A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
71. All these constants are in the first instance given as transcendental functions 
of the parameters, or say rather of exp. a, exp. h, exp. b, which exponentials cor 
respond to the q of the single theory: viz., in a notation which will be readily 
understood, the constants c, c'", c iv , c v of the even functions are 
in-fa, 11 + ß 
7 8 
^ I in + a, n + /3\ 
2eX H 7 8 
/ 
- r 2 l (to + a) 2 , 2 (to + a) (11 + /3), (n + /3)-’, exp. ( 
and the constants c\ c" of the odd functions are 
i -s' / , x / . 0 \ fm + a, n + /3 
|7rt2(w + a), (n + /3), exp. ^ g 
72. It is convenient for the verification of results and otherwise, to have the 
values of the functions, belonging to small values of exp. (—a), exp. (— b); if to 
avoid fractional exponents we regard these and exp. (— h) as fourth powers, and write 
also 
exp. (— a) = Q 4 , exp. (— h) = R 4 , exp. (— b) = S 4 , 
QR 2 S = A, QR~ 2 S = A', and therefore AA' = Q 2 S 2 , 
then attending only to the lowest powers of Q, S we find without difficulty 
'A, (w') = 1, and therefore also c 0 = 1, 
Si = 2 Q cos 7m, 
So = 2S COS 17TV, 
5 3 = 2A cos £ 7r (u + v) + 2A' cos r(u — v), 
5 4 = 1, 
5 5 = — 2Q sin \ntu, 
5 6 = 2S cos tv, 
5 7 = — 2A sin\ir (u + v) — 2A' sin \nr (u — v), 
5 8 = 1, 
5 9 = 2 Q cos 7m, 
S 10 = — 2S sin \ntv, 
S n = — 2A sin 17T (u -1- v) + 2A' sin \nr {u — v), 
5 12 = 1, 
5 13 =-2Qsin^7m, 
5 14 = — 2S sin \ 7tv, 
5 15 = — 2A cos ^7T (u + v) + 2A' cos \rr{u — v), 
Ci — 2 Q, 
Co = 2 S, 
c 3 = 2A + 2A', 
c 4 = 1, 
c 6 = asf, 
c 8 = 1) 
c 9 =2 Q, 
c 12 — 1, 
c 15 = — 2A + 2A'.