PARTICULARLY THOSE OE TWO VARIABLES. 
809 
Section II. 
21. If 0^ K be the general single theta-function, then 
m-oo (2m+/u.) a , Avx 
e^{x)= t 
m= — oo 
[This notation, since it is already in use, is adopted in preference to which 
would better agree with the definition of the double theta-function in (1) and with 
that of the “ r” tuple function to be given later.] 
As is well known, there are three even functions and uneven ; these are 
and 
(even) 0 Qt0 (a) = 1 + 2p cos 2a^+ 2ja 4 cos 2 P 9 cos 6 ^2K~*" * * * 
(even) 0 QA (x) = l — 2p cos 2x^+2p*cos 4a^—2p 9 cos 6aJ~ + . . . 
(even) 6 h 0 (x) = 2p l cos a?^r+ 2p f cos 3a?“+ 2p ¥ cos 5x—fi- . . . 
(uneven) t 2 («)=2p* sin as^j- — 2p f sin Sx sin 5a?— + . . . 
1 
sn x— -r-i - 
l\/ K 
cri 
K ^o,l(*) 
cfoi a?= 
a/ k' 
Aq» 
22. Writing in the definition of <3> given in (1) 
17T 
v=ezK 
11T 
w=e 2 a 
we have the following series of expressions for the S’b. 
4 0 = 1 + 2 !> cos cos ^ 
m=l W- p=l 
A 
m=ao W=C0 
; Y 2 jr 2 * w cos cos . . (64) 
and therefore 
+ 2 2 2 ©»Y* ] f*** cos 7T vp+Y 
m=l w=l L \ JV A 
5 L 2 
