VI. A second memoir upon quantics

Abstract

The present memoir is intended as a continuation of my Introductory Memoir upon Quantics, t. 144. (1854) p. 245, and must be read in connexion with it ; the paragraphs of the two Memoirs are numbered continuously. The special subject of the present memoir is the theorem referred to in the Postscript to the Introductory Memoir, and the various developments arising thereout in relation to the number and form of the covariants of a binary quantic. 25. I have already spoken of asyzygetic covariants and invariants, and I shall have occasion to speak of irreducible covariants and invariants. Considering in general a function u determined like a covariant or invariant by means of a system of partial differential equations, it will be convenient to explain what is meant by an asyzygetic integral and by an irreducible integral. Attending for greater simplicity only to a single set ( a , b , c ...), which in the case of the covariants or invariants of a single function will be as before the coefficients or elements of the function, it is assumed that the system admits of integrals of the form u = P, u = Q, &c., or as we may express it, of integrals P, Q, &c., where P, Q, &c. are rational and integral homogeneous functions of the set ( a , b , c ...), and moreover that the system is such that P, Q, &c. being integrals, ϕ (P, Q..) is also an integral. Then considering only the integrals which are rational and integral homogeneous functions of the set ( a , b , c ...), integrals P, Q, R,.. not connected by any linear equation or syzygy (such as λP + μQ + γR.. = 0*), are said to be asyzygetic; but in speaking of the asyzygetic integrals of a particular degree, it is implied that the integrals are a system such that every other integral of the same degree can be expressed as a linear function (such as λP + μQ + γR.. = 0*) of these integrals; and any integral P not expressible as a rational and integral homogeneous function of integrals of inferior degrees is said to be an irreducible integral.