of an Elliptic Transcendent Identity, 443 



"17. (l+2# + 2# 4 H-...)(.r* + ^ + ..0 = (* i + * f +---) 2 - 

 "18. (l+2# + 2* 4 + ...) 2 +(2**-}-2^+...) 2 



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

 "19. (l+2# + 2# 4 + ...) 4 =(l-2# + 2* 4 -...) 4 



+ (2a* + 2a* + ...) 4 " 



But for the word Beweis at the beginning, it would appear as 

 if this was merely a list of five equations allied to one another in 

 form ; and even with the heading it is not easy to see at once 

 what the property proved is, or how the proof is effected. The 

 identity, however, the mode of demonstration of which is 

 sketched in these formulae, is the elegant theorem 19, which 

 results from substituting for k and k' in terms of q in k? + k t2 = \ ; 

 and the four formulae 15-18 are subsidiary results required in 

 the process. As an algebraical demonstration of 19 is valuable, 

 I proceed to expand the proof the steps of which are thus briefly 

 indicated by Gauss in the above extract. 



The truth of 16 is very readily seen ; for all the terms on the 

 left-hand side involving an exponent which is the sum of an even 

 and an uneven square vanish, while the sum of two even squares 



4m 2 + 47r = 2{(m + 7i) 2 + (wi-?i) 2 }, 



and the sum of two uneven squares 



(2m + l) 2 +(2n + l) 2 =2{(m + ?i-fl) 2 + (m-w) 2 }, 



which together give the doubles of the sums of all pairs of 

 squares ; and the accuracy of the coefficients is evident on consi- 

 deration. 



Formula 16 being thus established, for brevity write 



a= l+2tf 4 + 2tf 16 -K.., 



/3=2* + 2a 9 + 2# 25 -r...; 

 and let c& x and ft x represent respectively « and ft with x* written 

 for x; then 16 is 



(«-/3f+(«+/3)3=2(« 1 +,9 i )3, 

 or 



«"+0»* («, + &)« (A) 



Herein write xs/ — 1 for x, and ft 2 becomes — ft 2 , so that 



J-^^-^Y, (B) 



which, writing a 2 — /3 2 as («— ft) (u-\-ft) y is 15. 

 By subtraction of (B) from (A), 



^ 2 =2« 1 /3 1 , 



which, on writing x* for x, becomes 17. 



