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

COMPOUND DETERMINANTS (REISS, 1867)

183

The complementary minor of this in its own compound being
I 12 13 23 1
| 134 1 35 4511,
and in the complementary compound being
1 1345 245 145 1
-1 1251 1124 123 1,
the result obtained from the theorem is
12345 1 345 245 1145
12345 1251 124 1123 1
The second factor here, however, being equal to
345 1245 145
512 412 312 1,
is observed to be an ' extensional' of
321
5431,
and therefore to be equal to
32145 45 2
54312 12.
Reiss thereupon goes a little beyond Franke and gives
14 15 24 25 34 35 45 1_ 12345 2 45 2
1S12 13 14 15 23 24 25 i 12345 12.
An investigation then follows (pp. 37-50) for the purpose of enabling
us in every case where it is possible to take this additional step and
others like it; in other words, for the purpose of resolving any
minor of any compound of C into factors which are C and minors
of C. It is duly recognised that on the application of Franke's
theorem we may not always be led to a minor like
345 245 145
125 124 123 1,
which is fully resolvable,-that, for example, the said application
might bring us down to such a compound determinant as
12 34 I