1899-1900.] Dr Muir on the Theory of Skew Determinants. 211 
have been already dealt with. These are in the first place (§3, 8 ; 
p. 12) Jacobi’s theorem regarding the vanishing of a zero-axial 
skew determinant of odd order, and Spottiswoode’s theorems 
regarding conjugate elements of the adjugate or inverse of a zero- 
axial skew determinant, the mode of proof for all being that used 
by Jacobi for his own theorem, viz., the multiplication of all rows 
or all columns by - 1, and then comparing the resulting deter- 
minant with the original. In the second place (§3, 10; p. 13) 
we have Brioschi’s theorem regarding the differential-quotient of a 
zero-axial skew determinant of eyen order, and a suggestive proof 
of the same which it is desirable to note. It is as follows : — Let 
the determinant 
a n . . 
• • a \n 
a n l • * 
. . a nn 
be denoted by A , and the cofactor of a rs in A by A rs . Then, bear- 
ing in mind that A is a function of a rs and that a sr is not indepen- 
dent of a rs , we have 
0A _ A , * da sr 
-c± rs T sr J 
oa rs ca rs 
= A rs — A sr1 because a sr = - a r s . 
But when n is even we know from Spottiswoode, as above, that 
A rs = - A sr ; consequently we have in this case 
3 A 
da rs 
2A 
r& j 
as Brioschi affirmed.* In the third place (§ 7, 5 ; pp. 28, 29) he 
applies Jacobi’s general theorem 
A rr A rs ^ 0^ A 
A sr A ss 'dct r7 Bci ss 
* It ought to be noticed also that Baltzer uses the equation 
to verify Spottiswoode’s theorem for the case where A is odd-ordered, the 
reasoning being that as A is then known to be zero, so also must 0A /dars, and 
that therefore A rs =A sr . 
