492 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
[704 
57. In the foregoing formulae, (be, ad), (ac, bd), and (ab, cd) denote respectively 
1, 
x + y, 
xy 
> 
1, 
x + y, 
xy 
y 
1, 
x + y, 
xy 
1, 
b + c, 
be 
1, 
c + a, 
ca 
1, 
a + b, 
ab 
1, 
a +d, 
ad 
1, 
b +d, 
bd 
1, 
c + d, 
cd 
and substituting for 21, 23, (£, 3) their values, and for a, b, &c., writing again a — x, 
b — x, &c., we have moreover 
A 2 u = fc — b .b — d.c — d (a — x), 
A°-v = fi „ (fl-y), 
Bru = Vc— a.c — d.a— d (b—x), 
B' 2 v = V „ (b — y), 
Chi = Va — b.a— d.b — d (c — x), 
C 2 v = V „ (c - y), 
Dhi = Vc — b .c — a. a — b (d — x), 
D 2 v = V „ (d- y), 
A 2 (u + v) = V „ (a — z), 
B- (u + v) = V „ ; (b - z), 
C 2 (u + v) = fi „ (c - z), 
D 3 (u + v) = V „ (d - 2), 
the constant multipliers being of course 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 transformation 
such as 
(b - c) (be, ad) = b — x.b - y, c — 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 to make them expressible as linear 
functions of the squared functions Ahi, &c., and A 2 v, &c., respectively: the formulae then 
give the quotient-functions A (a 4- v) -f- D (u + v), &c., in terms of the quotient-functions of 
u and v respectively. 
Doubly infinite product-forms. 
58. The functions Au, Bu, Cu, Du may be expressed each as a doubly infinite 
product. Writing for shortness 
m + n = (m, n), 
7n 
m + 1 + n . a . = (m, 11), 
7Tl 
m + (n + 1) ; = (m, n), 
7Tl 
m + 1 + (n + 1) —. = (m, n),