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

186 HISTORY OF THE THEORY OF DETERMINANTS

each set of column-suffixes into its two parts, we obtain a quite
simple and satisfactory notation: for example, the above determinant may be appropriately denoted by
12345 12 345 1345 12 345 12 345 12 345 1
12,567 13,568 14,569 23,578 24,579 34,589,
if the requisite convention be adopted with reference to the use of
the comma. Both kinds of compound determinants appear in the
simple identity
bi cl b4c5 e4act, l 45 5 a J C4C6 alb6C7 a1lbc IS
b2 c2 I b6c7 I I C6(7 I 67 ( 2b4cs 5 I c2b 7 I I a2bsc
b 3 c ]C 69 | [ CSa I (|3b9 j 3c5 ( 3b c 7 a 3b c
and this in the notations referred to is
a lbc3 | b1 ca1 a, 11b | b | abb6 6c7 5 7 bN, a, C b9 l.
On occasion the invariable set of row-suffixes may be temporarily
dispensed with: this is in effect what is done when the principal
minors of an oblong array are denoted by the numbers of their
columns only. It is the latter contracted notation which was used
in 1851 by Bazin in his statement of the first theorem regarding such
determinants (Hist., ii. pp. 206-208).
Reiss' notation is of course very different from the foregoing, both
in principle and in appearance, and considerable space is again
devoted to it. Six pages have to be gone through before his first
theorem is reached. This theorem, however, is fundamental, and
deserves every attention. It may be viewed as giving the resolution
of a special compound determinant of the second kind into two
factors which are compound determinants of the first kind. We
shall take it in a slightly different light, and for the sake of clearness
give a formal enunciation of it in words, detailing fully the construction of the determinants involved:-If the combinations of n
letters k at a time be taken, and each combination with any k suffixes
be used to denote a determinant, and the whole of such determinants
be made the diagonal of a compound determinant and be understood
to suggest as usual the other elements of the latter, then, D being this
determinant and D' a determinant similar in construction save that the
combinations are taken n-k at a time and are furnished from a