964 
PROFESSOR A. CAYLEY ON THE SINGLE 
90. I re-arrange these in sets of 16 equations, the equations of the first or square- 
set of 16 being taken as they stand, but those of the other sets being combined in 
pairs by addition and subtraction as will be seen. And I now drop altogether the 
characteristics, retaining only the current numbers : thus in the set of equations next 
written down, the first equation is 
^ 0 (w+A).\(w-A)=XX / +YY / +ZZ / +WW / : 
in the second set, the first equation is 
)} = X 1 X 1 / -fZ 1 Z 1 / , 
and so in other cases. 
First or square-set of 16. 
u+vJ U—vJ 
3 3 (Suffixes 0.) 
0 
0 
= 
X 
X' 
+ Y 
Y' 
+ Z Z' 
+ W 
AY' 
4 
4 
= 
X 
X' 
-Y 
Y' 
+ Z Z' 
-w 
AA" 
8 
8 
X 
X' 
+ Y 
Y' 
-Z Z' 
-W 
AY' 
12 
12 
= 
X 
X 
-Y 
Y' 
-Z Z' 
+w 
AY' 
1 
1 
— 
Y 
X' 
+ K 
Y' 
+ W Z' 
+z 
AY' 
5 
5 
=z ■ 
-Y 
X' 
+ x 
~STt 
—W Z' 
+z 
AY' 
9 
9 
Y 
X' 
+x 
Y' 
-W Z' 
-Z 
AY' 
13 
13 
= • 
-Y 
X 
+ x 
Y 
+ Y r Z' 
-Z 
AY' 
9 
9 
— 
Z 
X' 
+w 
Y' 
+ X Z' 
+ Y 
AY' 
6 
6 
= 
Z 
X' 
-W 
Y' 
+ X Z' 
-Y 
AY' 
10 
10 
= 
-Z 
X' 
-W 
Y' 
+ X Z' 
+ Y 
W' 
14 
14 
= ' 
-Z 
X' 
+ w 
Y' 
+ X Z' 
-Y 
AY' 
3 
Q 
O 
— 
w 
X' 
+ z 
Y' 
+ Y Z' 
+x 
AV' 
7 
7 
— • 
-w 
X' 
+ z 
Y' 
-Y Z' 
+x 
AY' 
11 
H 
XT! ■ 
-w 
■ X' 
-z 
Y' 
+ Y Z' 
+x 
AY' 
15 
15 
ay 
X' 
-z 
Y' 
-Y Z' 
+x 
AY' 
91. 
Si 
:cond set of 16. 
14 +It' 
u — vJ 
1+vJ 
U—If 
IS 
21 
3 . 
3 
+ 
3 
. 3 
i 
I 
(Suffi: 
ses 17 
) 
4 
0 
0 
4 
— 
X X' 
+ z 
Z' 
12 
8 
8 
12 
X X' 
-Z 
Z' 
5 
1 
1 
5 
Y X' 
+w 
Z' 
13 
9 
9 
13 
Y X' 
— AY 
Z' 
6 
2 
2 
6 
Z X' 
+x 
Z' 
14 
10 
10 
14 
-Z X' 
+ x 
Z' 
7 
3 
o 
o 
7 
W X' 
+ Y 
z 
15 
11 
n 
15 
-W X' 
+ Y 
Z' 
11 + vJ 
u—u' 
L + U' 
v,—v 
IS 
2 l 
3 . 
3 
— 
3 
. 3 
1 
J 
(Suffi: 
s:es 17 
) 
4 
0 
0 
4 
— 
Y Y' 
+ AY 
AY' 
12 
8 
8 
12 
Y Y' 
-AY 
AY' 
5 
1 
1 
5 
X Y' 
+z 
AV' 
13 
9 
9 
13 
X Y' 
-z 
AY' 
6 
2 
2 
6 
W Y' 
+ Y 
AY' 
14 
10 
10 
14 
-W Y' 
+ Y 
AV' 
3 
3 
7 
Z Y' 
+x 
AY' 
15 
11 
n 
15 
-Z Y' 
+x 
AY' 
