FUNCTIONS IN THE THEORY OF IMir.HAL KQUATIONS. 131 



it follows from the lemma just established that 



converges uniformly to zero in any finite interval. 



The same remarks being applicable when f(s + t) is replaced by either of the 

 functions f(st),f(s + t),f( st), we have the theorem : 



With the hypotheses of the first corollary of 10, each of the four functions 



dt 



i r' 



- f(st) 



n Jy 



converges uniformly to zero for values of s in any finite interval, as the positive 

 iiiti'i/er n increases indefinitely. 



It will be clear that a variety of results may be obtained from the corollary referred 

 to by a similar process. 



III. A METHOD OF REPRESENTING AN ARBITRARY FUNCTION IN TERMS OF 

 SOLUTIONS OF A STURM-LIOUVILLE EQUATION. GENERAL THEOREMS ON 

 THE BEHAVIOUR OF STURM-LIOUVILLE SERIES. 



1. We proceed to apply the foregoing results to the theory of Sturm-Liouville 

 series. With this in view, we shall commence by considering those solutions of the 

 differential equation 



O 



which, by a suitable choice of the parameter r, can be made to satisfy a certain pair 

 of boundary conditions at the ends of an interval (a, 6). In what follows it will be 

 assumed that in the closed interval (a, b) (1) I is a continuous function of x, (2) g and 

 / are continuous functions of x which never vanish, (3) k possesses a continuous 

 differential coefficient, and (4) gk has a continuous derivative of the second order. 



The pair of boundary conditions above referred to will be supposed to be one of the 

 following four : 



(i) fa-hv =0 at :r = a 



' dx 



v TT n r h 



-= + rn u ,, x u 

 d.i- 



(ii) v = x = a 



~ + Hi' = x = b 



a 9 



^ - 



