1899-1900.] Dr Muir on the Theory of Skew Determinants. 197 
How the said latter portion — that is to say, the deduction from 
this — can be justified is a mystery ; but of course if it he granted 
there is no objection to the cogency of the next step in the reason- 
ing, which is worded as follows : — 
“But since it is found on trial that for n= 1, 3, . . ., A 
vanishes, while for n = 2, 4, . . ., it does not, the following 
theorems may be enunciated : — 
“ Theorem XIV. A symmetrical skew determinant of an odd 
order in general vanishes , and the system has for its inverse 
an unsymmetrical skew system. 
“Theorem XY. A symmetrical skew determinant of an 
even order does not in general vanish , but the system lias foi 
its inverse a symmetrical skew system.” 
The name, however, given to the “ inverse system ” in the first 
case when, as we have seen, [rs] = [sr] is clearly inappropriate; 
and it is not improved in the second edition by alteration into 
“quadratic skew,” the fact being that the system is not skew at 
all, but is symmetric with respect to the principal diagonal, or, in 
later phraseology, is axisymmetric. 
The treatment of the next theorem taken up is happier than the 
foregoing, and is after the outset no less fresh. Taking an even- 
ordered skew determinant with zeros in the principal diagonal he 
develops it according to products of an element of the first row 
and an element of the first column, the result being written in the 
form 
* 12 . . 
21 * 
. In 
2 n 
= (12f 
* 34 . . 
43 * . . 
. 3 n 
. 4 n 
+ 2 (12) (13) 
34 
* 
35 .. 
45 .. 
. 32 
. 42 
n\ n2 
* 
n3 ni . . 
* 
ni 
n*b . . 
. n2 
where, be it observed, the second typical term on the right has 
been altered from 
-2(12) (13) 
32 34 ... 3 n 
42 * ... in 
•* 
■> 
n2 ni . . 
