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



which is the result referred to. The series on the right is, of course, both absolutely 

 and uniformly convergent. 



It will be recalled that a direct application of SCHMIDT'S method of proof imposes 

 narrower restrictions upon /(*) ; it breaks down, for instance, if [/(s)] 8 is not 

 integrable in (, b). 



3. Returning to the formula (2), it is evident that 



& (4) 



This relation is true for all values of X, other than the singular values of K (s, t), but 

 in what follows it will not be necessary to consider values of X which are positive or 

 zero. .4* the assumption that X is always negative will make our work somewhat 

 simpler tee shall adopt it throughout this section. 



We proceed to investigate the behaviour of the right-hand member of this equality 

 as X tends to t. For this purpose we shall suppose that the order of the numbers 

 X|, X, ..., X., ..., is that of non-decreasing magnitude. Thus far this order has not 

 been material, but it will appear that the result obtained below turns upon the 

 hypothesis stated. 



Let tr. (s) be the sum of the first n terms of the series 



The sum of the first m terms of the series on the right-hand side of (4) is 

 _ \ m _ \ 



K=j-W+5.j : ^[-.W-.-,(.fl 



which is 



Now, suppose that we can choose a positive integer N\ such that 



<r.(a)<h, 

 for all values of w * N,. If m > N,, the function of X (6) may be written 



Since the coefficients of the numbers r.(.) are all positive when X is negative, we 

 thus see that (6) is less than 



1 "' For 



