974 
PROFESSOR A. CAYLEY ON THE SINGLE 
111. The first set, 24 equations. 
This is derived from the second, third, and fourth sets, each of 16 equations, by 
writing therein u— 0. Taking cq, fi 1} y x , for the zero-functions corresponding to 
X l5 Y 1; Z l5 W 1} then on writing u— 0, X/, Y+ Z+ W/ become oq, y 1? S 1 . In the 
second set of 16 equations, the first equations thus are 
+ u . 3- 0 u= a 1 X 1 + y 1 Z 1 , 0=/3 1 Y 1 +8 1 W 1 , 
d 13 w ^• s zt=:a 1 X 1 —y 1 Z 1 , 0=/3 1 Y 1 — S 1 W 1 , 
viz., the equations of the column require that, and are all satisfied if, /3j = 0, 8j = 0 : 
hence the zero functions are oq, 0, y l5 0 ; and this being so we have only the equations 
of the first column. And similarly as regards the third and fourth sets; the zero 
values corresponding to 
X l3 Y 1? Z lf W L j X* Y* Zo, W 3 X 3 , Y s , Z 3 , W 3 
are oq 0 0 j cq /3. : 0 0 a 3 0 0 S 3 ; 
and we have in all 8 + 8 + 8, =24 equations. These are 
3u 
3u 
(Suffixes 1.) 
X Z 
3u 
3u 
(Suffixes 2.) 
X Y 
3u 
. 3u 
(Suffixes 3.) 
X W 
4 
0 
— 
a 
7 
8 
0 
— 
cl 
P 
12 
0 
= 
CL 
C 
12 
8 
= 
cl 
7 
12 
4 
=Z 
CL 
~P 
8 
4 
Z= 
oc 
— C 
6 
2 
zz 
7 
CL 
9 
1 
=Z 
P 
CL 
15 
8 
z=: 
h 
CL 
14 
10 
7 
— a 
13 
5 
= 
P 
- CL 
11 
7 
= 
— c 
CL 
Y 
w 
z 
W 
Y 
Z 
5 
1 
— 
a 
7 
10 
2 
= 
CL 
P 
13 
1 
= 
CL 
s 
13 
9 
= 
CL 
7 
14 
6 
— 
CL 
-P 
9 
5 
— 
CL 
— s 
7 
3 
= 
7 
CL 
11 
3 
= 
P 
CL 
14 
2 
8 
CL 
15 
11 
7 
- 
15 
7 
— 
P 
- CL 
10 
6 
— 
- O 
CL 
£0 
£0 
£0 
^0 
£0 
30 
4 
0 
— 
O 
+ 7 3 
8 
0 
= 
O 
CL'" 
+ P~ 
12 
0 
zn 
CL" 
+ S 2 
12 
8 
— 
o 
a" 
O 
7 
12 
4 
ZZZ 
CL" 
-P~ 
8 
4 
— 
O 
CL" 
— £2 
6 
2 
= 
2 
a 7 
9 
1 
= 
£ 
'.up 
15 
3 
= 
\u8 
