804 
MR. A. R. FORSYTH ON THE THETA-FUHCTIOHS, 
$n 3 k„ 
V KgK, 
{y/ft 3 (l — a?i)(l— ■%(! — ^ 2 )(1— ig^Xl — jcfaz)} 2 
( t ? 1 —A’g ) 3 
■V _ *9*8 
£ 12 3 « / 3 /c , 3 K 2 K 3 
?aAi( 1 — ^Xl — ^AiXl — «9^a)(l — * 3 ^ 2 ) + V 7 ^ 2 (1 —^ 2 )(1 —/Ci 2 ^ 2 )(1 —/c 2 2 ,^i)( 1 —Ay^)} 2 
V _ K \ 
V~~ ^ / 2« / 3 K 2 K 3 
{ a?i(l — %)(1 — gjfcgXl —/c 3 X)(l— ^ 3 %)+ \/^ 8 (l— a^)(l — /c^agjXl— /c 2 2 a; 2 )(l — /c 3 8 .a? 2 )} 3 
(aq —. 0 3 ) 3 
which correspond with Rosenhain’s formula (97). 
17. It is now necessary to find relations between x l3 a ? 3 and x, y. Let 
dX 
dX 
dx 0 
dX 
, dX 
d V o 
° 7 ~¥o 
where ~ imply that, after the differential of the function has been taken, both 
ax 0 cly Q 
the variables are to be put zero. Differentiating the equations (49), (50) with regard 
to f and then putting f, 77 zero, and noticing that 
d f(%+£) _ df(x+% ) 
d% dx 
we have from (49) 
df{x - g) _ _ df(x — g) 
dg “ dx ’ 
or 
CgCg ^13 ^ 5- 13 — C 6 C 7^3^3 C 4 C 6*VK 
$ 0 $1 
d A 3 \ 3 0 X S 3 4 
C ^9 ) =C ^ 3 ^ o- 
Ll/tO \ / ry l O 1 g rvAio rvA-j 
and similarly from (50) 
'12 "12 
-9. X 
'12 "12 
X X 
°9 C 13^ (^ 9 )~ C 0 C 5 ~ C 2 C 7$“ $ 
aX\r? 21 / r? 12 r? 12 ^12 
(53) 
(54) . 
Differentiating the same equations with regard to y and proceeding in the same 
manner we obtain 
