502 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
[704 
these are of the forms 
Va = Vaa,, Va6 = ^ {Vabfc / d / e / + Va^f/jde}. 
I remark that to avoid ambiguity the squares of these expressions are throughout 
written as (Va) 2 and (hab) 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= oo, 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) 
■y 
becomes =A—. 
{xyf 
afterwards explained) 
This being so, we write (adding a second column which will be 
^ _ 
0 = BD = coc 0 y/bd, 
c 0 = A v 7 bd, 
1 = CE — „ c x y/ce, 
Cl = „ v 7 ce, 
2 = CD = „ c 2 v 7 cd, 
c 2 =„v / cd, 
3 = BE = „ c 3 be, 
c 3 =„y / be, 
4 = AC — ,,Ci \ / 'ac, 
c 4 = „ v 7 ac, 
•5 = C = ,,k 5 y/c, 
h 
= „Vl 
6 = A B = ,,c 6 v/ab, 
C 6 = „ v 7 ab, 
i »© 
> 
if 
K) 
II 
ky 
8 = BC = „ c 8 y/be, 
Cs =„^ / bc, 
9 = DE = „ c 9 v 7 de, 
C 9 =,, ^de, 
10= F = ,,k w Vf, 
ho 
-«n 
11 —A = a, 
hi 
= „y / a, 
12 = AD = „ c 12 v/ad, 
Ci 2 = ,, y/ad, 
13= D = ,,k 13 \/d, 
hz 
14 = E = ,,k u y/e, 
hi 
15 = AE = „ Cis y/ ae, 
Ci 5 =„^ / ae; 
viz. here, on writing x, y = oo, oo, each of the functions Vbd, &c. becomes — 2^^: 
x ~y 
and each of the functions Va, &c., becomes =^lxy; hence by reason of the assumed