Combinantal Forms of a Quadric Net Applied to Irreducible Concomitant Systems

A complete system of combinantal forms (generalised and ordinary) of a pencil of quadrics fλ ≡ λ1f1 + λ2f2 can be chosen such that the coefficients of the various power products of λ1, λ2 in the former give a complete irreducible system of concomitants of the two quadrics f1, f2, and conversely. This result was proved by Todd (1), who used it in conjunction with Schur function analysis (2) (3) to derive the complete irreducible system of concomitants of two quaternary quadratics (4).

The Complete Irreducible System of Invariants of Three Quadrics

The present account is an application of the principles of combinantal forms and Schur function analysis given in a previous paper (A), the references therein being henceforward denoted by A1 to A9, and the complete irreducible system of invariants of three quaternary quadrics will now be obtained from the complete system (not necessarily itself irreducible) derived by Turnbull (A5, p. 483). This latter system comprises 47 invariants, viz. 15, 1, 6, 6, 1, 15 and 3 members of total degrees 4, 6, 8, 10, 12, 14 and 18 respectively in the coefficients of the quadrics. It will be proved that all of these are irreducible except for the one of degree 12 and the three of degree 18, the former being of especial interest as it is a real combinant and moreover, involves unusual features in the proof of its reduction and also in the derivation of the form expressing it in terms of irreducible invariants.

Ternary quadratic types

The first part of this paper deals with the determination of the complete system of concomitants of five or fewer ternary quadratic forms. In the second part of the paper it is shown that this system is irreducible, and that from it may be deduced the irreducible system of ternary quadratic types, thus giving a classification of the irreducible concomitants of any number of ternary quadratics.

Combinants of a pencil of quadric surfaces. II

1. In the first paper of this series I have explained a method by which the complete irreducible system of combinants of a pencil of quadric surfaces may be obtained, and have determined explicitly the combinantal invariants and covariants. The present paper deals with combinantal contravariants. The notation used is that of I, a knowledge of which will be assumed.

A Geometrical Interpretation of the Symmetrical Invariant of Three Ternary Quadratics

In the paper, “Sul sistema di tre forme ternarie quadratiche,” Ciamberlini has derived the complete irreducible system of concomitants for three ternary quadratics and has given a short treatment of their geometrical interpretations. Among the concomitants is the invariant (abc)2 which is symmetrical and linear in the coefficients of each quadratic. The purpose of this note is to give a geometrical interpretation of the invariant, and to extend the result for symmetrical invariants of forms in higher dimensions.

II.—The Irreducible System of Concomitants of Two Double Binary (2, 1) Forms

Little is known of the details of systems of concomitants belonging to double binary forms. The cases of the single ground form of orders (1, 1), (2, 1), (2, 2) respectively, together with the simultaneous system of any number of (1, 1) forms, are the only four cases, which have been published. The following pages establish the simultaneous system of two (2, 1) forms.This system is fundamental for the geometrical treatment of two twisted cubics lying upon a quadric surface and having four common points.

XIV.—The Minimum System of Two Quadratic Forms in n Variables

The following pages deal with the simultaneous system of two general quadratic forms in n homogeneous variables. It is a special case of Gordan's Theorem which proves such systems to be finite, for the general projective group of linear transformations. While several works have dealt with the cases when n = 2, 3, or 4, nothing seems to have been written on the general case except a memoir in the year 1908. We continue, and simplify, the results there obtained, and now establish that(1) All rational integral concomitants of two quadratic forms in n variables and any number of sets of linear variables may be expressed as rational integral functions of (3n + 1) concomitants, and forms derivable by polarisation.(2) These (3n + 1) forms, called the H system, constitute a strictly irreducible system.This system is exhibited in §7 astogether with polars.The work is divided into three chapters: I §§ 1–4 is introductory notation, II §§ 5–17 provides a proof of these theorems, while III §§18–21 gives the non-symbolic and canonical forms of the results.