1 J 
480 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
35. As an instance, taking a , a , = ^^, the product-equation is 
^ ] (u + u'). ^ ? (m - m') = © l (2u) . © J (2m') + © f (2w). © | (2%'), 
[704 
0 
= ¿0 J (2m). © 2 (2m') - ¿0 I (2m). © g (2m'), 
= iP.P' 
which agrees with the before-given value. 
- iQ • Qf, 
36. The following values are not actually required: but I give them to fix the 
ideas and to show the meaning of the quantities with which we work. 
u = 0 
X = © q (2u) = 1 + 2q 2 cos 27m + 2q 8 cos 4>7ru + ..., 
Y = © q (2u) = 2q^ cos 7m + 2q% cos 37m + ..., 
X t = © ^ (2m) = 1 — 2q 2 cos 27m + 2q 8 cos 47m — , 
a =1 +2q" + 2q 8 + ..., 
ß = 2^ + 2^+..., 
ol / = 1 — 2q 2 + 2q 8 — ..., 
Y t — © ^ (2it) = — 2q* sin 7m + 29^ sin Sttu — ..., j ß' = 2ir ( — q* + 39^ — ...) 
= -=- Y for u — 0. 
du ' 
P = © 0 (2m) = 9“ (cos ^7tm + 4 sin \tt%i) + 9* (cos §7m — i sin f 7m) 
4- 9 ’ 5 " (cos f 7TM + 4 sin 17TM) + ..., 
3 j , . 9 
Q = © * (2m) = q s (cos '2 7TM — 4 sin |7rM) + 9 ? (cos §7TM + 4 Sin f 7m) 
+ 9^’ (cos f 7tm — 4 sin 17m) + ..., 
P t = © ^ (2m) = ~^1-19^ (cos \mi + i sin \ im) — 9“ (cos §7m — i sin f 7m) 
1 — i [ X. 
— 9^’ (cos §7TM + 4 sin f 7TM) + ... j , 
Q / = © y (2m) = —-y- 4 -19^ (costu — i sin \ntu) — q% (cos §7m + i sin f 7tm) 
1 V2 ( 
— 9" s " (cos f 7m — i sin |7tm) + ... j ; 
and therefore also 
p = q = 9* + ç* + 9'^ + 
1+4 
P/ = 
V2 
9! _ 9I _ + 9-V- + 9V- _ _ ... j, q y = -^=-j Do.j ; v, = i( \r