AND DOUBLE THETA-FUNCTIONS. 
913 
giving 
1 —8g+48^ 3 —224g 3 + • • , 
and thence 
E = i7r{l-4?+209 2 -64(/ 3 + . . }. 
30. Other formulae are 
l=A^/q 
l + g\l + T.. 
1 + q.l Ay 3 . . 
4 
k'= 
1-q.l-q*.. ]* 
1 + q.l + qT. J ’ 
K= 
f l + g.l + y 8 .. l-g s .l-g*.. I 2 
2 [l— ^.1 — g 3 .. l + ^.l + y 4 .. J 
31. Jacobi’s definition of q is from a different point of view altogether, viz., we 
rlv 
K 
have q= exp. —where 
K= 
d(f> 
)X /1 -k* sin 2 0 5 
ttK' 
and K' is the like function of k'; the equation gives log q— — ; viz., we have 
K = log q, 
and, regarding herein K as a given function of q, this equation gives Eh as a function 
of q. 
The product-theorem. 
32. The product-theorem is 
J 
(«+<)•£( yR 
to 
■U'): 
:ed ( “ + 02,,ef i( ““2')2 !1 '+ef Ka+ h' +1 h«-e(' i( “““ ,, . + 1 
7+7 / \ 7”7 / V 7+7 / \ 7-7 
and here giving to , their different values, and introducing unaccented and 
accented capitals to denote the functions of 2 u and 2 v! respectively, the 16 equations are 
MDCCCLXXX. 6 B 
