504 
A MEMOIR ON THE AUTOMORPHIC LINEAR TRANSFORMATION 
[153 
or as we may_ write it, 
T 0 = — (№-' + tr. - tr. Or*}YC„ 
and thence 
T = - atil- 1 + tr. - tr. il-ijyr, + T„ 
where T, is an arbitrary skew symmetrical matrix. 
16. This includes the before-mentioned special cases; first, if il is symmetrical, 
then we have simply T = T / , an arbitrary skew symmetrical matrix, which is right. 
Next, if il is skew symmetrical, then T = — 0 -1 i2 -1 T / + T /} which can only be finite 
for T, = 0, that is, we have T = — 0 _1 il _1 0, and (the first part of T being always 
symmetrical) this represents an arbitrary symmetrical matrix. The mode in which this 
happens will be best seen by an example. Suppose 
and write 
then we have 
n-' = ( A , H+v ), tr. O" 1 = ( A , H-v), 
| H-v, B I \h + v, B | 
t, = ( o, 0), 
1-0, o| 
T = -(A, H)~'( 0, v)( 0, 0) + ( 0, 6) 
| -H, b | j - v, o ! ! - e, o | - e, o | 
v6 
AB— H 2 
(-B, 
H, 
H) + ( o, 0) 
-a\ I - 0, 0 I 
( vB6 
AB-H 2 
-vHQ 
AB-H 2 
-e, 
-vH6 
AB - H 2 
+ 0 
vA6 
AB-H 2 
0 )■ 
o I 
When il is skew symmetrical, A, B, H vanish; but since their ratios remain arbitrary, 
we may write kA, kB, kH for A, B, H, and assume ultimately re = 0. Writing k6 
in the place of 6, and then putting re = 0, the matrix becomes 
( vB6 — vB6 ) 
AB- H 2 ’ AB-H 2 
— vH6 vAQ 
AB — H 2 ’ AB-H 2 
which, inasmuch as A : 6, B : 6, and G : 6 remain arbitrary, represents, as it should do, 
an arbitrary symmetrical matrix.