852 
MR. A. R, FORSYTH OH THE THETA-FUNCTION'S, 
2 %p 0 =s 0 p 0 +‘s S 0 (-1 )^p 0 +‘s' is 0 ( - i)*" + ^p 0 
t — X 
t=r s=ru=r 
+ S S SS 0 (-1) 2 ^^“P 0 +.(229) 
t=ls=l w =l v 
in which 
and the summation on the right-hand side is taken for all values of the /Ps, without 
restriction, from — oo to + 00 ; the factor 2 r is prefixed to the left-hand side because 
there would remain 2 r terms of the initial value of 2 0 P 0 were the right-hand side 
written out at full length. Now 
P 0 =general term in IKE> 
( —1) 2 mi P 0 = „ „ nd> 
Aj» Ag, • • • j A,, • • • > A r 
N lr N 2 ,..., N w .. ., N. 
Aj -f-1, At,,. . ., A S} .. . , A 
Xj ,n 2 ,...,n w ...,n 
X„ x*... 
Xi,x 2 ,. 
, X,. . . , X,} 
..,X„...,X,} 
and therefore 
/Ai, A 2 , i > ■ , A s -j-1,... , A 
IAN,, Nj,..., N, .... ,N,/ Xl ’ X » ■ • • > X - 
A 1? A 3 ,. .., A, + l, A ( + 1,. . . , AAy .y. 
Nj, N 2 ,.,., Nj J, 
+ 2ik> 
*=i 
*=ri=r 
8=1 t=l 
«=»” t=r u=r _ J / A ls A 2 ,. .., A s -f 1, A*+l, A„+l,. .., AA v v 
+ !i2iii M lW 1) N 2 ,...,N J ,N, ,N m 
, x,j 
+ 
.,x, 
(230) 
where in that expression on the right-hand side which has k of the upper row of 
r\ 
numbers in its characteristic of the form A s +1 there will be ’ - independent terms, 
and these of course are the only ones to he included in the “ k” tuple summation. 
Thus 2C£ 0 P 0 is equal to the sum of 2 r products of four functions, to each product being 
prefixed a positive sign. 
Again 
2^P,=SP t -^(-l)^+S r ^(-l)^P,-¥s("l)^ + ^P < 
8=1 8=1 
+ s ’¥ s (- i ) s ^ p t “ ... 
8=1 M=1 
(231) 
