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