856 
n^> 
MR, A. R. FORSYTH OH THE THETA-FUNCTlOHS, 
Xi, X 2 ,. . . , \ t 
*1» /v 3 , 
v v 
VrJ 
'l+'sntl'" V..., x,+i,..., 
7 t =1 \V V Vo, .... v t , . . . ,v, 
_|_ S g" g r n<jA’ ^2» * • * ’ ^+lj • ’ • > A;+ 1, . . . , \A 
v v . • • I v s , . . . , Vf - 1 
s=lt =1 
V. 
j ^ "t b A 2 -f1, X s 4 1) • • • j Xs,. . . , Xfo . . . , \ r +1\ 
s=l i=I \^i j v 2 > ^3 , . . . , V s , . . . , V(, . . . , v r J 
+ Sn$(^ +1,>,+1, * ‘ * ’ Xw ‘ ’ * ,Xr+ 1 \-\-ji<$>( x i + 1 ’ x z +1 ’ ■ 
\ v i 
;same expression with A written for X 
N 
> •'a 
, X,. +1 
, V r 
^|throughout 
(235) 
ft /// /// 
... X r, X i, X o 
. . . X 
the variables on the left-hand side being 
Xu tTgj « • • Xf } X X 2) < • . X r} X u X 
and the variables on the right-hand side being 
X 1} X 3J . . . , X,, X' l3 X* . . . , X' r> X" l3 X" 2J . . . , X",, X'" ls X\ ...J' 
Again 
/x 15 x 2 ,. 
■ • * ■ 
. , X^ + l, . . 
..,x,\ 
■ ■ 5 Vt, . 
■ • , V r ) \V V V % , . . 
.,v t , . , 
■ ■ , V r ) 
+ sn$( 
% X 2 , .. 
. , \ +1, . , 
■ * »X,\ 
S = 1 
\»1’ v» . . 
■ J V'f , . 
. . ,v r ) 
— s’ n<f>( 
^X 15 X 2 , . . 
• s x,-f lj . 
> • »X<+1> . 
S=1 
\v Xi v z , . . 
• y Vs 
■ • j Vt t ■ 
s=r u=r /\ 
+ S SnW 
« = 1M=1 \ V V V % 
... -ssW^ 1 '^ 1 ’ 
S=1 M=1 
Xo, . . . , X 4 + l, . . . , X„-f-l, 
V s , . . . , V u , 
- J Xj, . . . , X M , 
• % V#> . . » * V<i/* 
Vo 
+ Sm>^ 1+1 ’* ,+1 ’ 
s=l 
Vo 
s=l 
-¥iW Xi+1 ’ Xa+1, 
Vi 5 v. 2 , 
Oq + 1, X 2 +l, . • 
\1>| } v 2 , • • 
+n$ 
..., ... 
• • • > V h . . * 
. . . , X,. + l\ 
. ... Vr 
> x, 
. ^ 
...V 
• •, v, 
. . , X,.+1\ 
. . ,v r ) 
, X, + l\ 
»*v / 
X^, . . . , X,.-J-1\ 
V-r 
—n<T( 1+1, * • * ’ v+i N 
J \ v l >V>2 > • • • , V r 
(236) 
— 1)H A!+A,(+V ' i+V "'}[same expression with A written for X and N for v 
throughout except for v (i for which write N*-}-1; 
the variables being as in (235)] 
This equation (236) is equivalent to r equations by substituting in succession 
s—r 
t=l ) 2, 3, . * , , r ; and a summation indicated by S implies that s is to have every 
S = I 
integer value 1, 2, 3, ... , r, except t. Again 
