442 
A SUPPLEMENTARY MEMOIR ON THE THEORY OF MATRICES. [357 
and 
(T + il) (T - il)“ 1 = ( 1-2 d, -2 h, -21, -2p ), 
— 2c, 1 — 2g, — 2k, — 2 o 
2b, 2/, 1+2\j, 2 n 
2a, 2e, 2 i, 1 + 2m 
so that these matrices are composed of terms which, except as to the signs, are the 
same in each. 
11. Now in general if 
© = ( 
a , 
/3 , 
7 > 
S X 
a' > 
/3' , 
/ 
7 » 
S' 
/3", 
// 
7 » 
S" 
then it is easy to see that 
a'", 
/3"', 
/// 
7 > 
S'" 
il- 1 ©il = ( 
S'", 
/// 
7 > 
-r, 
- a 
8", 
// 
7 > 
-/3", 
— a" 
-v, 
/ 
“7 » 
/3' , 
a' 
-s , 
“7 » 
/3 , 
a 
and hence, from the foregoing values of (T — il) (T + il) -1 and (T + il)(il — T) x , 
find 
n = -il- 1 (T-n)(T + il)- 1 il = ( -1 +2d, 
2c, 
-2b, 
— 2a 
2h, 
-i + 2g, 
-2/, 
- 2e 
21, 
2k, 
CM 5 
1 
r—1 
1 
- 2 i 
2p, 
2 o, 
— 2 n, 
— 1 — 2m 
n- 1 = -il" 1 (T + il)(T-il)- 1 il = ( — 1 — 2m, 
-2 i, 
2e, 
2a 
— 2 n, 
1 
t-i 
1 
2/, 
2b 
-2 o, 
-2k, 
-1+2g, 
2c 
— 2p, 
-21, 
+ 2 h, 
-1 + 2 d 
we 
this shows that the matrix II for the automorphic transformation of the function 
xw' + yz — zy' — wx' is such that writing 
n = ( A, 
B, 
c, 
D ) we have II“ 1 = ( 
P, 
L, 
-H, 
-D ), 
E, 
F, 
G, 
H 
0, 
K, 
-G, 
-C 
I, 
J, 
K, 
L 
-E, 
-J, 
F, 
B 
M, 
N, 
0, 
P 
-M, 
-I, 
E, 
A 
which is the theorem in question.