82 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
— (A 3 H 3 + A 2 G 3 + AjFg) • Rj 
+ (A 3 H 2 + A 2 G 2 + AjFg) • 
+ (A 3 H 1 + A 2 G 1 + A 1 F 1 )-M 1 
Ai A 2 Ac 
Fi G, Hj 
F 2 G s 
H, 
G q He 
M, 
R ]: 
as before. Similarly, having reached this we can show that the cofactor of 
I «i h c : 
is 
$1 
$ 2 
«8 
*1 
Vi 
fl 
ffl 
\ 
m 2 
m 3 
ft 
9* 
h 2 
n i 
n 2 
n B 
ft 
<Js 
r i 
r i 
i is 
the product of 
.i> 
\fl 92 
, | m 1 
»2 r S 
1 » 1*1 
A, 
^2 
^3 
x, 
Y, 
Zi 
*1 
G i 
M, 
M„ 
m 3 
f 2 
g 2 
h 2 
N, 
n 2 
g 8 
h 3 
^3 } 
and so on. Consequently, as the restriction in the foregoing to three-line 
determinants is readily seen to detract in no way from the validity of the 
procedure, we have the following general theorem — The coj actor of any 
element in the determinants 
a n $22 
, | l>\\ l > 2 
being denoted by the corresponding capital letter similarly suffixed, the 
cofactor of any element in the determinant which is the product of those 
determinants is got by changing every letter in the expression for the said 
element into the corresponding capital letter. 
From this we have the corollary that the adjugate of the product oj 
any number of determinants is not only equal to but is identical in form 
with the product of their adjugates. 
4. The theorem of § 3, it will be seen, may be described as asserting 
that the minors of highest order (the principal minors) and the minors of 
lowest order (the elements) in the product-determinant are expressible in 
one and the same form. Let us now consider the minors of intermediate 
order; but, in so doing, let us forsake the classification of minors into 
principal, secondary, etc., and group them instead according to the number 
of their rows and columns. 
5. Beginning then with minors of two rows we note first that one case 
has already been considered, the principal minors of a determinant of the 
