308 Proceedings of the Royal Society of Edinburgh. [Sess. 
and thence by addition, whatever s may be, 
a n(A s i + A ls ) + • • • + ct, rn (A sn + = 2zA rs . 
Writing co sr for (A sr + A rs ) 20 he thus has the set of n equations in a> sl , 
0) S 2 , • • • , W sn > 
+ G>i n u) sn 
CO 
II 
^21 <0 sl “t ' ’ 
‘ ' + a in M sn 
= a 2s> 
<^l w sl + ' ’ 
• • +a nn o) sn 
II ■ 
> ' 
a 
the peculiarity of which is that the right-hand members are cofactors of 
a column of elements of the determinant formed from the coefficients on 
the left. The solution is thus 
_ A^ s A^ r -I - A 2 s A 2r -f - ■ “t A. ns A nr 
0) sr ^ ) 
whence 
f A A A ft A A A ^ Asr + A rs ^ 
y -“-Isj -^*-255 • • • j -“-2n • • • > -“■ nr J ^ j 
as desired. Using this n 2 times, he, of course, obtains for the square of the 
adjugate the expression 
A n . 
(On 
o>12 . . 
• • w ln 
W 21 
w 22 ’ * 
. co 2n 
(x) n2 
1 • M nn 
and, it being known otherwise that the square of the adjugate is A 2 n ~ 2 , it 
follows that 
I W 11 w 22 • * ' w nn I ~ A n " , 
which is the other result wanted. 
In regard to the elements to one fact is noted, and is worth noting. 
Since A sr may be expressed as an aggregate of terms in z°, z l , z 2 , , 
namely, say 
A sr = ©o ®]2 + ©2 $ + • • • • 
and since A rs is got from A sr by altering the signs of all the (n— l) 2 
elements and then changing — z into 0 , there results when n is even 
A sr + A rs = 2©-^ -t- 2© 3 z 3 + • • • •; 
in other words, A sr + A rs is then divisible by 2 0 . 
Two “ observations ” are added, the first in regard to the case where 
0 = 0, and the second in regard to an alternative proof of the first part 
of the foregoing. The latter is interesting in that the expression for 
(A sr + A rs ) A is not found at once as a whole, but is viewed as consisting 
