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

COMPOUND DETERMINANTS (BAZIN, 1854)

207

the determinant whose columns are the rth, Sth, tth, Uth columns
of the array, if the array consist of four rows and six columns,
we have
5234 6234
1534 1634 = 1234-5634;
1534 1634
and if it consist of four rows and seven columns, we have
5234 6234 7234
1534 1634 1734 = (1234)2.(5674).
1254 1264 1274
K being the compound determinant involved, Bazin's mode of
proof consists in expressing K/D7' in the form of a determinant
of the same order as D, then multiplying D by K/D7I, and with
the help of a lemma finding the product to be E. This at once
gives
K = Dh-1E
as desired. Thus, taking the first of the two examples just
given, we have
a1 a2 a. a4 5234/D 6234/D
b1 b2 b3 b4 1534/D 1634/D
c1 c2 c3 c4 1254/D 1264/D 1
d1 d2 d3. d4 1235/D 1236/D. 1
a5 a6 a3 a4
b5, b b3 b
C5 C6 C3 C4
d5 d6 C73 d4
that is to say,
D. 5234 6234 D-=
1534 1634
and practically nothing more is wanted.
and thence
1274 3564 + 1734.2564 - 7234. 5614 = 1234 5674,
or, on removing the ' extension,'
127 356 - 137 256 - 237 156 = 123 567.
A relation is thus established between two very unlike theorems.