed number, for all positive integral 

 values ofn, and (2) 



for each limited function $ (s) which lias a Lebesgue integral in (a, b). Let f(s) be a 

 function, defined for all values of s, which has a Lebesgue integral in any finite 

 interval. Let x (s, t) be a limited function which, for each value of s in an interval 

 (a,, &j), has a Lebesgue integral with respect to t in an interval (a lt 6,)((t :< c^ < 6j ^ b) ; 

 and let x (s, t) be a uniformly continuous function of s in (a 2 , 6 2 ) for values of t 

 belonging to (a,, 6,). Then, as n tends to oo, each of the four integrals 



f 



Jo, 



converges uniformly to zero in the interval (a 2 , 6 2 ). 



The theorem we have just enunciated is of very general character, and may be 



