AND DOUBLE THETA-FUNCTIONS. 
929 
57. In the foregoing formulae (be, ad) (ac, bd) and (ad, be) denote respectively 
1, x +y, xy ; 
1, a -j- b, ab 
1, c-\-d, cd 
and substituting for @1 their values, and for a, b, &c., writing again a — x, 
b — x, &c., we have moreover 
A 2 u = fie — b.b — d. c — d 
(a — x), 
A+=f 
33 
(a-y), 
r *>> 
1 
1 
o 
1 
o 
II 
PQ 
(b—x), 
B*v=f 
33 
(b-y), 
C hi = fia — b.a—d.b — d 
(c—x), 
C*V=y/ 
33 
( c ~y)’ 
D ~u= fc — b.c — a.a—b 
(d—x), 
D+=f 
33 
(d—y), 
1, x+y, xy 
1, b -j- c, be 
1, a+d, ad 
l, x+y, xy 
1, c + a, ca 
j 1 , b-\-d, bd 
A z (u+v) = f „ (a—z), 
B 2 (u+v) = f „ (b—z), 
C ~(u,+v) = fi „ (c—z ), 
D 2 (u+v)= x / „ (d—z), 
the constant multipliers being of coarse the same in the three columns respectively. 
According to what precedes, the radical of the fourth line should be taken with the 
sign —. The functions (be, ad), &c., contained in the formulae require a transforma¬ 
tion such as 
(b—c) {be, ad)— b — x.b — y, e—x.e—y 
b — a.b — d, c—a.c — d 
in order to make them separately homogeneous in the differences a — x, b — x, c — x, 
d — x, and a — y, b — y, c — y,d — y, and therefore expressible as linear functions of 
the squared functions A hi, &c., and A.+, &c., respectively : and the formulae then give 
the quotient-functions A(u-b'r) -f-D^+v) &c., in terms of the quotient-functions of 
u and v respectively. 
Doubly infinite product-forms. 
58. The functions A u, B u, C u, Du may be expressed each as a doubly infinite 
product. Writing for shortness 
6 D 
MDCCCLXXX. 
