6 Mr Pollard, On the term by term integration 



and I ^ {x) I dx converges; then both sides 0/ (1) exist and are 



J a 



finite and equal. 

 B. If either of 



% i \un(cc)\dx, 2 I M„ (x) I dx, 



w = l J a J a n = l 



eadst and are finite; then both sides of (1) exist and are finite and 

 equal. 



rX m rx <x> 



C If lim S Un (x) dx, 'S. Un (^) dx, 



m-»-oo .' a«,= l ■!an=l 



exist and are finite and equal, and 



2 Un (x) dx 



J a 

 converges uniformly for a^^ x, and each 



u^ (x) dx 



J a 



converges; then both sides of(l) exist and are finite and equal. 



V 



D. 7/ I 2 M„ («) I < -^x (^) for a^ x^ X and all v, where -^x 

 n=l 



is summable in {a, X), and 



rx 



S Un (x) dx 



J a 



converges uniformly for a^ X, X being arbitrary, and each 



Un {x) dx 

 J a 



converges; then both sides of {1) exist and are finite and equal. 



D is a special case of C obtained by making use of (C 2). 

 A, B, D may be regarded as generalisations of theorems A — C, 

 pp. 452-455 of Bromwich's Infinite Series. 



Proofs. A. If m' > m, 



m' m' m 



we have 2 /„ {x) = X fn{x)-X fn {x), 



and therefore 



m' Til' m 



I 2 fn{x)\^\ Xfn{x)\ + \ ^ f,{x)\^2G. 



