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

66 HISTORY OF THE THEORY OF DETERMINANTS

It is convenient to state here that in the following year, in the
course of a short remark on Tanner's paper of 1878, Cayley asserts
the existence of the very important theorem afterwards known as
the Law of Complementaries. (See under Skew Determinants.)
GUNTHER, S. (1877): SALMON, G. (1877):
SERRET, J.-A. (1877).
[LEHRBUCH DER DETERMINANTEN... Zweite durchaus umgearbeitete, vermehrte und durch ein Aufgaben-Sammlung bereicherte Auflage. xii+209 pp. Erlangen.]
[VORLESUNGEN ZUR EINFUHRUNG IN DIE ALGEBRA DER LINEAREN
TRANSFORMATIONEN. Deutsch bearbeitet von Dr. Wilhelm
Fiedler. Zweite Auflage. xiv+477 pp. Leipzig.]
[COURs D'ALGEBRE SUPERIEURE. 4e ed. Tome i. xiv+649 pp.
Paris.]
GUnther's second edition is all that the title-page proclaims it
to be. The historical sketch is improved and lengthened by insertions in reference to Hindenburg, Reiss, Grassmann and others,
and by giving deserved attention to Studnicka's views of the preceding year. In the chapter on general determinants additional
readable notelets on minor matters are introduced; for example,
the paragraph (~ 5, p. 38) on the effect which is produced on a
determinant by reversing the order of its rows or the order of its
columns or the order of both. The bibliographical references at the
end of this chapter are also largely increased; the same, indeed,
is true of the whole. The Collection of Exercises (pp. 195-207)
includes sixty-six items; and it is followed by a two-page List of
Text-books in a variety of languages.
The second German edition of Salmon follows closely (pp. 1-66)
the third English edition, the additional matter consisting almost
entirely of illustrative examples.
Serret devotes a chapter (pp. 527-598) to determinants and certain
of their algebraical applications. The latter are the more interesting, the subjects dealt with being the discriminant (still called
' l'invariant') of a quadric, the transformation of a quadric, Sturm's
theorem and its application to Lagrange's determinantal equation.