1908-9.] Superadj ugate and Skew Determinants. 
675 
16. For the purposes of (xviii.) the most suitable expression for A pq is as 
a series of terms arranged according to powers of a, it being then possible 
to make, by the application of (x.), an immediate simplification of the 
expression for \(A pq + A qp ). To obtain the former expression let us pass 
the p th and q th rows over all the others so as to become the 1st and 2nd, and 
then do the like with the p th and q th columns. The result 
| Cl pp Cl qq CL^^'22 ‘ ' ’ %>-l , P— 1%>+1 .2H-1 • ’ • a <l- 1 . l^h+l . 2+1 ’ * * ®nn | > 
which is known to be still a skew determinant with univarial diagonal, has 
the element a pq the place (1, 2), and its cofactor A pq is seen to be 
— | (Xq p dyP 22 • • • * * • ®q—\ , q—\®$jr \ , 2+1 • • * | • 
We thus learn that If \ a n a 22 . . . a nn | be a skew determinant with univarial 
diagonal, any primary minor of it is expressible as the result of “ bordering ” 
a skew determinant of like kind ; or, more definitely, is the determinant 
obtained by prefixing to the secondary coaxial minor 
the row 
I ^11^22 ' - 
. • • • 
a q-l,q-\ a q+l . 2+1 • • • 
^nn 1 
Cl qp , O q -y , (Zq2 , 
• • • 5 ^q,p — 1 ) ^q,p + 1 5 • 
• • 3 ®q , g-1 5 , q+\ > • 
• • i %n 
column 
O'qp j ^1 p •) ^2 P ’ 
• * • 5 ^p— 1 ,p ) @ J p + 1 ,p } 
• ' • » ^q — 1 ,p > ®q + 1 ,p 5 
• • • j ® np 
king the sign - 
. 
(xx.) 
From this it follows that in A Pq the terms containing the product of all 
the original diagonal elements is 
- a qp . a n ~ 2 , i.e. a pq . a n_2 ; 
and that the cofactor of the product of any other number of the said 
elements is a “ bordered ” zero-axial skew determinant, and therefore is 
expressible as the product of two Pfaffians. Thus the cofactor of 
is 
* • • a p~l ,P~\®P+\ ,P+1 • • • ®q-l , q-l^q+l , 2+1 • • • a nn 
l.G. Cty^fcty^Ctq^ Qj2 p Cl q y H - CtqpCt^y) j 
Ctqp 
Mql ^22 
a 
IP 
a 
12 
(1*2 p 
and the cofactor of 
is 
a 44: • 
■ • Mp—ltP- 
■l^P+l ,2>+l 
• • a a-i,<z- 
l^'ff+l.tf-Hl • * • a nn 
Ctqp 
Ctqy 
Ctqo Ctq£ 
i.e. - | 
Ctqi Ctq2 
a q3 
• 1 Cl p y Ctp2 
a pz 
Ctyp 
• 
Cfj- ^2 ®]3 
^12 
^13 
a \2 
«1 3 
a 2 P 
a 21 
. CLc)q 
a 2Z 
a 23 
^3 P 
«31 
« 32 
