976 
PROFESSOR A. CAYLEY ON THE SINGLE 
113. Third set, 32 equations. 
We again change the notation, writing 
I + K, z'(I-K), J+L, i{ J—L) 
= X, Y, Z, W 
Ii+K,, {(I.-K,), (J.+L,), iOj-LJ 
X„ Y„ Z 1; W, 
Ijj —|— &Ko, I., — iK.-i, Jo-b^Lo, Jo— iL . 2 
= Xo, Yo, Zo, Wo 
ig-TiKg, Io tKo, jo+iLq, Jo tLq 
x 3 . 
Y 
3’ 
J 3> 
w, 
the zero values being 
a, 0, y, 0 a l9 0, 0, S 1 | a 3 , 0, y 3 , 0 | a 3 , o, 0, § s 
Then equations are 
(Suffixes 0.) 
(Suffixes 1.) 
(Suffixes 2.) 1 
(Suffixes 3.) 
3 u 
3m 
X Z 
3-u 
3;t 
X 
w 
3m 
3zt 
X z 
3i« 
X 
W 
2 
0 
= « 7 
6 
0 
_ 
CL 
— b 
2 
8 
= 
— lot, —I7 
6 
8 
— 
— lot. 
—lb 
6 
4 
— ct —'1 
2 
4 
— 
CL 
b 
6 
12 
— /a + I7 
9 
12 
= 
- lOL 
'lb 
3 
1 
= 7 « 
15 
9 
— 
b 
CL 
3 
9 
= 
—17 — [cl 
15 
1 
b 
OL 
7 
5 
= 7 —a 
11 
13 
= 
— b 
CL 
7 
13 
= 
— 2/7 -}- /a 
11 
5 
= 
— b 
CL 
Y W 
Y 
Z 
Y W 
Y 
Z 
10 
8 
= a 7 
14 
8 
CL 
b 
10 
0 
= 
a 7 
14 
0 
— 
OL 
s 
14 
12 
= OL -7 
10 
12 
— 
CL 
— b 
14 
4 
= 
a —7 
10 
4 
= 
CL 
— b 
11 
9 
= 7 a 
7 
1 
— 
— b 
CL 
11 
1 
7 a 
7 
9 
- 
— iS 
—la 
15 
13 
— 7 —* 
3 
5 
= 
b 
CL 
15 
5 
= 
7 —a 
3 
13 
= 
ic 
—la 
30 
30 
30 
30 
30. 
30 
30. 
30 
2 
0 
_ 2 i 2 
— cl"-\- 7 
6 
0 
= 
0 
OL" 
- C* 
2 
8 
= 
z(a 2 + 7 2 ) 
6 
8 
= 
-i(a 2 + ^ 2 ) 
6 
4 
O 0 
= a- —7- 
2 
4 
= 
O 
CL" 
+ c 3 
6 
12 
= 
t (a 2 7") 
2 
12 
- 
l(a 2 (5 2 ) 
3 
1 
— 2a7 
15 
9 
= 
2ao 
O 
O 
9 
— 
— 2/a 7 
15 
1 
= 
2a£ 
