PARTICULARLY THOSE OP TWO VARIABLES. 
789 
5. In the general definition of <£> substitute 
v—e^ 
w=e 2 a 
so that 
(5) 
( 6 ) 
J / P\ x y\ — ^ ^ / -^\m\+iipgl^(2?M+M) 8 log/3+(2//+O a log q+2(M+iJ-)(-2n+v) log r j + 2 |(2w+^)jj+(2m+v)^ j 
’ \H<> V J ' ^ J m=-oo ' 
Obviously 
<I> 
{fcMHCtfr+M 
icc, 2/ + 4A 
^ V 
00 
so that 4K and zero, zero and 4A, form two pairs of actual periods, conjugate in 
x and y, for d>. 
Since 
77-2x2 
e~4K2 
2 x 2 f /A n\ I W-J# n—x> 1 c 7TSS 2 m+/x ") 3 . r .n-y 2 m+/u, ) (2n+v) 2 
i^ 3 >J ( y l = 2 2 { — l) mX+np e^pV2K+-2r^p} +^){*2S+-a“iog**}+- T - io g s 
l\ r ’ / J 7)i= —OO 7l = —30 
and the right-hand side is unaltered by writing 
and 
cc+^logjp for x 
log 7’ for y 
4K 4A 
r log p, —r log r form a pair of quasi-periods for <&, conjugate in x and y. Again 
7 T% TTX 
7 r 2 ?/ 2 f /\ n\ 1 Qn=co n =oo i c py 2?i+v ->2 , y 1 rx 2n+v , ■> (2m+u.) a 
e“4A2I^<I>J( £ £ (_.l)™*+ w Pe^p2l+~2 logff} +(2 W H-M){*. iI H— 2 - log 5 -}+—— log; 
l \ f ^> V ) J 771= — X 71= — 00 
and the right-hand side is unaltered by writing 
4K 
as-1- 
and 
xJ r~ log t for aTj 
2/+^ !og 2 for Vj 
4K 4A 
so that —. log r, —. log g form another pair of quasi-periods for <5, conjugate 
in x and y. 
