, 90 ML J. MERCER: S TURM-UOUVILl,E SERIES OF NORMAL 



Since each of the four functions 



converge, u,,ifor m .y to ro in any finite interval* (II., .0), we see fro this that, 

 for any positive value of less than IT, 



7?7ro 



convey uniformly to zero in (0, ), as the positive integer increases "definitely^ 

 lTtfs suppose tLt . is a fixed point of the closed mterval (0,,) at wh.ch F(+0) 

 a nnite value. When any positive number . is ass,gned, we may then ehoose . 



,,,,s 



small enough to ensure that 



at all points of (0, ). With this choice of a, we have 



. 



which, since 



1 /sin 7. _ 



hm - . \- a* - 



and e is arbitrarily small, leads to 



. (*) 



The reader will have no difficulty in seeing that this inequality is valid when 

 Ff+0) has one of the improper values oo (cf. III., 13). It may be proved in the 

 same way that 



li ra *> + S2 s + --- + M s * F ( + 0). 

 n 



From these inequalities it is evident that, at any point of (0, ir) where F( + 0) 

 exists, 



n 



exists and is equal to it. Moreover, in virtue of our definition of /i (s),\ it will be 



wen that at a point of the open interval F( + 0) is lira i [/(*-)+/(* + *)] > that at 



*-o 



* = it is /(O + O) ; and that at s = rr it is f(ir 0). 



* This seems to have escaped notice hitherto. 



t The simplest method of obtaining this result is to apply CAUCHY'S theorem to (35) above. 



