792 
MR. A. R. FORSYTH ON" THE THETA-FUNCTION'S, 
indicating that there are four functions having numbers X, /a, p, v with subscript 
indices 1, 2, 3, 4. Taking the general definition given in (l) for <1>, let 
Mj -f- 2 m 1 =M 3 -{- 2to 3 =M 3 +2 m 3 =M 4 ■+ 2m 4 = m 1 +m 3 +m 8 + 
Ni + 2%! J=Kg+^% = N 3 +2^3 = ^4 +&%*=»1 +% +«3 +% 
2(Ai+Xi) =2(Ag-f-X 3 ) = 2(A 3 -fX 3 ) = 2(A 4 +X 4 ) =X 1 +X 3 +X 3 +X 4 
2 (o-]+/*i) =2(<r 2 +/A 3 ) =2 (ct 3 H-/a 3 ) = 2(o- 4 +/a 4 ) =/^i+/x 3 +/a 3 +/a 4! 
2 (Pi+Pi) =2(P 3 +p 3 ) =2(P 3 +p 3 ) = 2(P 4 +p 4 ) =Pi+p 2 +/ 5 3+Pi 
2(o-' 1 +i/ 1 ) =2(^2+^) =2(cr , 3 -l-v 3 ) =2 (o- 4 +%) =v 1 +*' a -f*'3+*'4 
which contain the assumption that XX, X/a, Xv, Xp are all even; and 
2(X 1 +a? 1 )=2(X 3 +a? 3 )==2(X 3 +^ 3 )==2(X 4( +x 4 )=ir 1 +a? 3 +a3 3 +^ 4 ] 
2 ( Y i+2/i)= 2 (Y 3 +2/ 3 ) = 2(Y 3 +2/ 3 )=2(Y 4 -h2/ 4 )=2/ 1 +2/ 3 + 2 / 3 +2/ 4 J ' 
In the course of the proof the equivalent of the algebraical identities 
A/iA/+A/+A/=X/+V+V+V . . . . . 
Y 4 P i+A 3 P 3 +A 3 P 3 +A 4 P 4 =X 4 p 4 -fiXgpg+Xgpg+X 4 p 4 , . . . 
will be used. We have 
(16) 
(17) 
(18) 
(19) 
{ /\ n \ 1 (2»h +Mi) 2 4- . . ■ -f (2?)?4+/x i ) a 
l 3 H J x, y j = 4X( — 1 ) %,ll+ ’ 11 +w ** +n fr+ ■ ■ -x p 4 
,q (2ni 1 +Mi)»‘i+ . . . +<2iH 4 +f*4)* ! 4 
. . ■ +(2rt 4 +i/ t ) 8 
S' 4 
(2w 1 +|u- 1 )(2wi+»'i)+ • . ■ +(2i)i4.+Hj)(Zn i +v i ) 
X ' y / 2ni + Vl > 2 ' 1 + • • • +( 2 »*+ i » 4 ) y * x T 2 
( 20 ) 
where the summation is over all integral values of the m’s and ri s from — oo to + oo. 
Using (IS), (19), the indices on the right-hand side of (20) can be at once trans¬ 
formed ; and the following equations give these transformed values :— 
m^d- . . . -fm 4 X 4 =^(M 1 A 1 -f . . . +M 4 A 4 ) 
n ipi+ * • * +n 4 /> 4 5=J'{NiP 1 + • • • +N4P4} 
(2m 1 +p 1 ) 3 + . . . + (2m 4 +p 4 ) 2 =(M 1 -f o-j) 3 + • * • + (M 4 +<r 4 ) a 
(2 n l -\-v 1 ) 2 + . . . + (2%+v 4 ) 3 =(N 1 +cr / 1 ) 3 + • ■ • +(N 4 +ct , 4 ,) 3 
(2m J +/A 1 )a7 1 + . . . +(2%+/A 4 )a! 4 =(M^+<r 1 )X 1 + • • • + (M 4 -fo- 4 )X 4 
(2 n 1 +v^jy 1 + • • . + ( 2 n 4 +v 4 ) 2 / 4 =(N 1 +cr' 1 )Y 1 + . . . + (N 4 + o-' 4 )Y 4 
(2m 1 +/A 1 )(2w 1 +v 1 )-f . . . +(2m 4 +/A 4 )(2?i. i( +v 4 )=(M 1 +o- 1 )(N 1 4-cr , 1 ) +.-• 
+(M 4 +cr 4 )(N 4 +<r 4 ) 
