946 Proceedings of Royal Society of Edinburgh. [sess. 
aa 
ad 
ad' b c \ 
da 
do! 
a a" b' c 
a” a 
,, , 
n n i n a 
a a 
a a DC 
P 
p 
P" ■ ■ 
y 
y 
y" ■ ■ 
pointing out by way of demonstration that the former of these is 
arrived at hy transforming the given determinants into 
a 
b c . 
a (3 
y 
a 
b' c . 
and - 
a /S' . 
7 
a" 
b" c" . 
a" (3" . 
7 
. . 1 
. 1 
and applying the ordinary rule of multiplication, and similarly 
that the latter is got by multiplying 
a b c 
a .. (3 y 
a b' c . 
d .. (3' y 
a" b" c" . . 
b y 
d' . . /r / 
. . . 1 . 
. i . . . 
. . . . 1 
. . i . . 
He thereupon leaves the subject with the remark : — 
“This rule is interesting as exhibiting .... a complete 
scale whereby we may descend from the ordinary mode of 
representing the product of two determinants to the form 
.... where the two original determinants are made to 
occupy opposite quadrants of a square whose places in one of 
the remaining quadrants are left vacant, and shows us that 
under one aspect at least this latter form may be regarded as 
a matrix bordered by the two given matrices.” 
The second lemma is the identity — 
a \2 a iZ 
a i n 
1 - 
An 
-^12 • 
. . . A ln 
1 
«S1 
(%22 • • • • 
a 2 n 
1 
An 
■^22 • 
■ • • -^~2 n 
1 
U„2 . . . . 
&nn 
1 
A 
-A j -w2 • 
• • • Kn 
1 
1 
1 . . . . 
1 
1 
1 . 
. . . 1 
