942 
PROFESSOR A. CAYLEY ON THE SINGLE 
and between the products a system of 6 + 9 equations 
a'a"+b'b"+c'c" = 0, 
a"a +b"b +c"c =0, 
act +11/ +cc = 0, 
be +b'c +b"c" = 0, 
ca + ca + c"a" = 0, 
cib +a'b' +a"b" = 0, 
a, b, c — b'c"—b"c, 
a, b', c b"c —be", 
a", b", c" be —b'c, 
that is 
e a —c a , a b —a b , 
c"a — ca", ctb" — ci'b, 
ca' —ca, cib' —ctb, 
c 2 
c 3 
c 3 
c 3 
c 2 
c 2 
9 
2 
+ 4 
15 
_ 
3 
8 
2 
12 
-15 
1 
— 
8 
6 
12 
9 
- 1 
4 
+ 
6 
3 
1 
6 
- 4 
3 
+ 15 
8 
6 
12 
+ 3 
9 
— 
8 
2 
12 
1 
- 9 
4 
— 
2 
15 
-0 
12 
+ 4 
8 
+ 
3 
15 
-0 
1 
+ 3 
2 
+ 
8 
9 
-0 
6 
— 9 
15 
+ 
2 
4 
-0 
9 
-15 
6 
+ 
8 
1 
+ 0 
4 
- 8 
12 
— 
6 
2 
-0 
3 
-12 
15 
— 
1 
2 
— 0 
2 
+ 1 
3 
+ 
4 
6 
+ 0 
15 
+ 6 
9 
— 
3 
12 
+ 0 
8 
-12 
4 
9 
1 
each of the first set of 15 giving a homogeneous linear relation between four terms c 4 ; 
and each of the second set giving a homogeneous linear relation between three terms 
c 3 . c 3 , formed with the 10 constants c. Thus the first equation is c 13 4 +c 1 4 + c G 4 —c 0 4 = 0 ; 
and so for the other lines of the two diagrams. 
79. I form in the two notations the following tables:— 
Table of the 16 Kummer hexads. 
A 
A 
A 
A 
A 
B 
B 
B 
B 
C 
C 
C 
D 
D 
E 
A ' 
B 
C 
D 
E 
F 
C 
D 
E 
F 
D 
E 
F 
E 
F 
F 
B 
AB 
AC 
AD 
AE 
AB 
BC 
BD 
BE 
AB 
CD 
CE 
AC 
DE 
AD 
AE 
c 
CD 
BD 
BC 
BC 
AC 
AD 
AC 
AC 
BC 
AB 
AB 
BC 
AB 
BD 
BE 
D 
CE 
BE 
BE 
BD 
AD 
AE 
AE 
AD 
BD 
AE 
AD 
CD 
AC 
CD 
CE 
E 
DE 
DE 
CE 
CD 
AE 
DE 
CE 
CD 
BE 
BE 
BD 
CE 
BC 
DE 
DE 
F 
11 
11 
11 
11 
11 
7 
7 
7 
7 
5 
5 
5 
13 
13 
14 
11 
7 
5 
13 
14 
10 
5 
13 
14 
10 
13 
14 
10 
14 
10 
10 
7 
6 
4 
14 
12 
6 
8 
0 
3 
6 
2 
1 
4 
9 
12 
15 
5 
2 
0 
8 
8 
4 
12 
4 
4 
8 
6 
6 
8 
6 
0 
3 
13 
1 
3 
3 
3 
12 
15 
15 
12 
0 
15 
12 
2 
4 
2 
1 
14 
9 
9 
1 
2 
15 
9 
1 
2 
3 
3 
0 
1 
8 
9 
9 
10 
