204 Proceedings of the Royal Society of Edinburgh, 
which from Laplace’s expansion-theorem we know to be equal to 
or 
2 
1 <*A 
CL-J-Tq 
a A 
a 6 h 1 
1 «A 
a 2pQ 
«<*A a A ■ ■ ■ 
a A “6*1 
. . . d 9 \ 
^2^6 • * * 
$2^9 
. . . a 9 b 2 
[Sess. 
Further, it is seen that the other elements of the compound determinant 
take like forms, and that in fact the said determinant is 
A 1 -B 1 
A x -B 2 . 
• • • 
A 2 -Bi 
A 2 .B 2 . 
... A 2 .B 6 
, 
Af,' Bj 
A 6 -B 2 
. . . A 6 -B 6 
. . . A : ' 1 
1 \ 
if A and B be taken to denote the arrays j 
“A 
a A 
«A 
a A a A 
. . . d 9 b-^ 
a A 
a 2 b Q 
. . . a 9 b 2 
a A 
a A ■ ■ ■ 
a 3 b 9 
a rJ J 2 a A 
. . . a 9 b z 
aA 
$q& 9 , 
a A a A 
. . . a 9 b±. 
As for the right-hand member of the general identity, the first determinant 
in it becomes 1, and the second becomes 
• 
• 
a A 
aA . . . 
a A 
• 
• 
a A 
a A • • • 
a 2 b 9 
«A 
aA ■ ■ ■ 
a 2 ,b 9 
• 
* 
• 
a A 
aA ■ ■ ■ 
a A 
a b \ 
a b b 2 
1 
■ 
■ 
a 6 b 2 
a fA 
1 
aA 
a A 
a A 
« 9 ?>4 
l 
which equals A • B ; so that the member in question becomes 
(A-B)» 
as it ought. 
The important thing to note in connection with the deduction here 
made is the fact that Sylvester must at this date have known how to 
