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

192 HISTORY OF THE THEORY OF DETERMINANTS

same memoir established the theorem that the determinant of
any " derived system " of a product-determinant is equal to the
product of the determinants of the corresponding "derived
systems" of the two factors.
Those are all the general results that fall to be noted prior to
the middle of the nineteenth century; and, as is readily seen,
they all concern what at a later date came to be called the
"compounds" of lanl. With one exception they are due to
Cauchy.*
The fact has also to be recalled, however, that compound
determinants of a special type were considered by Jacobi in
1841, namely, those whose elements are functional determinants,
his main theorem being
J (1) (2).() - fl af2 fm"nl' + fl af2 Dfn+
wh ere1 2 f ' x 3 Za -n c to '' ' xm+m
where fl, f2,.., f.n+, are functions of x1, x,,..., x+m and
',= - x1 'a2 ' ' ' xn+r.'
This theorem and certain deductions therefrom have been already
dealt with in another connection (see pp. 381-385 of History, i.).
SYLVESTER, J. J. (1850).
[On the intersections, contacts, and other correlations of two
conies expressed by indeterminate co-ordinates. Cambridge
and Dub. Math. Journ., v. pp. 262-282; or Collected Math.
Papers, i. pp. 119-137.]
In a footnote to this paper the name "Compound Determinants" first appears. The passage is (p. 270): "... a theorem
given by M. Cauchy, and which is included as a particular case
in a theorem of my own relating to Compound Determinants,
i.e. Determinants of Determinants, which will take its place as
an immediate consequence of my fundamental theorem given
* They are numbered xx., xxi., xli., xlii. in History, i.