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

190( HISTORY OF THE THEORY OF DETERMINANTS

Similarly it may be shown that
111234 11235 11245 13451 1 2345 I 12345 12345 112345 12345
|1234 123a ly 12 8e lept nr0oK | 1234a ' 1233y 128eI lnOlK
and so on in general, a first requirement being that the k of the
theorem be equal to n-1. Again, the given determinant being
112 13 14 23 24 34 |
112 13 14 56 57 58
Reiss selects the twin form
34 24 23 i14 13 12
56 26 251 16 151 12,
which, being a second compound, is equal to
i1234 3
1256,
and, finding that the product of the two has a vanishing column,
concludes that
|12 13 14 23 24 34 1 _
1 12 13 14 56 57 58
Much of the space that follows (pp. 67-89) is occupied with the
question of the vanishing of compound determinants of the two
kinds, the requisite criteria being found for a number of special
forms. The results are mainly deductions from the fundamental
theorem or its corollaries.
The concluding portion (pp. 89-99) of the Section is devoted
to the establishment of a theorem regarding any minor of the
determinant
c+, kl V a' 1. a.. K
which we have seen to be the subject of the so-called Picquet's
theorem. The result obtained is related to the latter theorem
exactly as Franke's is related to Sylvester's, and, as Reiss points
out, degenerates into Franke's theorem simultaneously with the
degeneration of Picquet's into Sylvester's.
In an Appendix (pp. 100-113) Reiss makes a variety of
applications to Geometry.