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