1907-8.] Dr Muir on the Theory of Skew Determinants. 307 
worth noting. In the first place, there is his opening proposition that in 
any zero-axial shew determinant, conjugate minors, if of even order, are 
equal, and if of odd order differ only in sign : this is a slight generalisa- 
tion of a previously known result. In the second place (p. 80), Jacobi’s 
general theorem 
•o 
A rr 
A. sr 
A rs 
A qq 
A x compl. minor of 
a rr a rs 
Q-sr ^ss 
is applied to the case where A is a zero-axial skew determinant of even 
order, A 2to say, and where, therefore, A rr = 0 = A ss and A sr = — A rs , and the 
said complementary minor is a determinant of the same kind as A 2to but 
of the order 2 m — 2 : and it is thus seen that if A 2to _ 2 be a square, so also 
must A 2 m. The use to which this is put is evident. 
Janni, G. (1863). 
[Teorica di determinanti simmetrici gobbi. Giornale di Mat., i. pp. 
275-278.] 
Janni’s final result is a troublesome rule for finding the expression 
whose square is a skew determinant of even order, the line of thought, so 
far as it goes, being similar to Scheibner’s (1859). 
Cremona, L. (1864). 
D’Ovidio, Torelli, Magni (1865). 
[Quistione 32. Giornale di Mat., ii. p. 62 ; iii. pp. 5-7, 7-10, 10-14.] 
Cremona’s theorem is that if A be a shew determinant having its 
diagonal elements a n , a 22 , . . . , a nn each equal to z, then the product of any 
two rows or any two columns of the adjugate determinant contains A as 
a factor, and the determinant of the n 2 cofactors equals A” -2 . Proofs are 
given by E. D’Ovidio, C. Torelli, and A. Magni ; but the second alone need 
be attended to here, as the two others are less direct, being connected with 
the subject of orthogonal substitution. 
Starting with the known result 
«nA s i+ ••• +a rs A ss + ••• + a rn A sn = A when-s = r> 
= 0 when s =t=r f 
Torelli by subtraction of 2a rs A ss and change of signs obtains 
a n A ls + • • • + a rs A ss + . . . + a rn A ns = - A + 2zA rs when s = r) 
= 2^A rs when s 4= r ) 
