178 PR. J. MERCER: STORM^IOUVILLE SERIES OF NORMAL 



Let I. (a) denote the value of 



sn 



for integral values of m. It is easily seen that 



sn 



Substituting * + * = w, and then replacing w by , we obtain 



which, owing to the periodicity of/, (s), leads to 



hence we have 



In a similar way it may be proved that 



* ....... (34> 



Let us now suppose that s is a point of an interval (y, 8) lying wholly within (0, ir) '> 

 and let a be a positive number less than both y and ir-8. We have 



/,(_,_)=/,(_* + ) = (y<s=ES, OS*a) 

 Thus 



as 



Since, by a known theorem,* the right-hand member converges uniformly to zero 

 n tends to oo, it follows that I^+ils) converges uniformly to zero in the interval 



(y, S). It may be shown in a similar way, by using (34), that 1^ (s) has the same 



property. 



14. Let l' m (s) be the value of 



, 



II., 6. 



