PARTICULARLY THOSE OF TWO VARIABLES. 
787 
Section I. 
3. The general double theta-function is defined by the equation 
T . (1) 
VI —QO ?l=c 
m =—go n= —co 
in which X, [i, p, v are given integers (afterwards taken to be each either zero or unity) 
and P ) is called the characteristic; x, y are the variables; p, q, r, v, w are known 
vW 
constants, called the parameters; and the double summation extends to all positive 
and negative integral values (including zero) of m and n. To ensure the convergence 
of the doubly infinite series it is necessary that the real part of 
(2m+p,) 3 log p-\-{2n-\-v) 2 log q-\-2(2m-\- p)(2n-\-v) log r 
should be negative for all real values of m and n ; beyond this restriction, there is no 
limitation to the form or the values of p, q and r. 
4. It follows at once from the definition that 
• • • ( 2 ) 
. . . (3) 
the variables being the same throughout. Hence there are, in all, sixteen distinct 
functions, obtained by assigning to the four numbers of the characteristic the values 
zero and unity and taking all possible combinations. 
Also from the definition 
H 
[A+2, p\] 
Lv* > v )\ 
>=<£ j 
p.p\ l 
Xft "/J 
!tr ! )i 
\(* >p\~\ 
\\ji + 2 ,v) J 
H 
A />\1 
V, "/J 
M 
[(A*)! 
{ l\, n\ 1 m=co «=» , ( 2 ( 2 / 
L P v )~ x < ~y\= $ t (-!)«-> * q 
\r'> V ] J Vl= — cx> 71= — go 
(2w2r+jm)(2w+i/) 
4 V S 
V 
ra=oo n=& 
(—2}a+ju,+/i) 3 (—2n+v+v) 2 
= ^_]\v+vp ^ ^ ^-4- q -4- 
m=—aa n= —00 
(—2m+u.+u)(— 2n+v+v) , _ - . . -- . 
Y -2--2w+ 2;s+ v+v) 
m-oo n'=oo (2m +u)* (2n'+v)* (2m'+p)(2w'+v) 4 x 
— ^^ ^ ^^ yh'^+u’pp — 4 — q — 4 —^-g-^*(2 w'+h),^(2?j +o 
m'= —oo n'——co 
= (-!)*+•>* {£')**}.(4) 
Hence there are ten even and six uneven functions, the latter being denoted by an 
asterisk in the following table, showing the correspondence between the notations 
