Part III gives an introduction to the symbolic notation
of Aronhold and Clebsch. The notation is first explained at
length for a simple case; likewise the fundamental theorem
on the types of symbolic factors of a term of a covariant of
binary forms is first proved for a simple example by the method
later used for the general theorem. In view of these and
similar attentions to the needs of those making their first
acquaintance with the symbolic notation, the difficulties usually
encountered will, it is believed, be largely avoided. This
notation must be mastered by those who would go deeply
into the theory of invariants and its applications.
Hilbert’s theorem on the expression of the forms of a set
linearly in terms of a finite number of forms of the set is proved
and applied to establish the finiteness of a fundamental set
of covariants of a system of binary forms. The theory of
transvectants is developed as far as needed in the discussion
of apolarity of binary forms and its application to rational
curves (§§ 53-57), and in the determination by induction of
a fundamental system of covariants of a binary form without
the aid of the more technical supplementary concepts employed
by Gordan. Finally, there is a discussion of the types of sym
bolic factors in any term of a concomitant of a system of
forms in three or four variables, with remarks on line and plane
coordinates.
For further developments reference is made at appropriate
places to the texts in English by Salmon, Elliott, and Grace
and Young, as well as to Gordan’s Invariantentheorie. The
standard work on the geometrical side of invariants is Clebsch-
Lindemann, Vorlesungen über Geometrie. Reference may be
made to books by W. F. Meyer, A Polarität und Rationale Curve,
Bericht über den gegenwärtigen Stand der Invariantentheorie, and
Formentheorie. Concerning invariant-factors, elementary divi
sors, and pairs of quadratic or bilinear forms, not treated here,
see Muth, Elementartheiler, Bromwich, Quadratic Forms and
their Classification by Means of Invariant Factors, and Bócher’s
Introduction to Higher Algebra. Lack of space prevents also
the discussion of the invariants and covariants arising in the
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PREFACE