XXIII. A fifth memoir upon quantics

The present memoir was originally intended, to contain a development of the theories of he covariants of certain binary quantics, viz. the quadric, the cubic, and the quartic; ut as regards the theories of the cubic and the quartic, it was found necessary to conider the case of two or more quadrics, and I have therefore comprised such systems of wo or more quadrics, and the resulting theories of the harmonic relation and of invoition, in the subject of the memoir; and although the theory of homography or of the inharmonic relation belongs rather to the subject of bipartite binary quadrics, yet from cs connexion with the theories just referred to, it is also considered in the memoir, The paragraphs are numbered continuously with those of my former memoirs on the subject: Nos. 92 to 95 relate to a single quadric; Nos. 96 to 114 to two or more quatrics, and the theories above referred to; Nos. 115 to 127 to the cubic, and Nos. 128 o 145 to the quartic. The several quantics are considered as expressed not only in erms of the coefficients, but also in terms of the roots,— and I consider the question of he determination of their linear factors,—-a question, in effect, identical with that of he solution of a quadric, cubic, or biquadratic equation. The expression for the linear actor of a quadric is deduced from a well-known formula; those for the linear factors of a cubic and a quartic were first given in my “Note sur les Covariants d’une fonction uadratique, cubique ou biquadratique à deux indéterminées,” Crelle, vol. L. pp. 285 o 287, 1855. It is remarkable that they are in one point of view more simple than he expression for the linear factor of a quadric. 92. In the case of a quadric the expressions considered are (a, b, c)(x, y)2, (1) ac — b2, (2) where (1) is the quadric, and (2) is the discriminant, which is also the quadrinvariant, atalecticant, and Hessian.