846 
MR. A. R. FORSYTH ON THE THETA-FUNCTIONS, 
58. Again, if 
a -\-b +c H -cl =0 
a '+ b' + c' -f- d !—0 
and in the general theorem we put 
Xi=u-\-ci x 2 =u-\-b £Cg— u-\~c x±—u-\-d 
yi=v+d y^=v+b’ y z =v-\-G y±=v + d' 
then X^u—ci, Y^v—ci, and so for the others : and it is not difficult to prove 
that 
&o ( u d~ + b)S- 2 (u -j- c) r9 - 3 (u -f- d) - 1 - d- 3 (w+&) $3 {u + b) 3- Q (u -j- c)&i (11 -j- d) 
+& 0 ( u—a)$- L (u—b)S- 2 (ii—c)3- d (u—d) -f S- 2 (u — a)^(u — b)S- 0 (a —c) S-^u — d) 
—^3 {u fi- ci) S- 2 (u -f- b) 3-^ (u -}- c) 3-q(u -f~ cl) -{- S-^u -f- ci)S- 0 (u -|- b) r9- 3 (w-|- c)$-g(tt -j- d) 
+ ~~ ci)S- 2 (n — — d) -^-S-^u—djS-^u — b)3- s (u—c)S- 2 (u + cl) 
3-(u-\-d), . . . denoting 3-(u-\-a, v~\-d), . . . , with other relations of the same kind 
between the theta-functions. 
Section IV. 
The u r” tuple theta-functions. 
59. The general “ r ” tuple theta-functions is defined by the equation 
' * * ’ X 'W x,\ = $$ . . . (— i)W • • * . . . p r { mr+v iY 
l\V|> V 2> • • • , V r ) J 
Pi, g 2 ( Wl+ 2 )( Wz+ 2) . . . Ps, !f 2 ( m,+! |)( WH+ l) . . . Vj- 2mi +Vl) . . . V pm r -Yv,)x r ' ' (220) 
in which \ lt X 2 , . . . , \ r , v\, v 2 , . . . , v r are given integers (afterwards taken to be either 
zero or unity) and ( 15 ‘ ‘ ‘ ’ ^ j is called the characteristic; x lt x 2 , . . . , x r are the 
variables ; p 1} p z , ... , p n p 1>2 , . . . , p s j, . . . , v x , v 2 , ... , v r are — constants and 
are called the parameters; and the “ r ” tuple summation extends to all positive and 
negative integral values from — co to + co (including zero) of m lf m 2 ,... , m r . To 
ensure the convergence of the series it is necessary that the real part of 
(2OT 1 +j- 1 ) 8 logp 1 + . . . +‘M2m 1 +v l )(Zm 2 +v i ) log^ 2 + . . . 
