The theory of determinants in the historical order of development, by Sir Thomas Muir.

COMPOUND DETERMINANTS (SPOTTISWOODE, 1853) 201

SPOTTISWOODE, W. (1853).
[Elementary theorems relating to determinants. Second edition,
rewritten and much enlarged by the author. Crelle's
Journal, li. pp. 209-271, 328-381.]
In his first edition Spottiswoode had a section (~ vi.) headed
On Inverse Systems and Determinants of Determinants; but
in it, as we have seen, he dealt merely with the adjugate
determinant and its minors. Now, this is supplanted by a
section (Q x.) of considerably greater extent (pp. 350-372) with
the short Sylvesterian title On Compound Determinants.
Although it is the original definition of a compound determinant which is given on starting, the name afterwards
seems to be unconsciously limited to compound determinants
whose elements are minors of a given determinant of the nth
order. This? limitation leads to the introduction of the word
class in connection with compound determinants to indicate " the
degree of minority of the constituents" (i.e. elements), a compound determinant of the ith class being one whose elements are
minors of the (n-i)tlh order; thus, the adjugate determinant is
a compound determinant of the nth order and 1st class. If a
compound determinant of the ith class be of the highest possible
order, namely, the
n(n-1)... (n-i+)th,
1.2... i
-that is to say, contains all the minors of the (n -i)th orderSpottiswoode calls it the "comnplete determinant of the ith class,"
a name equivalent therefore to the more modern "(n — i)th
compound."
One of the notations employed is essentially the same as
Sylvester's-that is to say, he uses
f o21 h.. a1 12 22 w ' 2 l h ab 2b ee. w t 1
12.2.. U 1 22....u 2 in 2n... n J
for what at a later date would have been written
11 2I1..... I 2 22. u?12 276... u ~ 1 2
1\ 21... l 2... U1 2,... I 1 21