The Product of Any Determinant and its Bordered Derivative. 91 
and a quite similar mode of proof suffices. Thus, taking the aggregate 
of the first three of the nine terms on the right, namely, 
| KVsl K& 3 ! — KVsl \ a A\ + KVsl \ a A\ [ • \ c \ c k e ?>\ 
we see that it 
a l a 2 a s a 4 a 5 
b Y bo b$ fr 4 6 5 
c 3 c 4 c» 
a 1 a% a§ . . 
b, bl b, . . 
| Cl d 2 e 3 | 
• • • 
■ • • hh 
C l C 2 C 3 C 4 C 5 
a>] aiQ a^ . , 
h l &2 &3 • • 
= I «A I • I «l^2 C 3 I • I C l C h e S I > 
which is also one of the three terms got on the left by using Laplace's 
expansion on the five-line determinant. 
The order of the square array of the bilinear depends on the order of 
the initiating determinant and the breadth of the border ; for example, 
a, a 2 a s a 4 
h h h h 
d x d 2 d 3 d 4 
a l ao a s a 4 a 5 a 6 
' b i h h h fc 5 h 
, C '\ C 2 C S C 4 C 5 C 6 
' d^ do d s d± d 5 d 6 
is equal to the bilinear whose square array is 
I a A I I a A I 
a A I 
a 3 c 4 I 
and whose laterals are 
|aA C Al»~l a 2 6 4 C 5 d 6l 
and 
|a 2 & 3 c 5 c? 6 |, la^CgcZg | , — |«ifr 3 c 5 c7 6 
l a 1^2 C 5^6 I » 
The general theorem may with useful enough fulness be enunciated as 
follows : The product of an i\-line determinant by the determinant got from 
it by bordering with r rows and r columns is expressible as a bilinear function 
whose quasi variables are the n-line minors of the multiplier that contain 
the bordering rows and the n-line minors that contain the bordering 
