671 
1908-9.] Superadjugate and Skew Determinants. 
this where A besides being skew is unit-axial to see that Cayley s orthogo- 
nant is resolvable into determinant factors — that, in fact, it is the row-by- 
column product of A -1 and A, where A is the basic determinant, and A -1 
is in the form which is got from the adjugate of A by dividing each 
element by A- (vii.) 
7. Confining ourselves now to skew determinants which have a univarial 
diagonal, let us denote the repeated element of the diagonal by a, so that, 
A being | a n a 22 . . . a nn | , it is understood that a rs = — a sr and a rr = a ss = . . . = a. 
The first property we note is that In any skew determinant with uni- 
varial diagonal the product of any two rows is the same as the product of 
the corresponding columns. (viii.) 
This is seen on writing the p th and q th rows 
ctpi > eCp 2 ? • • • ? eipp , ... , ctqq , ... , ctp n 
®>q\ 5 ^qi j • • • > ®qp J • • • j d'qq j • • • ) ™ qn 
and the p th and q th columns 
d\p 5 \p 5 • • • 5 ^pp » • • • j O'qp ®np 
q •> Ct.qq , ... , CLpq , ... , CLqq , ... . (V n q 
and then noting (1) that a pr . a qr — a rp . a rq because of the skewness, and (2) 
that 
G/pp.Giqp -j- CLpq.Gjqq CLpp.Llpq -1- Ctqp.C^qq 
because each of them = a(a pq + a qp ) = 0. 
8. In any skew determinant with a univarial diagonal the conjugate 
of any m -line minor is got from the latter by changing the signs of the 
elements of it that belong to the or iginal d iagonal and prefixing to it 
the sign ( — l) m . (ix.) 
For we can multiply the given minor by ( — l) m by changing the signs 
of all its elements, and we can change all the said signs by altering the 
elements of the form a pq into a qp , and those of the form a into —a, that is 
to say, by taking the conjugate minor and altering the signs of the elements 
that belong to the original diagonal. 
The number of 2-line minors of A that are independent of the diagonal 
element is 3C n>4 . (x.) 
For there are \n(n— 1) pairs of rows to be considered, each pair having 
n — 2 ){n — 3) minors of the kind mentioned, and each minor occurring 
twice. 
9. If A be skew and have a univarial diagonal, the product of the 
p- 7t row of the superadjugate of A by the (f h row of A is a pq A whether p and 
q be different or the same. (xi.) 
