196 Mr. J. W. L. Glaisher on Arithmetical In •ationality. 



but this method either does not succeed at all, or if it does, the 

 process is not so convenient, when there are coefficients ^(1), 

 i/r(2), &c. to the powers of q. Applying (3) to the series 

 1 + q + q 4 + . . . ) we have 



l + J+^U — H--^- - • W 



'/' — ______ n-o 



r 3 +l 



r b +l 



r 7 + l~&c. 



This, and the corresponding fraction when the alternate terms 

 of the series are negative, was given by Eisenstein in vol. xxvii. 

 p. 193 of CrehVs Journal, where he has applied it to prove the 

 irrationality of the scries. He has also proved by means of a 

 continued fraction a more general theorem, viz. that 



x x* 2 x s 

 r r- r a 



is irrational whenever r is an integer, and x a rational fraction 

 not greater than unity, while it has been shown above that the 

 series is irrational if x is an integer not greater than r 2 ; so that 

 the methods give results which, though they overlap, do not co- 

 incide. Eisenstein has not stated how he obtained the fraction 

 (4) ; but the manner in which he has stated his result leads to 

 the inference that it was by means of treating the series 



-+ * +^ + ... and l+- + ^4^+... 



in a manner analogous to that of finding their greatest common 

 measure. 



In a letter to Jacobi (Crelle, vol. xxxii. p. 205), Heine has 

 proved Eisenstein' s theorem (4) by transformations from Eider's 

 formula 



#— b + c— d+e— ... = r— 



1 "+ 1 , aC 7. 7 



N a — o + i — - , bd 



o — c-\- 



c — d + hc. 



which really comes to the same thing as using (2), although the 

 work is much longer, reductions &c. being required. 



I may also mention that the usual way of treating series of the 

 same form as that in (3), viz. by means of the formula 



1 l +l- 



*• b * h > » I+ ._j__. i 



b 2 -b l -\- 



b* — bc,-\-lkc. 



