in Rectangular Order upon a Medium. 491 



The analytical question is accordingly that of the expansion 

 of log 1 (x) in ascending powers of x. Now, Jacobi * has 

 himself investigated the expansion in powers of x of 



6 x [x) = 2 {gW sin x-q^ sin 3x + q 2 '^ sin 5x-...\, (37) 



where q= e -" K '' K (38) 



So far as the cube of x the result is 



d^ =i -£{ 3KE - (2 -* 2)K2 } + ---> < 39 > 



D being a constant which it is not necessary further to specify. 

 K and E are the elliptic functions of k usually so denoted. 

 By what has been stated above the roots of 0i\x) are of the 

 form 



7r(ro + m'iK'/K); (40) 



so that 



Z{m + im'K'/K\- 2 =$\3KE--(2-k 2 )K 2 }, . (41) 



the summation on the left being extended to all integral 

 values of m and m, except ra = 0, m' = 0. 



This is the general solution for 2 2 . If K'=K 7 & 2 = J, and 



Z{m + im'\- 2 = 2\2Kft-K 2 \ = >7r, 



since in general f, 



EK' + E'K-KK'=i7r. 



In proceeding further it is convenient to use the form in 

 which an exponential factor is removed from the series. 

 This is 



in which • • • • v / 

 A= 2K B= 2E_^2K 



7T ' 7T 7T 7 V y 



So =/3, , 1=a /3, S2 =/3(« 2 -2^), s 3 = a /3(« 2 -6/34), 

 the law of formation of s being 



s w+1 = 2m (2m + 1) /3 4 s TO _! + */3 dsjdfi - S^dsjdci, (44) 



* ' Crelle,' Bd. 54, p. 82. 



t Cayley'9 ' Elliptic Functions/ p. 49. 



