704]
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
543
115. It will be noticed that the pairs of theta-functions which present themselves
in these equations are the same as in the foregoing “ Table of the 120 pairs.” And
the equations show that the four products, each of a pair of theta-functions, belonging
to the upper half or to the lower half of any column of the table, are such that any
three of the four products are connected by a linear equation. The coefficients of
these linear relations are, in fact, functions such as the a- + 8 2 , a 2 — S 2 , 2a8 written
down at the foot of the several systems of eight equations, and they are consequently
products each of two zero-functions c.
Thus (see “The first set, 24 equations”) we have
Qu
(Suffixes 3.)
X w
Su
(Suffixes 3.)
Su Y Z
£0
(Suffixe
£0
4
8
— a
- 8
5
9 = a
- 8
4
8 = a 2 - 8 2
0
12
— a
8
1
s
II
CO
rH
8
0
12 = a 2 + 8 2
3
15
= 8
a
2
14 = 8
a
15
3 = 2aS.
7
11
= -8
a
6
10 = - 8
a
116. In the left-hand four of these, omitting successively the first, second, third,
and fourth equation, and from the remaining three eliminating the X 3 and W 3 , we
write down, almost mechanically,
Xu
4
Xu
8
4- 2aS,
- 8 2 - a 2 ,
a 2 — 8 2
0
12
— 2aS,
- 3 2 4- a 2 ,
a 2 + 3 2
3
15
a 2 4- 8 2 ,
S 2 — a 2 ,
2a8
7
11
- a 2 4- 8 2 ,
8 2 4- a 2 ,
— 2a8
and thence derive the first of the next following system of equations; read
CyCisAAio CyCi'jA/As + C4C8 "A'fà'n — 0.
03^15^4^3 4“ C4C8 ^'3^15 C(>Cl2^Y^ll — 6,
C(fiv2^4^S C4C8 Ao^12 4- Cÿpis^jXn — 6,
C4C8 Ai^8 4” C 0 C 12 ^oA 12 CjCjsAAis = 6,
where the theta-functions have the arguments u, v.
Observe that, on writing herein u = 0, v = 0, the first three equations become each
of them identically 0 = 0; the fourth equation becomes
. /1 2/j 2 1 p 2p 2 . ri 2p 2 —— A
W ^8 ^ °0 KL2 ^3 ^15 —
which is one of the relations between the c’s and serves as a verification.
But in the right-hand system, on writing u = v — 0, each of the four equations
becomes identically 0 = 0.
