704] 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
473 
and we thus of course identify the functions Au, Bu, Cu, Du with the H and the © 
functions of Jacobi. 
20. If in the above-mentioned four equations we write first u = 0, and then u' = 0, 
and by means of the results eliminate from the original equations the quantities 
X, Y, X', Y' which occur therein, we obtain expressions for the four products such 
as A (u + u) A (u — u'). One of these equations is 
C-0 .C(u + u) C (u - u') = Chi Chi' - D-uD-u. 
Taking herein u' indefinitely small, we obtain 
GuG"u - (Cu) 2 _ G"0 fD'OA Dhi 
~ CO \Go) G 2 u ’ 
Chi 
where the left-hand side is in fact 
d 2 
dii- 
log Cu, or this second derived function of the 
D u 
theta-function Cu is given in terms of the quotient-function : hence, integrating 
twice and taking the exponential of each side, we obtain Cu as an exponential the 
if Dhi 
argument of which contains the double integral I j - (du) 2 , of a squared quotient- 
function. This, in fact, corresponds to Jacobi’s equation 
@ w = J*Kk' (i - - ft2 / 0 J 0 dw sn 2 u ' 
21. From the same equation 
C-’O. C (a + u ) C (u — u r ) = C 2 uC 2 u! — DhiDhi, 
differentiating logarithmically in regard to u' and integrating in regard to u, we obtain 
(y (xt T 'll} 
an equation containing on the left-hand side a term log X) 7-, and on the right- 
(j \%l *4" 'll) 
hand an integral in regard to u; this, in fact, corresponds to Jacobi’s equation 
©'a 
©a 
+ 2 logrw^ a x = nO, a) 
2 6 © (u + a) v ; 
k 2 sn a en a dn a sn 2 u du 
1 — k 2 sn 2 a sn 3 u 
22. It may further be noticed that if, in the equation in question and in the 
three other equations of the system, we introduce into the integral the variable x 
in place of u, and the corresponding quantity f in place of u, then the integral is 
that of an expression such as 
dx 
TVa- 
- x. 
b — x 
. c 
— X. 
any one 
of 
three 
forms 
T 
x + 
x% 
1, 
a + 
b, 
ab 
1, 
c + 
d, 
cd 
C. X. 
60