516 ME. J. W. L. GLAISHEE ON THE THEOEY OF ELLIPTIC FUNCTIONS. 
Put #=0, and 
1— ia? 2 
1 2 n _ir 2 _l -I t 2 ^ 
4 sin 2 x~ 4 • * 2 (1-^ 2 ) - 4a? 2 ^ 2 ^ 1"3^ ; 
while 
“ 4 * 2 24! 
w e' + e ' 
2^ 2 [t z -e-~] 
= ±(l . 
! 4« 2 y 2 j «, 2 3 |«. 2 / 
so that 
1 _i i _ 
“ 4a? 2 ^ 2 4 jU. 2 _ 
7T 2 7 t 2 f c' + e-*’ e 2l ' + e~ 2 ‘' 
2 ^+ !+<**+ 1 + e ; v +& c - — ai ^ ^ ^_ e -,)2 ( e 2,_ e -2^a 
+ &c 
•} 
+ e~ av )(l + 2 e~ 21 ' + 3e~ 4v -j- 4e~ 6v +&c.) 
_ ( e -** .j. e~ 6v )(l + 2e~ iv + Se- Sv + ±e~ l * v + &c.) 
+&c. \ 
, 7T 2 7T 2 f e~ v 3e~ 3v 5e~ 5v 0 
1 +e -v + 1 +C -3V+ 1 + c -5» +«C. 
_ 1 tt 2 tr 
~ — fJ? 
pH IP +&C. 
l+e* 1+e ^ 
•} 
whence, on integration with regard to 
„ P 3^2 
-^-log(l + e^)-log(l+e- 3 ^)-&c.= — ^ 4 - -l°g(lH-e *)— log(l + e 14 )— &c . 
viz. 
+ const., 
6 24(i_j_ e -^(i . . . =C . e 24 ^(l +e ^)(l + e“ ^ ) 
and C=l, as is seen by putting ; so that (52) is established. 
The other identity (51), or rather the generalization of it, 
4 / nr C ^ nr'Y' Anrr 'i 
e~ l2 +e~ (x ~* )2 -\-e~( x+tL)2 -\-tkc. = — < l+2<?~^cos — +2e - ^ cos — +&c. > (53) 
(which is much more difficult to prove by elementary methods than any of the identities 
discussed in this paper), I deduced by algebraical processes from the equation in § 12 , 
viz. from 
a a a 
,x 2 + ffl 2 ^"" (x—^ + a 2 ' (x + p) 2 + a* 
+&c. = -<M + 2 e~~cos — + 2 e" 
- 0 
COS h&C. 
V- 
(54) 
in the Philosophical Magazine for June 1874 (ser. 4, vol. xlvii. p. 437 et seq.) ; but it 
perhaps is worth while to note here what is the most natural way of obtaining it from 
