478 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
[704 
31. Jacobi’s definition of q is from a different point of view altogether, viz. we 
have q = exp. — , where 
K = 
clef) 
o Vl — k 2 sin 2 </> ’ 
ttK' 
and K' is the like function of Jc'; the equation gives log q = — ^ , viz. we have 
K' = — — log q, 
7T 
and, regarding herein K as a given function of q, this equation gives K' as a function 
of q. 
The product-theorem. 
32. The product-theorem is 
& Q + (u - u) 
= @ /H« + a ')\ 2u . © fh ( a a 'A 2v! + © ft (a + a '\ + 2u. © C* ( a a '\ + lN ) 2u'. 
\ 7+7 / \7—7 / \ 7+7 / V 7—7 / 
OL v! 
Here giving to , their different values, and introducing unaccented and accented 
capitals to denote the functions of 2u and 2u' respectively, the 16 equations are 
A. A 
cx 0 /cv 0 
qU + U rr qU — u = 
XX' + YT, 
(square-set) 
B.B 
^ 1 ^ 1 = 
0 ” 0 ” 
YX'+ IF, 
G. C 
— 
»° 2 v ^ J » 
XX'- YY', 
D.D 
+ a 1 
^ | >> ~ ff 
YX'+ XY'; 
G .A 
+ v! ■= 
*,x;+ yj;, 
(first product-set) 
A.G 
^ 1 ^ = 
- 0 ** ^1 9} 
x,x;- yj;, 
D.B 
§ 1 § 1 — 
^ j 
Y t x; +x,y;, 
B. D 
^ ^ ^ 1 = 
^ Q >} ^ » 
y,x; - xj; ;