1*1 DR. J. MERCER: STURM-LIOUVILLE SERIES OF NORMAL 



for all values of t in the interval (0, ). We have therefore . ,, . , K 



Since 



e-*dt=-l, 



P+.CC JO 



we see from this that 



or, as is arbitrarily small, 



- - w 



This inequality is obviously true when X( + 0) has the improper value + 06, 

 provided that we interpret it as suggested in II., 4 ; and the reader will be able to 

 prove that it also holds when X( + 0) is - . It follows that the inequality as 

 written above holds in all cases. 



It may be shown in a similar way that 



lim -x['K A (M)/(0<fo=X(+o). . .:,.';:. (so) 



T -I ft 



The function X (t) depends upon Xi (t), Xs (0 and so x ( + 0) may have different values 

 when we replace these functions by others satisfying the requirements of 11, 12. 

 Let us denote by oT(s) the lower limit of the set of values of X ( + 0) obtained by 

 taking all possible pairs of functions Xi (t), Xa (') > an d let w_(s) be the upper limit of the 

 corresponding values of X( + 0). We shall speak of w () as the upper bilateral limit of 

 f(s) at the point s, and of &>(*) as the lower bilateral limit of f(s) at this point. 

 When oi(x) is equal to M (.s), the common value will be referred to as the bilateral limit 

 of f(s) at the point s, and will be denoted by o> (s). 



It will be obvious from (29) and (30) that 



= Imi -X (' K, (s, t) f(t) dt 5: lim -X f K, (s, t)f(t) dt>a> (s). . (31) 



A-*--te JO **.- JO 



In particular, it will be clear that, when the bilateral limit of /(s) at the point s 

 exists, 



lim -x[*K x (s, t)f(t)dt . . (32) 



X oo o 



exista, and is equal to it. 



From the inequality just written, and the definitions given above, we have 



(33) 



