542
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704
114. Fourth set, 32 equations.
Again changing the notation, we write
Mo + iQ,, Mo — iQo, N 2 + iP 2 , No — iP o M :i + Q 3 , i (M :i — Q 3 ), N 3 + P 3 , i(N 3 — P 3 )
= X, F, Z, W X u Y u Z xt W u
M+Q, i(M-Q), N+P, i(N-P) | M x + iQ x , M 1 - iQ x , N x + iP u X x -iP x
= Xo, F, Zo. Wo ' X. F,. X. IF,.
the zero values being
a, 0, 7, 0 I «!, 0, 0, 8 1
The equations then are
Su
(Suffixes 0.)
Su X Z
Su
(Suffixes 1.)
X W
0
3 =
a
y
3
4 =
— ia
i8
15
12 =
— a
y
15
8 =
ia
i8
2
1 =
y
a
6
1 =
8
a
14
13 =
-y
a
10
13 =
- 8
a
F
TF
Y
Z
4
7 =
a
-y
7
0 =
a
8
8
11 =
a
y
11
12 =
a
- 8
6
5 =
y
— a
2
5 =
i8
— ia
10
9 =
y
a
14
9 =
i8
ia
S0
so
£0
£0
0
3 =
a 2 + y 2
3
4 = — i (a 2 — S 2 )
15
12 =
-(a 2 -y 2 )
15
8 = i (a 2 + S 2 )
2
1 =
2ay
6
1 = 2aS
&2 J
o
O
1 o,
ßs,
7 s» 0.
«
(Suffixes 2.)
(Suffixes 3.)
X
Su
. Su Y
X
15
4 = ia
— iy
15
o=~
y
3
8 = — ia
-iy
3
12= ß
y
14
5 = iy
— ia
10
5= y
-/9
2
9 = — iy
— ia
6
9~y
-/?
Y
TF
X
TF
11
0 = a
y
11
4 = ^
y
7
12 = a
- y
7
8 = -/?
- y
10
1= y
a
14
1= y
-/?
6
13 = y
— a
2
13= y
/5
£0
£0
£0
. £0
15
4 = i (a 2
-y 2 )
15
o=- (P
-y 2 )
3
8 = - i (a 2
+ y 2 )
3
12= ß 2 + y 2
2
9 = — 2zay
6
9 = —
2ßy.