850 
MR. A. R. FORSYTH ON" THE THETA-FUNCTION'S, 
Let 
f f\, • • 
•, V\ 
1; 
1W> v 2> • • 
. . , Vr) 
(226) 
M^+2m^=M / i *+2m' ! f =M"+2m"* =W" t -\-2m" t = m t + m't +m" -f- m" t 
2 (X*+a?/)=2 (X 7 ^+^) = 2 (X"*+ x"t)=2(K'" t -\- x"'t) =x t -\- x't ++ w"t 
2(A;;4-^) = 2 (A'^-J- = 2(A"t-\-\"t) =2 (A =X jf +X^+X / ^+X /,/ i{ =2L^ 
2(N,+^) = 2(N',+^)=2(N^+A) =2(W" t +v'” t ) = vi+v ' t+v " t+v '" i 
in which for t are to be substituted, in succession, tbe values 1, 2, 3, ...» r. Then 
m^+w / ^+m / ^X y/ ^+J(M*A*+M'^A' t +M"*A"*+M'^A'"*) 
(2w i! d-V i ;) 2 d-(2W / i ;+ I/ ^) 3 + (2wi // / + ^ // i ;) 3 -j- (2 Vl'" t -\-v"t) 
= (M,+N,) 3 + (M'*+N ',)»+ (M"*+N",) 2 +(M^+r,) 
(2w«+^)(2w s +^)+ . . . +( 2m /,/ t -{-v ,/ t )(2m ,, ' s -\-v / ' s ) 
= (M,+N<)(M S +N S )+ . . . +(M'\+W'\){W'\+W'\) 
H+^-f • * ■ +(2m'" = (M,+N,)X t + . . . +(M'",+N'")X'". 
These, substituted in <E>d> / d> // # /,/ , give 
t % r M,A<+M'fA'j4-M"< M't +M'", X'" t 
$>&<$>"&" = %t . . • ( — l)f=l 2 ^ 1 i{(l«i+N 1 )*+ . . . +(M/" 1 +U" } .^} . , 4 
^9 ? .i{(M,.+N r )2+ .. . + (M"' r +N w 1 -) 2 } p l 3 »{(M 1 +N 1 )(M 2 +N 2 )+ ... +(M"' 1 +N'" 1 )(M: / " a +N'" 2 )} 
^{(M s +N,)(M r +N r )+ . .. +(M , " s +N" , ,)(M"' r +N'"r)} Vl <M 1 +N 1 )X 1 + . . , +(M'" 1 +N'" 1 )X'" 1 # 
^(Mr+N r )Xr+ . . . + (M"' r +N'"r)X"', _ ... ..... ... (227) 
the summation being taken for all values of the M’s defined by the preceding equations. 
Now the difference between any two of the M’s with the same suffix is even, so that 
all the M’s with the same suffix are either even or uneven. In the former case let 
M,= 2^ M',= 2 fi\ M"=2/*" W" t =2n'" t 
and it will be found that if the equations are satisfied 
Pt+p't+p"t dV"*=e-ven. 
In the latter case, let 
M,= 2^+l M'^2^+1 W'i=2[*" t +l M/%=2 i a ,// i +l 
and then it will be necessary that 
fa~\~Pt~\~ uneven. 
