1899 - 1900 .] Dr Muir on the Theory of Skew Determinants. 195 
reality with Cayley’s determinant solution of the problem of 
orthogonal transformation. 
In a sense the mode of treatment is indirect, the general skew 
determinant being viewed, not as a separate entity, hut in its 
relation to a set of linear equations, the coefficients of which are 
its elements. The set of equations is 
(11)^ + (12)ar 2 -t- . . . + (1 n)x n = u x ] 
(21)^ + (22);r 2 + . . . + (2n)x n = u 2 
I 
(nl)x 1 + (n2)x 2 + . . . + {nn)x n — u n J , 
where it has to be remembered that in every instance (rr) = 0 and 
(rs) + ( sr ) = 0. The right-hand members of what he calls the 
“derived” set are v v v 2 , . . ., v n m } that is to say, there exists 
simultaneously with the original the set 
(ll)#q + (21 )z 2 + . . . + ( nX)x n = v-^ ] 
(12-K + (22)^ 2 + . . . + (n2)x n = v 2 ^ 
x • ' I 
(1 n)x 1 + (2n)x 2 + . . . + ( nn)x n — v n J 
whose determinant is got from the determinant of the former set 
by the change of rows into columns, and may therefore he de- 
nominated by the same symbol A . Solving the two sets of equa- 
tions we have 
x 1 A =| 
[UK 
+ 
[12K 
+ . , 
. • + 
[i*k, 1 
II 
<1 
03 
P IK 
+ 
[22K 
+ . 
. . + 
[2 n]u n , 
X n /\ = 
[wljzq 
+ 
[» 2 K 
+ . . 
, . + 
1 
[nn]u n , J 
and 
x 1 A = 
[n>i 
+ 
[21> 2 
+ . . 
. + 
[nl K, 
Xq A =| 
: 12 
+ 
[22]e 2 
+ . . 
. + 
[n2]v ni 
«»A = 
[1»>1 
+ 
[2 n]v 2 
+ . . 
- . + 
\nn]v n 
where, he it remarked, it would have been much better if in every 
case the coefficients of u r and v r had been interchanged, for then 
* There is herein used the fact, first noted by Rothe in 1800, that the 
cofactor of rs in any determinant is equal to the cofactor of sr in the conjugate 
determinant. 
