484 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704 
43. We can from each set form two fractions (each of them a function of u + v! 
and u — u'), which are equal to one and the same function of u! only: for instance, 
Y' 
from the first set we have two fractions, each : putting in such equation u = 0, 
we obtain a new expression for the function of u' involving only the theta-functions 
Aii!, &c., which new expression we may then substitute in the equations first obtained : 
we thus arrive at the six equations 
u+u'u—v! u+u' u-v! u+u' u—u' u+u' u-u' 
G.A-A.G D.B-B.D Du'.Bu' 
D.B + B.D~C.A+A.C~Cu'. Av! * 
B.A-A.B D.G-G.D Du’ .Gu 
D.G+G.D~~ B.A+A.B~ Bu'.Au” 
B.G—G.B_D.A — A.D_ Du’ .Av! 
D.A+A.D~B.C+G.B~ Bv! . Gu’ ’ 
where observe that the expressions all vanish for v! = 0. 
44. Taking herein v! indefinitely small, we obtain 
Au. C'u — Gu. A'u Bu. D u — Du. B'u D O . BO 
Bu. Du Gu . Au GO. AO 
An. B’u — Bu. A'u _ Gu. D'u — Du. C'u _ D'O . GO 
Gu. Du Au . Bu AO . BO 
Gu. B'u — Bu. C'u _Au. D'u — Du. A'u _ D'O. AO 
Au . Du Bu. Gu BO. GO 
where the last column is added in order to introduce K in place of D'O. 
45. These formulae in effect give the derivatives of the quotient-functions in terms 
of quotient-functions : for instance, one of the equations is 
d Du _ £ B u Gu 
du Au Au' AiG 
= -K 
= -K 
= -K, 
B* 0 
A* 0’ 
C* 0 
A*6' 
substituting herein for the quotient-fractions their values in terms of x, this becomes 
d /d ~ x _ Ir /^8® 7b — x.c — x 
du v a — x v 213) a — x 
= -K 
7 
f 7b — x . c — x 
a — x 
or the left-hand being 
this is 
Kdu = 
— £f dx 
(a — x)% 7d — x du ’ 
Vaf. dx 
7a — x.b—x.c—x.d—x 
where on the right-hand side it would be better to write 7— af in the numerator 
and x — d in place of d — x in the denominator.