292 
ON THE THEORY OF ELLIPTIC FUNCTIONS. 
[45 
or 
whence 
i.e. 
n K' 
4 K . 
mU = iJk e 
C'Y tv 
7r(JT -2 ili) 
4K 
© (u + iK') 
©zt 
*Jk sn u 
H(zz) 
© (u) 
(3); 
and the equations (1), (2) and (3) may be considered as comprehending the theory of 
the functions H (u), © (u). The preceding process is, in fact, the converse of that made 
use of in the Fund. Nova; Jacobi having obtained for snu an expression in the form 
of a fraction, takes the numerator of it for H (u) and the denominator for © (u), and 
thence deduces the equations (1), (2), the intermediate steps of the demonstration 
being conducted by means of infinite series; the necessity of which is avoided by the 
preceding investigation. 
I proceed to investigate certain results relating to these functions, and to the 
theory of elliptic functions which have been given by Jacobi in two papers, “ Suite 
des notices sur les fonctions elliptiques,” Creile, t. HI. [1828] p. 306, and t. iv. [1829] 
p. 185, but without demonstration. 
In the first place, the equation 
cZ 2 2 
did 
- 2 u k' 2 - 
E\d% 
K) du 
+ 2 kk' 2 
dZ 
dk 
= 0 
(4) 
is satisfied by 2 = © (u) or 2 = H (u). It will be sufficient to prove this for 2 = © (u), 
since a similar demonstration may easily be found for the other value. The following 
preliminary formulse will be required: 
k 
dK 
dk 
= E- K, 
h d E _ _ K k 3 
dk ~ k' 2+ k' 2 ’ 
KK' - EK' - E'K - - tt7t, 
which are all of them known. 
Now, writing © (u) under the slightly more convenient form 
©m = 
„iodu¡ 0 dudn 2 u—iu 2 - 
6 41 
E 
K u 
©w = I u \ k' 2 — + k 2 J 0 du cn 2 itj ©m, 
d 2 ©w 
du 2 
d ©ii 
dk 
dn 2 u — g + I u I'k' 2 — jy"j + k 2 Jo du en 2 u 
1 dKk' . , d E r t r ï d j * 
iKF-dT-i u 'dkK +f " du f ° du dk in ' u 
©îi, 
©Zi. 
we have