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



It appears, therefore, that with the convention we have explained (7) is true in all 

 cases ; and in the same way it may be shown that 



with a corresponding convention as to the meaning of x > - . Adopting the 

 hypotheses in 1, we may sum up these results in the theorem :- 



If /r,(s), /,(), ..., tMs), are a c m P lete system of normal functions of K(S, t), 

 arranged in such a way that the corresponding singular values are in non-decreasing 

 order of magnitude, and if U (s), L (s) are respectively the upper and lower limits of 

 indttei-minacy of the series 



then 



U (s) == lim -X f K A (, )/() eft S lim -X f K x (s, t)f(t) dt == L (s). 

 A^TTa, Jo A-.--* Jo 



In particular it is clear that, if the series (5) converges, its sum is 



r , . 



hm X K.i(s,t)f(t)dt; (8) 



while, if the series is non-oscillatory and divergent, (8) is + oo, or oo, according as 

 the series diverges to + oo, or to oo. 



5. Let us now suppose that ^i(s), fa(s), ..., ^ B (s), ..., is a set of functions which 

 are continuous in the interval (a, 6), and such that 



f ^(s)^ m (s)dg = 0, (nytm), 



Ja 



= 1, (n = m). 



For brevity, we shall refer to the set as a system of normal functions for (a, b). 

 We proceed to obtain two theorems which have important applications in the sequel. 



Let /(s) be any function whose square has a Lebesgue integral in (a, b). The 

 functions 



/() - I +. (s) j* fc (t) f(t) dt, (m = 1, 2, . . .), 



have then the same property in virtue of the continuity of the functions ^ (s). We 

 deduce at once that, for all values of m, 



n=ll_Jo 



