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.