540 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704 
112. The second set, 32 equations. 
To exhibit these in a convenient form, I alter the notation, viz. I write 
E+G, i(E — G), (F+H), i(F—H) 
X, Y, Z, W 
E 1 + iGi, E 1 — iGi, F l + iH l , F 1 — iHi 
x lt T u Z 1} If, 
(K + G 2 ), i(E 2 — (t 2 ), (F 2 + H 2 ), i (F 2 — H 2 ) 
= X„ F„ z 2 , W 2 
E 3 + iG 3 , E 3 — iG 3 , F 3 + iH 3 , F 3 — %H 3 
X t , F„ Z„ 
so that as regards the present set of equations, X, Y, Z, W, signify as just mentioned. 
And, this being so, the corresponding zero-values are 
a, 0, 7, 0 | a lf 0, y 1} 0 
a2, 0, 0, 8 2 | a 3 , 0, 0, S 3 . 
The equations then are 
(Suffixes 0.) 
. 9u X Z 
Su 
(Suffixes 1.) 
X Z 
Su 
Sic 
(Suffixes 2.) 
X W 
Su 
Sit 
(Suffixes 3.) 
X w 
1 
0 = a 
7 
1 
4 = 
— ia — iy 
9 
0 
— a 
- 8 
9 
4 = 
— ia 
— i8 
9 
8 — a 
-y 
9 
12 = 
— ia + iy 
1 
8 
— a 
8 
1 
12 = 
— ia 
+ i8 
3 
2 = y 
a 
3 
6 = 
— iy — ia 
15 
6 
= 8 
a 
15 
2 = 
8 
a 
11 
10 = y 
— a 
11 
II 
H 
— iy + ia 
7 
14 
= - 8 
a 
7 
10 = 
- 8 
a 
Y 
w 
Y W 
Y 
z 
Y 
z 
5 
4 = a 
y 
5 
0 = 
a y 
13 
4 
= a 
13 
0 = 
a 
8 
13 
12 = a 
-y 
13 
8 = 
a — y 
5 
12 
= a 
- 8 
5 
8 = 
a 
- 8 
7 
6 = y 
a 
7 
2 = 
y a 
11 
2 
= — 8 
a 
11 
6 = 
— i8 
- ia 
15 
I—* 
fi 
ll 
— a 
15 
10 = 
y - a 
3 
10 
= 8 
a 
3 
14 = 
¿8 
— ia 
£0 
£0 
30 
30 
30 
30 
30 
30 
1 
o 
II 
°» 
+ f 
1 
4 = 
- i (a 2 + y 2 ) 
9 
0 
= a 2 
- 8 2 
9 
4 = 
— i (a 2 
+ 8 2 ) 
9 
8 = a 2 
-y 2 
9 
12 = 
— ¿(a 2 —y 2 ) 
1 
8 
= a 2 
+ S 2 
1 
II 
CM 
H 
— i (a 2 
-S 2 ) 
3 
2 = 
lay 
3 
6 = 
— 2iay 
15 
6 
= 2aS 
15 
2 = 
2a8.