858 PROCEEDINGS OF THE AMERICAN ACADEMY. 



§4. 



Proof of Convergence. 



The convergence of the series (34) will now be proved ; and in this 

 proof it will also be shown that : 



(43) limit <A , = q > 0. 



We shall use in the proof the following functions. A function of a:, 

 B, is chosen having the same properties of integrability as b k J /} and such 

 that : 



i—m j=e, //. _ 1 9 ... 



i—i j—\ \ ' ' * 



Next consider all the differences Rr k — Rr K which are not zero, and 

 choose a positive constant d such that : 



< d < \Rr k - Rr K \. 



Then (J shall be a positive constant such that : 



tr=n 



d* 



t—n -J 



(45) C>n, C>2^=I 



We also define : 



X 



CBMoaxfx^dx. 



(46) M(x) = f 



The point c is chosen so that : 







(47) < c < 6, c S- (where loge = 1), M (c) < 1. 



This is the final determination of c to which we have referred on page 

 350 ; and this point c will, from now on, mark one end of the interval 

 for x. Instead of M(c), we shall, for the sake of brevity, write 

 simply M. 



The convergence of the series (34) will be proved by showing by 

 mathematical induction that the following inequalities hold for all 

 values of q: 



k=m l=e k i=m j=e t 

 * We might, for instance, take B = 2 2 2 2|6^|. 



k=\ 1=1 i=l j=\ 



