130 



DR. J. MERCER: STURM-LIOUVILLE SERIES OF NORMAL 



. 



; ' * c *' v -= f -= 8 (0 ^ y < S : TT). Then, as the positive 

 s and t in the rectan<jk y : x == b , y ^ > 



integer n increases infinitely, each of the eight ^ntegra 



converges uniformly to zero in (y', 8'). 



From the first corollary of the preceding paragraph, it is possible to deduce a 

 theorem winch we shall have to employ below. This theorem depends upon the 



following lemma : * , , , x 



Let g (, i<) be a limited function defined for all values of s in an interval (a, b), 

 and for all positive wtegral values of n ; also let this function converge to g (s), as n 

 increases indefinitely, uniformly in (a, b). Then, as n increases mdefimtely, the 

 arithmetic mean 



n 



converges uniformly to g (s) in (a, b). 



To prove this, we observe that, when any positive number e is assigned, a positive 

 integer N, may be chosen great enough to ensure that 



\9(s,n)-g(s)\ <e 

 for all values of n a N,, and of s in (a, b). It follows at once that we have 



n r = i 



As g(s, n) is limited, it is clearly possible to choose a positive integer N 3 in such' a 

 way that the first term on the right is less than e, for all values of n 2: N 2 , and of 

 s in (a, b). Hence we see that, when n is not less than the greater of N\ and N 2 , 



for all values of s in (a, b). The lemma is therefore established. 

 With the notation of the previous paragraph, let us write 



X (0 sin i C 2 ' 1 - 



Clearly g (s, n) satisfies the requirements of the above lemma, the function g (s) 

 being everywhere zero, and (a, b) any finite interval whatever. Since 



N 



2 sin (2r-l) t = sin 2 /i</sin $t, 



r = 1 



* Cf. HARDY, ' Proc. Lond. Math. Soc.,' ser. 2, vol. 4, p. 257. 



