442 Mr. J. W. L. Glaisher on the Verification 



whence 



Jo 



dx o « 2 7T 



and taking a?= -> 



x 



f 



the well-known result. 



It is scarcely necessary to remark that from (6) we at once 

 derive 



a formula, however, which for obvious reasons cannot often 

 admit of useful application to definite integrals. 



The elliptic-function identities that are given in the Funda- 

 menta Nova would, as a rule, be very difficult to verify in an 

 elementary manner; and the facility with which results of a 

 purely algebraical character, but which seem to transcend the 

 ordinary methods of algebra, are established affords one of the 

 most obvious illustrations of the great power of elliptic functions 

 as a branch of analysis. 



Of course some of the identities, such as those on p. 103 of 

 the Fundamenta Nova, are very easy to prove by mere expan- 

 sion ; but there is a very interesting one of the more difficult 

 kind which was verified by Gauss. 



This verification is to be found under the title ' Zur Theorie 

 der neuen Transscendenten,' Werke, vol. iii. p. 446; and the paper 

 containing it consists of a collection of seventy-three formulae 

 relating to elliptic functions (or, as they are there described, re- 

 lating to the arithmetico-geometric mean), with here and there 

 a few words of directions, and is rather the materials for a me- 

 moir than a memoir itself. It was taken from a note-book of 

 Gauss's, "finished April 28, 1809," and first published after 

 his death among his collected works in 1866. 



At the bottom of p. 447, immediately following fourteen for- 

 mulae of a different kind, there occur the following : — 



" Anderer Beweis 



"15. {l-2x + 2x 4 -...)(l+2x + 2x 4 + ...) 



= (l-2^ + 2* 8 -...) 2 . 



"16. (l-2# + 2* 4 -...) 2 +(l+2 < r-f2,* 4 + ...) 2 



=2(l + 2# 2 +2# 8 +...) 2 . 



