AND DOUBLE THETA-FUNCTIONS. 
937 
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 correspond 
to the q of the single theory): viz., in a notation which will be readdy understood, the 
constants c, c!", c w , c v of the even functions are 
% exp 
th dj r ti H - /3\ 
.7 S / ’ 
— (m+a) 3 , 2(m+a)(w+/8), (n+/3) 3 , exp 
/m + a, ?i + 8\ 
\ 7 g / ’ 
and the constants c', c!' of the odd functions are 
^rriZ (ra+a), (n+/3), exp 
7?l + U, 'll T" /3 \ 
.7 S /’ 
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 
exp ( — a) = Q i , exp (—A) = H 4 , exp (— b) — S 4 , 
also 
QR~S = A, QR~ 3 S=A 7 , and therefore AA 7 =Q 3 S 3 , 
then attending only to the lowest powers of Q, S we find without difficulty 
$~o{ u ) 
= 
1, and therefore also 
Cu =1 , 
— 
2Q cos \nu, 
<h = 2Q, 
h 
-- 
2S cos \ttv, 
Co=2S, 
$3 
= 
2A cos ^7r(u + y) + 2A / cos v), 
c 3 '— 2A+2A' 
= 
1, 
C*=h 
- = 
— 2Q sin ^7 ru, 
^6 
= 
2S COS TV, 
e 6 =2S, 
= 
— 2A sin ^rr(ii-\-v) — 2A 7 sin \tt(u—v), 
^8 
— 
1, 
Cg = I j 
3-q 
= 
2Q cos ^7 ru, 
C? 
CM 
II 
O 
^10 
= 
— 2S sin ^ 7 tv, 
dll 
= 
— 2A sin \tt(u-\-v)-\-2A.' sin r(u—v), 
s la 
= 
1, 
C l-3 — I> 
*18 
= 
— 2Q sin \ttu, 
*14 
= 
— 2S sin \ttv, 
*15 
--- 
— 2 A cos |77-(M+r) + 2A 7 cos \i t(u — v), 
c i5 = 2A+ 
6 E 
MDCCCLXXX. 
