1910-11.] The Less Common Special Forms of Determinants. 319 
the product becomes 
(a - y)(P - y) (P - y)(a - y) 
(a - S)(/3' - S') (P~S)(a'-S') 
and there is obtained 
(y - a )( 8 ' ~ a ') ( s ~ a )(Y - °0 
(y-m'-P) (8-p)(y-p) 
1 a a act! 
la P' a P' 
1 P P PP 
1 P a Pa 
1 y y yy 
1 y S' yS' 
1 S S' SS' 
S y' Sy 
= ( ( a - y)(P ~ s )( a ' - 8 ')(P - y) 1 2 
» - ( a - W - y)( a ' ~ y)(P - S') J • 
From this Cayley concludes (1) that '\f r is not changed by the transposition 
r« y\ 
V S'/’ 
and (2) that either form equals 
(a-y)(P-S)(a'-S')(P'-y') - (a-S)(/J-y)( a' - y')(/5' - S') . 
Sardi, C. (1864). 
[Quistione 39. Giornale di Mat, ii. p. 256, pp. 315-316.] 
On the determinant '\f r Sardi performs the operation which we may 
indicate by 
thus obtaining 
col 4 - p col 3 - S' col 2 + pS' colj 
1 a a (a — P)( a — S') 
1 P P? 
i y y (r - £)(y' - 8 ') 
1 S S' 
in which the cofactor of (a — B)(a — S') is 
1 
P 
P 
1 
y 
7 
or 
1 
s 
S' 
and the cofactor of (y — /3)(y’ — S') is 
1 a a 
- 1 P P' or - 
1 8 S' 
p~y p'-y 
S — y S' — y 
I p~a P' — a 
I S — a S' — a' 
where, be it observed, it is the rows 1, /8, f3' and 1, <5, S' that are diminished 
on both occasions. 
