704] 
29. We have 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
The constants of the theory. 
AO = 1 + 2q + 2q i + 2g 9 +..., 
477 
B0= 2q? + 2q î + 2q™ + 
CO = 1 — 2q + 2q i — 2q 9 + ..., 
DO = 0, 
D' 0= — Tr{(f — + 5q ? ^ — ...}. 
If, as definitions of k, k', K, we assume 
J № C 2 0 _J4) m 
A 2 0’ ^ ~A 2 0’ £0 ' CO ’ 
then we have 
k =4 Vq 
- f 1 + q 2 + q 6 + ... 
[1 + 2q + 2 ç 4 + ...J 
, =4 Vg(l — 4>q+ 14g 2 + ...), 
k' = 
ii+tw+fiÎ . = 1 - 8 Ï + - 96ÿ + .... 
jsf _,r(l + 2 g + 2g« + ...)(l-3ÿ + 5g'-...) , „ + 4 „. ,<>„» + , 
2(1-2j + 25*-...)(1 + 9 s + ?‘+...) ’ 1^0 + »? + «« +°2 + ••••>’ 
where I have added the first few terms of the expansions of these quantities. We 
have identically 
k 2 + k' 2 = 1. 
It will be convenient to write also, as the definition of E, 
C"0 
K (K — E) = 
CO * 
we have then 
E = K-l C "° 
moreover, 
giving 
and thence 
30. Other formulæ are 
k = 4 
K CO ’ AO. BO .CO. D'O ‘ A2 ° + 520 • 00 • ’ 
E _ 1 C"0 _ 2it 2 q — 4ç 4 + 9ç 9 — ... 
~K~ K 2 ‘ <70 ’ ’ = if 7 ' T - 2q + 2q*+ ... 5 
~ = 1 -8q + 4<8q 2 - 224 i q 3 + ..., 
E = \ r n {1 — 4g + 20g 2 — 64ÿ 3 +...}. 
- (1 + q 2 .1 +g ,4 ...| 4 
k' = 
1+ q. 1 + q 3 .. 
11 — q . 1 — q 3 ., 
(1 + q . 1 + q 3 ...j ’ 
[1 + q . 1 + q 3 ... 1 — q 2 . 1 — q* ...} 2 
[1 — q . 1 — q 3 ... 1 4- q 2 . 1 + q 4 ...}