( 89 ) 
NOTE ON THE PRODUCT OF ANY DETERMINANT AND 
ITS BORDERED DERIVATIVE. 
By Sir Thomas Muir, LL.D. 
(1) The fundamental result here obtained is the theorem that the product 
of two determinants, the second of which is got by bordering the first, is 
expressible as a bilinear function of which the quasi variables are the cof actors 
of the bordering elements in the second determinant, and the discriminant is 
the unbordered determinant. For example, when the order of the initial 
determinant is the 3rd we have 
6, h 
ebi a 2 a$ a^ 
b Y b. 2 b s b 4 
D, D, D, 
h bo b. 
A, 
B 4 
k 
d\ do d 3 
By way of proof we note that on the right-hand side the cofactor of A 4 
= a l D 1 + aoDo + a 3 D 3 
+ «2 
a -2 a 3 a \ 
bo b 3 b 4 
Co C s C 4 
(Xj a 2 an . 
a Y a 3 a s a 4 
b i h h h 
C l Co C 3 C A 
! a l a 3 a 4 
bi bn b. 
b } bo bo b. 
a l a 3 a 4 
b 1 h 2 & 4 
Cj Co C 4 
= - a 4 | a 1 bo c 3 | , 
= cofactor of A 4 on the left ; 
and the outcome is similar when the cof actors of B 4 and C 4 are considered. 
(2) As every bordered determinant is already known to be expressible as 
a bilinear, for example, 
d, d, do 
