482 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
[704 
Relations betiueen the constants. 
39. The formulae contain the differences of the quantities a, b, c, d; denoting 
these differences 
b — c, c — a, a—b, a — d, b — d, c — d, 
in the usual manner by 
a, b, c, f, g, h, 
so that 
. -h + g — a = 0, 
h . -f -b = 0, 
-g+f . - c = 0, 
a + b + c . = 0, 
and also 
af + bg + ch = 0, 
and then assuming the absolute value of one of the quantities 21, 33, (5, 2), we have 
the system of relations 
21 2 = - agh, 33(5a = 2l2)f, 21bcf = - 33(52), 2133(52) = abcfgh, 
33 2 = bhf, (52lb = - 332)g, 33cag = (5212), 
(5 2 = cfg, 2133c = - (53)h, (5abh = 21332), 
3) 2 = - abc, 2)fgh = - 2133(5, 
c 2 33 2 + b 2 (5 2 — f 2 2) 2 — bcf (af + bg + ch), = 0, 
— c 2 2l 2 . + a 2 (5 2 — g 2 2) 2 = cag ( „ ), = 0, 
-b 2 2l 2 + a 2 35 2 . — h 2 2) 2 = abh ( „ ), =0, 
- f 2 2l 2 + g 2 33 2 + h 2 (5 2 . = fgh ( „ ), = 0. 
It is to be remarked that, taking c, a, b, d in the order of decreasing magnitude, 
we have —a, b, c, f, g, h all positive; hence 2l 2 , 33 2 , (5 2 , 2) 2 all real; and taking as 
we may do, 2) negative, then 21, 33, (5 may be taken positive ; that is, we have 
— a, b, c, f, g, h, 21, 33, (5, - 2) all of them positive. 
40. We have 
A- 0 = a 2 4- /S 2 = 2lf, 
B-0 = 2a/3 = 33g, 
The foregoing equations 
9 
o 
II 
- /3 2 = (5h. 
B 2 0 
7/ C 2 0 
~X 2 0’ 
k A 2 0 ’ 
fcUOl'i-' 
II 
21f ’ 
give