170 Proceedings of Royal Society of Edinburgh. [sess. 
The continuation of the process, and the same treatment applied 
to the equation 
x * ■ ^ = yi 1 + 1/2 2 + 2/s 2 
1 . . fljj 
. 1 . x 2 
1 Z 3 
thus lead us to the result that, for any positive integer p, we 
have — 
x 1 x 2 x s 
*1 
X 2 
X?, 
GiV + GiV + w 
Not only therefore do we know that T^Gx^za 2 can be expressed 
A 
in terms of the a?’s, hut the actual form of the expression — and a 
beautifully simple form — is obtained. 
(3) If this result is to hold for negative values of p, some con- 
vention must be established as to negative powers of a matrix. 
Now according to Cayley the first negative power, M -1 , is 
defined by the equation 
«11 
£?12 
CO 
II 
7 
: a u 
A 
A 21 
A 
A 3 I ) 
A 
a 21 
^22 
a 23 
A-12 
"aT 
81 *- 
A gg 
X 
«31 
«32 
A-13 
A 
^23 
A 
A 33 
A 
where A = | a n a 22 a 33 | and A n , A 12 , . . . are the cofactors of 
a n , a l2 , . . . in A : consequently the p th negative power, M -p , 
may be viewed either as 
(< 
ffl n 
a \2 
a i3 ■ 
r 
) 
®21 
^22 
a 2Z 
> 
| 
a 3l 
a Z2 
^33 
( A u 
A 21 
; A^31 
A!, 
■ *. 
A -12 
A-22 
A -32 
A 
A A 
A-13 
A-23 
A 33 
A 
A 
A 
