PARTICULARLY THOSE OF TWO VARIABLES. 
813 
% r TT 
24. Writing log r= ^ log p, then 
2K 
m= oo n=on 
= t % 
W= — 00 W- - — GO 
= X (— l) w g"4—e 
(2ra+n) 2 (2n+v)» (£«,+*)»*■? &m+ii)iirf„j_ 0 n+v w A 
(_] yi^npp 4 4 g 2A e 2 IC V 2 / 
(2n+<.)5 (u+v)wry ^ / , 2w + I/, ,\ 
®+-^— l °gP • ' * ' 
• ■ (91). 
which is, in effect, Rosenhain’s definition of the double theta-functions. Taking a 
particular case, 
n = co {ir?/ 
£ 0 = 2 2 A^ 0i0 (x-f log p) 
n.= — oo 
=0o,o( x )+y cos 2 y^S e o,|§h- iogp)+^o,o(^— lo g p')} 
-fg 4 cos 4y ^{<9 0i0 (x+2 log p') + ^o,o(«— 2 lo g *>')} 
+g 9 cos $y^{e 0t0 {x+3 log p)+0 OyO (x—S log p)} 
+ . . . 
+iq sin 2y^{0 OtO (x+ log p) — 0 o , Q (x— log p )} 
+^ 4 sin 4y^{0 OtO (*+2log p^-O^x-Z log p)} 
+k 9 sin %^{0 o ,o( a; + 3 lo £pO — ^ao^— 3 lo &/>')} 
+ • • • 
Expanding by Taylor’s theorem and re-arranging, this gives 
—0o,oW 
77" 77" 
l + 2gcos2y—-f 22 4 cos4y—+2^ 9 cos6y~-f . . , 
2A 
7r 
2A 
+* 1 ogp'^^r22sm 2y^+2.2j 1 sm42/^+3.22 9 sm62/^+ . . . 
ifo? [_ 
+(logp')i^ } 
2gcos22/^- + 2 2 .22 4 cos42/^ r +3 2 .2^ 9 cos6y^ v H- . . 
2A 
7T 
2A 
2 ? sin 2y ^-+2 s .2g*8in 4y ^+3 3 .22 9 sin 6 y ^r-f . 
7r 
7T 
2A 
2A 
2A 
] 
