Mr. J. 'TV*. L. Glaisher gii Arithmetical Irrationality. 195 



It is not essential that the exponents of the powers of q 

 should be given by an algebraical formula <j>(n) ; the reasoning 

 is equally successful when they are the series of prime numbers, 

 or of their squares, cubes, &c. 



I need scarcely remark that the sole condition for the success 

 of the method is that the groups of O's or 9Vdo not circulate; 

 so long as this is the case the signs + may occur in the terms 

 in any order. A good example is afforded by the product 

 (1 — q) (1 — q 2 ) (l—q 3 ).,., which Euler proved to be equal to 

 l^ q - q i + q s + q 7_ q u- ttaj 



the general term being (— )»gi( 3 » 2±w ^ so that the product, when 

 q is the reciprocal of an integer, is evidently irrational. 



It will thus have been seen that the method is applicable to a 

 considerable number of series, the irrationality of which is not 

 seen otherwise. Lambert's principle, however, can be applied 

 to a good many cases by means of the following formula? : — 



1 + 



Co- 1 + &c. 



A + J- + JL, + ... = _L_ , <»> 



2 



+ 1 



+ 1 + &C. 



which are proved at once, since 



1 * 



a, a x a 





1 



so that the nth. convergent to the fraction is identically equal to 

 the first n terms of the series. 

 From (2) we have 



i + i. + ^+i+;;.i+-i— -' (3) 



r* rP ft a r a 



r _ _^ 



rv-P + l-^&a 

 since ^fc^„r0~ a , ? y = r «,^-«. 7 .Y-^ &c. ; 



so that if «, ft, 7, . . . are such that the differences ft — a, 7 — /?/.-.* 

 after some point always increase, Lambert's principle is appli^ 

 cable and the series is irrational — the same result as was obtained 

 above when the series was written 1 + qW 4- q®W + . . . The 

 same reasoning also applies when any of the signs are negative } 



2 



