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

COMPOUND DETERMINANTS (REISS, 1867)

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different collection of suffixes, the product DD' is expressible as a compound determinant in which the (i, j)t7 element is a determinant denoted
by all the n letters accompanied by the ith of the first sets of suffixes and
the (n —j)t' of the second sets. The proof is a mere application of the
multiplication-theorem, with the assistance of Laplace's expansiontheorem. Thus, if n= 5, k= 2 and the letters be a, b, c, d, e, we
first form the ten combinations ab, ac,..., de, then the determinants | a1b2, | a3c4,... by taking any suffixes whatever, and
finally from these the compound determinant
I ab2 1 a4 I ad6 I * I,
so that we have
D _ a11b62 | agc2 1 | al2 - d. e
a3b4 a3c 4 a3d4 1... I d3e4
Ia5b6 1 I a5c6g | a5d6g 1 d,e I
10'
Similarly, we form the combinations abc, abd,.., cde, writing them
for present convenience in the reverse order, then the determinants
c d e bde
1' 2' 3' ' 4' 5' 6',' and finally have
cde bde bce
- '3' 4' ' 6' '8' 9'
_ cdee bde bce
- 1 '2'3' 1'2'3'
c d e b d e b c e I
4/5'6' 4'5'G' 4'5'6'
c d e b d e bce
7' 8'9' 7' '9' 7' 8'9'
The result of the multiplication of these, after altering the signs of
the 2nld, 4th, 6th, 9th columns in one of them, is
DD' - abc d e abc d e
_ 121'2' 3' a12 45' 6
abc d e abc d e
341'2':' 3844' 5' 6.......... 10 '