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

COMPOUND DETERMINANTS (SYLVESTER, 1851) 197

We thus learn that Sylvester's general result may be accurately described in later phraseology as the " extensional" of the
theorem regarding the rt~ compound of a determinant, and that
the discovery of both the said theorem and of its "extensional"
is almost certainly due to him. At the same time it is hard
to believe that this "extensional" is the all-embracing theorem
referred to by him in a previous paper: for by no stretch of
imagination could we see comprised in it "all the heretofore
existing doctrine of determinants." His last words thereanent
are: "This very general theorem is itself several degrees
removed from my still unpublished Fundamental Theorem,
which is a theorem for the expansion of products of determinants."
SYLVESTER, J. J. (1852, Dec.).
[On a theorem concerning the combination of determinants.
Cambridge and Dub. Math. Journ., viii. pp. 60-62; or
Collected Math. Papers, i. pp. 399-401.]
The statement of the theorem referred to in the title unfortunately shows want of proper care,* with the result that it is
unnecessarily lengthy. It may be recast as follows:Tf from the array
all Ca12 *.. aln
a21 a22... (2n or A, say,
al,, 2 * * amn
we form every possible array of r voZvs (r > mi < n), calling the
said arrays Al, A2,..., w..., ere of course mu = Ce,,; and if the
corresponding arrays formed from
bl b12.. bl
b21 b22... b2, or B, say,
bml b?,2.... b,
* See especially line 8 from bottom of p. 61, where in every case m should be m - 1.