AND DOUBLE THETA-FUNCTIONS. 
939 
these are of the forms 
\/ a— \J aa /S \/ ab = - { \/ahfc / d / e / + \/ajb^cde} 
and I remark that to avoid ambiguity the squares of these expressions are throughout 
written as (v a) 2 and (\/ab)' 2 respectively. 
76. There is for the constant factors a like single-and-double-letter notation which 
will be mentioned presently, but in the first instance I use for the even functions the 
before mentioned 10 letters c, and for the odd ones six letters k. I assume that the 
values x, y=<x> , oo (ratio not determined) correspond to the values u=0, v—0 of the 
arguments; and that w is a function of (x, y ) which, when (x, y) are interchanged, changes 
only its sign, and which for indefinitely large values of (x, y) becomes = 
x-y 
(?yY 
This 
being so, we write (adding a second column which will be afterwards explained) 
0=BD= coc 0 Vbd, 
c 0 —\Vbd, 
1 = CE =,,c 1 v ce, 
A =„v / ce, 
2 = CD = „c a y/cd, 
Co —,,\/ cd, 
3 = BE = „c 3 Ybe, 
c 3 =„v . be, 
4=AC=„c 4 y/ac, 
c 4 =,,v / ac, 
5= C = „k 5 y/< c, 
h =„v / c, 
6=AB=,,c 6 v 7 ab, 
C G =„v / ab, 
7= B =„k 7 Vb, 
k 7 = „y/b. 
8 = BC=„c s y/bc, 
c 8 =„V be, 
9 = DE=,,c g \/ de, 
c 9 =„v / de, 
10= F = „k 10 y/f, 
= »yj, 
11= A =,,k u v-a, 
hi —’’Y'a, 
12 = AD = ,,c 13 V / cid, 
c ia = a d> 
13= D =„k 1B y/d, 
hs = > j Y' cl, 
I <^T 
if 
w 
II 
H 
hi, 
15 = AE=„c 15 V / ae, 
015 =,^ oe, 
viz.: here, on writing x, y=cc, go , each of the functions ^/bcl, &c. becomes =: 
/ay ) 1 
x—y 
and each of the functions a, &c., becomes = \/xy ; hence by reason of the assumed 
form of oj, the odd functions each vanish (their evanescent values being proportional 
6 e 2 
