AND DOUBLE THETA-FUNCTIONS. 
907 
u+u' U—vJ u+u' U—vJ 
B . A - A . B 
C . D + D . C 
= function 
(O, 
where for brevity the arguments are written above; viz., the numerator of the 
fraction is 
B (u + u) A(u — u )—A ( u + u) B (u—v !), 
and its denominator is 
C(u+ u)D(ii—u^+D^u+u^Ciu— u ')- 
Admitting the form of the equation, the value of the function of u' is at once found 
by writing in the equation u— 0; it is, as it ought to be, a function vanishing for 
u'— 0. 
18. Take in this equation u' indefinitely small; each side divides by u', and the 
resulting equation is 
AuBfit— YnoA'u 
QiiDu 
— const. 
where A'u, B 'u are the derived functions, or differential coefficients in regard to u. 
It thus appears that the combination Av+>'u — ¥>uA!u is a constant multiple of CADm: 
or, what is the same thing, that the differential coefficient of the quotient-function 
y- is a constant multiple of the product of the two quotient-functions y- and 
JAfU> JAU JAW 
19. And then substituting for the several quotient-functions their values in terms 
of x, we obtain a differential relation between x, u ; viz.: the form hereof is 
dn— 
M clx 
y/ a—x.b—x.c—x.d—x 
and it thus appears that the quotient-functions are in fact elliptic-functions: the 
actual values as obtained in the sequel are 
snKl(= — —yDu-r Cl£, 
V D? 
cnK u— \/j B u -F Cw, 
dnK?«= y/k'Au + Cu ; 
and we thus of course identify the functions A u, B u, C u, D u with the H and © 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 
6 a 2 
