704]
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
539
110. The first set of 16 equations is the square-set, which has been already
considered. If in each of the other sets of 16 equations we write in like manner u' = 0,
each set in fact reduces itself to eight equations; sets 2, 3, 4 give thus 8 + 8 + 8,
= 24 equations; sets 5 to 8, 9 to 12, and 13 to 15, give each 8 + 8+8 + 8, =32
equations; or we have sets of 24, 32, 32, 32, together 120 equations, the number
being of course one half of 256— 16, the number of equations after deducting the
16 equations of the square set.
111. The first set, 24 equations.
This is derived from the second, third, and fourth sets, each of 16 equations, by
writing therein v! — 0. Taking a 1} y u ^ for the zero-functions corresponding to
X u Y lt Z 1} W 1} then on writing u — 0, X Xi F/, Z x , W x become a u &, y l5 In
the second set of 16 equations, the first equations thus are
u. %u = a 1 X 1 + yi Z u 0 = ft Fj + 84 W 1}
% 2 u. %u = a l X 1 - y X Z U 0 = /3 1 Y 1 -8 1 W 1 ,
viz. the equations of the column require that, and are all satisfied if, ft = 0, = 0:
hence the zero-functions are 0, 7l , 0; and this being so we have only the equations
of the first column. And similarly as regards the third and fourth sets; the zero
values corresponding to
X xt
Fj, ft, W 1
X 2>
F„
^2,
W 2
X 3 ,
F„
Z>3,
Fa
a i
o
o
«2
Â
0
0
a 3
0
0
S 3 ;
we have in all 8 + 8+8, =
24 equations
These are
(Suffixes 1.)
(Suffixes 2.)
(Suffixes 3.)
Su
. Su
X Z
Su .
Su
X
Y
Su
•
Su
X
w
4
0
= a y
8
0
= a
ß
12
0
= a
8
12
8
i
Ö
il
12
4
= a
-ß
8
4
= a
-8
6
2
= 7 a
9
1
= ß
a
15
3
= 8
a
14
10
= y -«
13
5
= ß
— a
11
7
= -8
a
Y W
Z
W
7
z
5
1
= a 7
10
2
— a
ß
13
1
= a
~8
13
9
= “ -7
14
6
= a
~ß
9
5
= a
-8
7
3
= 7 a
11
3
= ß
a
14
2
= 8
a
15
11
= 7 -«
15
7
= ß
— a
10
6
= - 8
a
£0
. £0
SO .
SO
SO
SO
4
0
- a 2 + y 2
8
0
= a 2
+ ß 2
12
0
= a 2
+ S 2
12
8
II
1
12
4
= a 2
-ß 2
8
4
= a 2
- 8 2
6
2
= 2 ay
9
1
2aß
15
3
2a8.
68—2
%
i M
h
