488 
A MEMOIK ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
[704 
The square-set u ± u', v! indefinitely small: differential formulae of the second order. 
52. I consider the original form 
(7-0 C (u + u ) C (u — u') = ChiC 2 u' — DhiDhi, 
which is of course included in the last-mentioned equations. 
Writing this in the form 
n C (u 4- u) C (u — u') _ n , , DhiD 2 v! 
Chi Chi 
and taking u indefinitely small, whence 
C (u + u r ) = Cu + u'G'u -j- ¡¡u 2 C"u, Gvl = (70, 
C (u — ii) = Cu — u'G'u + \u 2 C"u, Du' = u'D'O, 
G (u + u ) C (u - u') = Chi + u' 2 [GuG"u - (C'u)% 
the equation becomes 
cs ° i 1+ {w - {m)}) - ° ! ° + “ ,s i coc "° - ^ mi • 
/C'u\ 2 
C"0 
(D' 0V 
\cf) 
~ CO 
\(70 j 
that is, 
/ ¿I \ 2 ^ Dhi 
viz. we have ) log Cu expressed in terms of the quotient-function , and conse- 
\cLuj u a 
quently Gu given as an exponential, the argument of which depends on the double 
integral fdu idu^J^. 
J J L/“U 
53. To complete the result, I write the equation in the form 
dff 
dti 2 
log Cu — 
C"0 1 /D'OV 1 (B 0\ 2 
(D' 0y 1 fi 
(70 k V(707 + ^V(70 
1 7 Dhi 
G hi 
7y o _ (7"o 
—A i s z=-fkK, and -7ÿ7T is = K(K — E); hence the equation is 
CO cu 
du 2 log Cu - Rl ( X K k C 2 u) ’ A2 ( 1 K A sn2Ru )’ 
or integrating twice, and observing that log Cu and log Cu, for u = 0, become = 0 
and log (70 respectively, we have 
log Cu — log (70 + |^1 — K 2 u 2 — k 2 J duj du K 2 sn 2 Ku, 
which is in fact 
log 0 (Ku) = log CO + | (^1 — gj Khi 2 — k 2 J du j" du K 2 sn 2 Ku,