198. Mr. J. W. L. Glaisher on Arithmetical Irrationality, 

 whicli by (1) 



r-1 



1 + -i-ifr r- 1 



r 4 — 2 + 



2 + &c, 



to which also Lamberts principle applies. 



I may mention that Eisenstein (whose chief results have been in- 

 cidentally reproduced above) has enunciated his theorems without 

 demonstration, and evidently intended to return to the subject, 

 though after an examination of his subsequent memoirs I feel 

 pretty certain he never did do so. He states that he was in posses- 

 sion of a more general theorem; andl think it likely that the result 

 he alludes to was that marked (3) above, or some other continued 

 fraction of kindred scope, from which the irrationality of a good 

 many series could be deduced. Eisenstein must have been aware 

 of the irrationality of 1 + q + q s + q 6 + . . . > as appears from his 

 theorems with reference to series of a similar kind given two 

 years previously ; but, curiously enough, he has omitted to state 

 it as a conclusion to be deduced from the continued fraction into 

 which he transformed it; and at the time of writing the paper 

 of which the abstract appeared in the British-Association Report, 

 I had not seen his previous papers, and in fact did not know he 

 had considered the subject of irrationality at all; so that I drew 

 the inference which it was probably merely an accidental omis- 

 sion that Eisenstein had not himself pointed out. The method 

 explained in this paper, however, is far more simple and appro- 

 priate in such cases. 



I may mention that Eisenstein also gave another continued 

 fraction for the series with squares as exponents, viz, 



14- 1 I * + 1 4- 1 1 



r 



r— 



: &c.' 



resembling that given by him for 1 + ■— ' + '-$ + ;-g + . . . 



It is interesting, in conclusion, to note that the method ex- 

 plained above gives the proper result for the geometrical series 

 1 + Q a + 9 ,2a + 2 3a + • • • > which would be written in the scale of 

 radix r l'OO . , . 100 ... 1 (a — 1 ciphers in each group), which 



does circulate, as of course it ought to do, since the sum is = . 



3 ° " ' ' 1— q a 



Cambridge, February 11, 1873. 



