AND DOUBLE THETA-FUNCTIONS. 
975 
112. The second set, 32 equations. 
To exhibit these in a convenient form I alter the notation, viz., I write 
E+G, {(E—G), (F+H), i( F—H) 
= X, Y, Z, W 
Ej-H’Gn Ej — iG v F 1 +i'H 1 , F x — 
X„ Y lf Z ls W x 
(E 2 +G 2 ), i(E. 2 - G 2 ), (F 2 +H 2 ), i(F 9 -H 3 ) 
= X 2 , Y 2 , Z 3 , W 2 
E 3 +iG 3 , E 3 -YG 3 , F 3 +iH 3 , V,-in, 
x 3 , y 3 , z 3 , w 3 , 
so that as regards the present set of equations, X, Y, X, &c., signify as just mentioned. 
And this being so the corresponding zero-values are 
a, 0, y, 0 I oq, 0, y lt 0 | a 3 , 0, 0, S 2 | a 3 , 0, 0, § s . 
The equations then are 
(Suffixes 0.) 
(Suffixes 1.) 
(Suffixes 2.) 
(Suffixes 3.) 
Sic . 
Sic 
X 
Z 
Sic 
Sic 
X Z 
Aw 
5 it 
X W 
Sic 
Sic 
X W 
1 
0 
— cc 
7 
1 
4 
— 
— toe - V'f 
9 
0 
= a — 0 
9 
4 
= — ioc — iS 
9 
8 
= cl 
— 7 
9 
12 
— la, + 17 
1 
8 
— cc 0 
1 
12 
= —'2a -}"$> 
3 
2 
= ry 
3 
G 
= 
— i 7 —Aa 
15 
G 
— C oc 
15 
2 
— S cc 
11 
10 
= 7 
— a 
11 
14 
= 
— i~j -f t’a 
7 
14 
— - 0 CL 
7 
10 
= — 6 a, 
Y 
w 
Y W 
Y Z 
Y Z 
5 
4 
= OC 
7 
5 
0 
= 
a 7 
13 
14 
— CL O 
13 
0 
— a 8 
13 
12 
— cl 
7 
13 
8 
= 
a —7 
5 
12 
= -6 
5 
8 
— cc — 0 
7 
6 
“ ry 
CL 
7 
2 
1= 
7 a 
11 
2 
= -0 06 
11 
6 
= — ic —ix 
15 
14 
— 7 
-56 
15 
10 
— 
7 —06 
3 
10 
= C oc 
3 
14 
= AO — ioc 
SO 
£0 
5-0 
. £0 
A0 
. SO 
SO 
. .5-0 
1 
0 
O 
— a" 
+ 7 2 
1 
4 
= 
— l*(a 2 + 7*) 
9 
0 
_ 2 22 
— cc" — c* 
9 
4 
= -i(7 + r) 
9 
8 
0 
— a" 
—7 
9 
12 
= 
— i(u6 3 y 2 ) 
1 
8 
= 06 2 + O 3 
1 
12 
= -iU 8-«3) 
3 
2 
= 2»7 
3 
6 
— 
— 2i’«7 
15 
6 
= 2 060 
15 
2 
= 2060 
