677 
1908-9.] Superadjugate and Skew Determinants. 
Thus the above expression for B 3 may also be written 
b^.x^ — 7'^T^.X" + + — Fgh 3 . 
D. If I ana 22 • • • a nn | be a skew determinant with univarial ■ diagonal , 
then 
i( A ^ + A qp ) = ~ a n ~ 5 -^[p a (3y][<i a fiy] -••• 
or 
= + + . ■ . 
according as n is odd or even. (xxii.) 
This follows at once from (xxi.) and (ix.). 
18. If A be a skew determinant with a univarial diagonal, the product 
of the p th and f h rows of the superadjugate of A is 0 when p and q are 
different, and is A 2 when p and q are the same. (xxiii.) 
For when p and q are different we have 
(Spi , S p 2 , . S pn () , b‘g2 ) • • • j Sqn ) 
4a 2 ( Ap 4 , Ap 2 j • • • ? A pn $ Ag 4 , Ag 2 > • • • j A qk) ~ A (j!(iA.qp + lu A p p , 
and therefore by (xvi.) 
= 4« 2 . J(A. P2 + A qp )— - 2aA(A^ + A M ) = 0; 
CL 
and when p and q are the same the product 
= 4« 2 (A i ,i + Ay, + . . . + A p l) — 4:(iA pp A + A 2 
and therefore by (xvi.) 
= 4a 2 .A ?) „- - 4a A™, A + A 2 
a 
= A 2 , 
as was to be proved. 
The theorem is Cayley’s for the case where a — 1. 
From it we have 
S 2 
A 2 0 0 . . . 
0 A 2 0 
0 0 A 2 . . . 
A 2n , 
which agrees with (iii.) and § 6. 
19. If | a n a 22 . . . a nn I, or A say, be a skew determinant with univarial 
diagonal, r p r q the product of its p //( and (ff rows, and F the Pfaffan of the 
elements on the right of the diagonal, then 
- a 2 rp 2 .... rp n 
*7i r n r 2 r n r n - a 2 
| 0 when n is odd. 
J (xxiv.) 
I F 4 when n is even. 
