AND DOUBLE THETA-FUNCTIONS. 
973 
X 
Y 
z 
W 
A 
11 = 
s 
y 
-p 
— a, 
B 
7 
8 
-y 
— a, 
AB 
6 
y 
-8 
a 
-A 
CD 
2 
y 
8 
a 
S, 
CE 
1 
fi 
a 
8 
y> 
DE 
9 
a 
8 
—y- 
viz.: it should thence follow that there is a linear relation between any four of the six 
squared functions 11, 7, 6, 2, 1, 9 : and it is accordingly seen that this is so. It 
further appears that in the several linear relations, the coefficients (obtained in the 
first instance as functions of a, (3, y, 8) are in fact the 10 constants c : the 15 relations 
connecting the several systems of four out of the six squared functions are given in 
the table. 
Read 
• 
< V'\" C 2 
2 + 2 +c 
2 a 2 
1 
2o 2_ 
°9 "^9 — 
0, 
c 0 v9- n 
3 . +C 15 2 + s -c 
2 a 2 
12 "D 
++ 2 V= 
o 
5+ 
P 
11 
7 
6 
2 
1 
9 
G 
_ 2 
1 
— 9 
6 
+ 15 
-12 
+ 4 
- 2 
-15 
+ 8 
- 0 
1 
+ 12 
- 8 
+ 3 
— 9 
- 4 
+ o 
- 3 
6 
3 
- 0 
+ 8 
- 2 
- 3 
+ 4 
-12 
1 
+ 0 
- 4 
-15 
— 9 
- 8 
+ 12 
+ 15 
-15 
+ 3 
+ 2 
- 6 
-12 
+ o 
+ 1 
— 6 
- 4 
+ 8 
+ 9 
- 6 
- 3 
+ 15 
+ 9 
- 1 
- 0 
+ 12 
+ 9 
- 2 
- 8 
+ 4 
+ 1 
- 2 
110. The first set of 16 equations is the square-set, which has been already con¬ 
sidered. If in each of the other sets of 16 equations we write in like manner u'= 0, 
each set in fact reduces itself to eight equations; sets 2, 3, 4 give thus 8 + 8 + 8, 
= 24 equations; sets 5 to 8, 9 to 12, and 13 to 15, give each 8+8 + 8 +8, =32 
equations; or we have sets of 24, 32, 32, 32, together 120 equations, the number 
being of course one half of 256 — 16, the number of equations after deducting the 
16 equations of the square set, 
