794 
MR. A. R. FORSYTH OH THE THETA-FUNCTIONS, 
(_.. . +M' 4 A 4 +N 1 P 1 + . .. +N' 4 P 4 Q_^ 
— pi{(2Mi+o- 1 )2+ . . . }^{(2N' 1 +<r'i) a + • • • } r *{(2M 1 +cr 1 )(2N’ 1 + ff ' 1 )+ . . . } /y (2M' 1 +<r 1 )X 1 + . . . |g , x +<r' 1 )Y 1 + .. . 
(_1^!^+ - .. +M' 4 A 4 +N' 1 P 1 + . . . +N 4 P 4 Q^ 
_^{(2M' 1 +<r 1 )3+ . . . }gi{(2N' 1 + l + o-i)2+ . . . } r %{(2M' 1 +or 1 )(2N' 1 +l + <r 1 )+ . ,.}^ 1 +^)X 1 + . . . ^(2N' 1+ 1+ *',)¥,+ . . . 
( _ 1 ^MiAi+ ; . . +M' 4 A 4 +N 1 P 1 + . . . +N 4 P 4 Q s 
-^j(2Mi+H(TiF+ . . . }^l{(2N / i+o- 1 )3+ . . . } T \ {(2M'!+1+cr I )(2N' 1 +<r 4 )+ . .. }^M'j+l +o- 1 )X 1 + . . . ^SN i +*',)Y 1 + . . . 
(_ iJMiA^ . . . +M' 4 A 4 +N' 1 P 1 + • . • +N '^Q 4 
— pi {(2 M 'i + 1 4- o-i) 2 + . . . }^a{(2N' 1 +1-H7' 1 )S+ . . . } r i{(2M 1 + l+oi)(2N 1 + lW l )+ . . . } <y (2M' 1 +l+ 0 - 1 )X 1 + . . . ^(2N 1 +1+<t' 1 )Y 1 + . . . 
and (21) becomes 
4n *{£ ty, 2/}=4S 1 .Q 1 +(-lf4S 3 .Q s +(-l) i '4S 3 .Q 3 +(-l) A+p, 4S 4 .Q + . (22) 
Consider these four sums separately ; then 
4^. Qi = ttQ + tt( — 1 JN'i+^+N's+NiQi _|_ 2£( _ 1 )M\+ M ' i! +M' s+ M' 4 Q 1 
+tt(— 1) N ' 1+ ' • ■ +N '4 +M4+ • • • +M '*Q 1 
where the summations on the right hand side are now taken without restriction for all 
integral values of the M’s and N’s between — oo and + oo; and with similar removal 
of the restrictions on the values of M and N to which the summation extends 
44Q 3 =^Q 2 -^(-l) N ' i+N ' a+N ' s+ ^Q 2 +^(-ir +M ^ M,s+ ^Q 3 
— tt(— l) Nl+ ‘ ’ * +N '4 + M 'l + * • • +M ' 4 Q 2 
4^3- Q 3 = 22Q 3 + — 1 ) N 'i +N 'a +N 's +N, 4Q 3 _ _ l) M 'i+ M ' S! +M's+ M ' 4 Q 3 
— tt( — 1) N ' 1+ • • • +N *+M' 1+ * • • + M '*Q 3 
42*0*= tt( -1 - tt (-1 ) m 'x+».+*.+»'4Q 4 
+Z$(—lf 1+ ■ * • +N '* +M ' 1+ ■ • • +m '*Q 4 
Thus 
^\x, y j = sum of sixteen double summations. 
4ii#> 
But each of these double summations is the product of four double theta-functions : 
thus, in particular, 
