972 
PROFESSOR A. CAYLEY ON THE SINGLE 
106. In the square set, writing u’—v— 0, and «, ft, y, S for X', Y', Z', W'; also 
slightly altering the arrangement, 
the system becomes 
and further writing herein u— 0 , v— 0 it becomes 
X Y Z W 0 
0 
4 
8 
12 
— 
OL 
ol 
ol 
OL 
rrp ccg 
" 1 1 
7 
7 
V 
7 
8 
— 8 
— 8 
8 
S 2 
0 
4 
8 
12 
- 
a 2 + /3 2 + y 2 + S 2 
J-/3 2 +r-s 2 
<z 2 —/3 2 — p — c 2 
a 2 —d 2 —7 3 + £ 2 
= 
0 
4 
8 
12 
1 
= 
d 
ol 
8 
7 
1 
20/3+7?) 
— 
1 
5 
d 
— ol 
8 
— 7 
5 
0 
9 
d 
a. 
— 8 
7 
9 
= 
IO 
“to 
1 
= 
9 
13 
= 
d 
— a. 
— 8 
7 
13 
0 
2 
_ 
7 
c 
tz 
d 
2 
izn 
2(-7+d«) 
— 
2 
6 
7 
— 8 
OL 
-d 
6 
= 
207 -d+> 
= 
6 
10 
— 
7 
8 
-£% 
-d 
10 
0 
14 
= 
7 
— c 
— cz 
d 
14 
0 
3 
- 
8 
7 
d 
CL 
3 
= 
2(*« + d7) 
“ 
3 
7 
= 
8 
7 
d 
- OL 
7 
0 
11 
= 
8 
7 
-d 
— cz 
11 
0 
15 
= 
8 
— 7 
-d 
OL 
15 
= 
20 s — d7> 
15 
viz. : this last is the before-mentioned system of equations giving the values of the 
10 zero-functions c in terms of the four constants a, (3, y, §. 
107. The system first obtained is a system of 16 equations 
$ 0 2 (u, v) = aX+^Y+yZ-j-SW, &c. 
showing that the squares of the theta-functions are each of them a linear function 
of the four quantities X, Y, Z, W. If the functions on the right hand side were 
independent (asyzygetic) linear functions of (X, Y, Z, W) it would follow that any 
four (selected at pleasure) of the squared theta-functions were linearly independent, 
and that we could in terms of these four express linearly each of the remaining 
12 squared functions. But this is not so; the form of the linear functions of 
(X, Y, Z, W) is such that we can (and that in 16 different ways) select out of 
the 16 linear functions six functions, such that any four of them are connected 
by a linear equation; and there are consequently 16 liexads of squared theta- 
functions, such that any four out of the same hexad are connected by a linear relation. 
The hexads are shown by the foregoing “ Table of the 16 Rummer liexads.” 
108. The d posteriori verification is immediately effected ; taking for instance the 
first column, the equations are 
