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

DETERMINANTS IN GENERAL (REISS, 1867)

23

'extended by (t... t'm, he arrives at Schweins' theorem for
1... * Tom
transforming an aggregate of products of pairs of determinants
into another aggregate of similar kind (Hist., i. pp. 165-172), and
then advances to the further conclusion that this theorem also can
be 'extended.' Further, he notes that a theorem of Sylvester's
in compound determinants (Hist., ii. pp. 60-61) is an instance *
of 'extension,' and that it may be deduced from Jacobi's
theorem regarding a minor of the adjugate. Lastly, he gives
what he calls an application of Sylvester's and Jacobi's theorems
to functions of the form.eaAaCal.. Ca,,,
where the c's are determinants, the A's are coefficients independent
of the c's, the e's are signs, and the function as a whole is, like the
individual determinants, homogeneous in the elements. Such a
'doppelt-homogene Function,' if it vanishes identically, he finds
has two interesting properties, namely, (1) it may be 'extended'
and still vanish, (2) the identity will continue to hold if the determinants be replaced by their coefficients in another determinant
to which they all belong as minors. It is also carefully noted
that the same properties belong to the difference of two such
functions if they be identical.
It is much to be regretted that these theorems of Reiss' received
no attention from his contemporaries. His work is not even
mentioned by any of the German text-books giving bibliographical
references, for example, Baltzer's editions of 1870, 1875, 1881 and
Gunther's of 1875, 1877. Had it been otherwise, the generalisations
known as the Law of Complementary Minors and the Law of
Extensible Minors would have been formulated much earlier than
they were.
The second and third Sections of the memoir (pp. 25-99) deal
with comnpound determinants, and the fourth Section (pp. 100-113)
with geometrical applications.
* As we have already seen, Bazin's theorem of 1851 is another instance: also,Chio's of 1853; indeed the latter may be viewed as an extreme case of Sylvester's,(see Hist., ii. pp. 206-207, 80).