Agnes Scott College

Annie MacKinnon

Concomitant Binary Forms in Terms of the Roots
The Annals of Mathematics, Vol. 9, No. 1/6 (1894), pp. 95-157


In the preface to Gordan's Lectures on the Invariant Theory [Gordan's Vorlesungen fiber Invariantentheorie-Kerschensteiner, Leipzig, 1885]. Dr. Kerschensteiner speaks of three books as containing the substance of the modern Invariant Theory-the books of Salmon, Clebsch, and Faa de Bruno. Adding to these Gordan's Lectures and Burnside and Panton's Theory of Equations, we include the principal works which give a general presentation of the In-variant Theory of the present time. In these five books, the prominence given to the expression of Covariants and Invariants in terms of the roots is various. Fag de Bruno's treatment of the root expressions is the most extensive ; he gives tables of the Invariants (not Covariants) of the lower binary quantics through the Sextic (omitting B and D of the Sextic), expressed as functions of root differences. There is no suggestion of any system for calculating the root expressions of these tables, aside from the use of coefficients expressed as symmetric functions of the roots.

Burnside and Panton approach the subject of Covariants and Invariants through the expressions of symmetric functions of, root differences, and make use of symmetric functions of the roots to establish connections between the root and coefficient forms.

Scattered through Salmon's book on Modern Higher Algebra are many of the simpler Covariant and Invariant root expressions. The methods presented by Salmon for the calculation of root forms are based upon symmetric functions of the roots, symmetric functions of root differences, and upon the use of any convenient geometric relation obtained through the coefficient forms, and also upon the use of transformed equations. There is in this book, no recognition of symbolic root forms nor of the possibility of the application of Cayley's symbolic operators to the calculation of root expressions for Covariants and Invariants.

Both Clebsch and Gordan touch upon a theory of symbolic root forms, the theory to be presented in this paper. Clebsch appears not to have recognized, or if he recognized has not made clear, the directness and simplicity of the connection which exists between root and coefficient symbolic expressions for Covariants and Invariants. Gordan fully recognizes the relation between the two forms of symbolic expression ; and the work which follows in this paper, though developed independently of Gordan's work in this line, is in reality an application of the underlying principle of the Gordan symbolism.

As far as I have been able to ascertain there has been in English writings no recognition of symbolic methods applied to the expression of Covariants and Invariants in terms of the roots, excepting (possibly) the following sentence by Sylvester [American Journal of Mathematics, Vol. II (1879), p. 329]: " Gordan's and Jordan's results concerning symbolic determinants are correlative and coextensive with theorems concerning root differences, so that the method of differentiants when fully developed would lead to the substitution of actual differences or determinants for symbolic determinants in the Gordan theory."

A realization of the substantial identity of the form of a Covariant root symbol with the form of its expression in the root differences, and of the directness of the interpretation of one form from the other, brings into clearer light the practical value of German Symbolism in Modern Algebra.

The following pages present the results of a study of root forms and of an attempt to systematize the calculation and comparison of Covariants and In-variants in terms of the roots.

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