, 71 DR. ... MERCER: STORM-UOUVTLI.E SERIES OF NORMAL 



/W .-.- -anr i 



r, , s, , : 



fli frii )/<*)* 



Jo 



converges uniformly to zero for all values of . in (0, ,). Denoting. by <r n (,) the sum 

 of the first terms of the series 



---' (24) 



and by ?. (s) the sum of the first n terms of 



TT.o 



it follows that, as n tends to oo, 



r, (*)-.(*) 



converges uniformly to zero, for all values of s in (0, TT). 

 If D (s) is any limiting point of the set 



<TI (), <r,(), .... <r.(a), ..., ........ ( 26 ) 



we can select an increasing sequence of integers n,, n,, w 3 , ..., n m , ..., m such a way 



that the sequence 



<r%()i %() ' "-( s )> '' 



tends to the limit D (s). It Mows from the result obtained above that the sequence 



also tends to the limit D (s). We have thus proved that each limiting point of the 

 set (26) is also a limiting point of the set 



?i(*). *a( s )> ?(*) ; 



in particular, the upper and lower limits of indeterminacy of the two series are 

 identical. It should be observed that this includes the result that, if either of the 

 aeries (24), (25) is convergent, so also is the other. 



Since 



<r ()-.(*) 



converges uniformly to zero in the whole of (0, IT), we see that, if either of the series 

 * This is, of course, FOURIER'S cosine series corresponding to / (*). 



