AND DOUBLE THETA-FUNCTIONS. 
977 
114. Fourth set, 32 equations. 
Again changing the notation we write 
Ms+tQ* Mo — t'Qo, 
N 2 +7P 2 , No— 7P 2 
M3+Q3, i(M 3 Q 3 ), N3+P3, t(N a P 3 ) 
= X, Y, 
z, w 
X lf Y lt Z l5 W lf 
M+Q, i(M — Q), 
N+P, i( N-P) 
M 1 +7Q 1 , —iQ 1? N 1 +i'P 1 , N x — iP l 
= x 2 , y 2 , 
Z 2 , Wo 
X 3 , y 35 z 3 , w 3 
the zero values being 
a, 0, y, 0 | 
«1, 0, 0 l5 § J 
* 2 , 0 , y z , 0 1 0, £3, y 3 , 0 
and the equations then are 
(Suffixes 0.) 
(Suffixes 1.) 
(Suffixes 2.) 
(Suffixes 3.) 
■Shi 
3« X 
Z 
3 it 
3 it 
X 
W 
3tt 
3it 
X Z 
3<t 
3 it Y Z 
0 
3 =' * 
7 
3 
4 
= — ia. 
id 
15 
4 
= ioL — 2*7 
15 
0 = '-p 7 
15 
12 = -* 
7 
15 
8 
— 
id 
3 
8 
= —to — {7 
3 
12 = P 7 
2 
1 = 7 
<% 
6 
1 
= 6 
at. 
14 
5 
= 2*7 —2 a 
10 
5 = 7 -d 
14 
13 = -7 
Ot 
10 
13 
= — s 
a 
2 
9 
— — {7 •— to 
6 
9 = -7 —/I 
Y 
w 
Y 
z 
Y W 
X W 
4 
7 = (fc 
—7 
7 
0 
— 
8 
11 
0 
= at, 7 
11 
4 = — P 7 
8 
11 = * 
7 
11 
12 
— a 
— 0 
7 
12 
— a —7 
7 
8 = ~/3 -7 
6 
5 = 7 
-a 
2 
5 
= io 
—ict 
10 
1 
= 7 « 
14 
1 - 7 -d 
10 
9 = 7 
at. 
14 
9 
= id 
lot. 
6 
13 
= 7 — * 
2 
13 = 7 d 
30 
30 
30 
30 
30 
30 
30 
30 
0 
3 = 
2 1 2 
+ 7 
3 
4. 
— — t(a 3 — ci 2 ) 
15 
4 
II 
V s 
to 
1 
to 
15 
O 
II 
to 
1 
to 
15 
12 = -(« 
2 „ 2\ 
' / / 
15 
8 
= i(» s + * 8 ) 
3 
8 
= -*V+7 2 ) 
3 
12 — /3 2 + T 
2 
1 = 
2*7 
6 
1 
= 
2«8 
2 
9 
= — 2to 7 
6 
9 — — 2/J 7 
G K 
MDCCCLXXX. 
