Mr. J. W. L. Glaisher on Arithmetical Irrationality. 397 



gives 



. 11 ,1 

 1--T- + -5— ...=1 + isr 



11 r * J : r 2/3 



+ r y_,* + &C.' 

 which on dividing out superfluous powers of r gives the same 

 form as (3). 



The product -^ — ^^ — ^-^ — ^ — is Known to be equal 

 1 (1-g) (l-? d )(l-2 J )... 



to l + + g 8 + § 6 + g lo + ... (the exponents being the triangular 



numbers), and this latter series Eisenstein (Crelle, vol. xxix. p. 96) 



developed into the fraction 



1 



1 













i - 



— T , 







a 



r 



1 



- 1 



r 2 - 



-r 2 

 -&c. 



(q= — ), whence (in the British-Association Report, 1871, Trans. 



Sect. p. 16) I inferred its irrationality by means of Lambert's 

 principle. The irrationality is evident at sight by the principle 

 explained in this paper ; also (3) gives us the means of expand- 

 ing the series into another continued fraction, to which also 

 Lambert's method applies, viz. 



i + I + i + i + ...=i + 1 



/v*fj i^»0 y 



V r 2 +l~ 



r+i 



r 4 + &c. 



In another memoir (Crelle, vol. xxviii. p. 39) Eisenstein has 

 developed Euler's product into a continued fraction as follows : 



l+r+- 



l—r 3 



l+r 2 ~&c. 



(the alternate denominators being l — r, l — r 3 , l—r 5 , ... and 

 1-fr, 1-H* g , 1 + r 3 ,..., and the corresponding numerators 1, 

 r 2 , ? A . . . and 1, r, r 2 . . . ), whence he has inferred its irrationality. 

 Formula (1), however, affords the means of obtaining a still 

 simpler form for the product in question as a continued fraction ; 

 for by a well-known theorem, 



1 



+ .... 



(r-l)(r»-l)(i*-l) 



