# Annie Louisa MacKinnon's thesis

Annie MacKinnon was awarded a Ph.D. by Cornell University in 1894 for her thesis Concomitant Binary Forms in Terms of the Roots. We present below a version of the Preface.

Preface

In the preface to Gordan's Lectures on the Invariant Theory, Dr Kerchensteiner speaks of three books as containing the substance of the modern Invariant Theory - the books of Salmon, Clebsch, and Faà di 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 Invariant 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. Faà di 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 recognised, or if he recognised 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 recognises 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:
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 realisation 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 systematise the calculation and comparison of Covariants and Invariants in terms of the roots. The subject is presented according to the following arrangement:

Part I. - General Theory.
1. Theory of Covariants and Invariants of binary quantics in terms of the roots.
2. Comparison of Root and Coefficient Symbols.
3. Tables of Covariants and Invariants of the lower quantics including the sextic, and of pairs of the first five quantics (including linear quantics).
4. Particular classes of Forms and Operators.
Part II. - Some Geometrical interpretations and applications.
1. Geometry of Binary Forms.
2. Particular Covariants and Invariants with geometrical interpretations.
3. Binary root forms in their relations to certain ternary forms.

Last Updated December 2021