166 



III. "A Fifth Memoir upon Quantics." By ARTHUR CAYLEY, 

 Esq., F.R.S. Received February 11, 1858. 



(Abstract.) 



The present memoir was originally intended to contain a develop- 

 ment of the theories of the covariants of certain binary quantics, viz. 

 the quadric, the cubic, and the quartic ; but as regards the theories 

 of the cubic and the quartic, it was found necessary to consider the 

 case of two or more quadrics, and I have therefore comprised such 

 systems of two or more quadrics, and the resulting theories of the 

 harmonic relation and of involution, in the subject of the memoir ; 

 and although the theory of homography or of the anharmonic rela- 

 tion belongs rather to the subject of bipartite binary quadrics, yet 

 from its connexion with the theories just referred to, it is also con- 

 sidered 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 quadrics, 

 and the theories above referred to; Nos. 1 15 to 127 to the cubic, and 

 Nos. 128 to 145 to the quartic. The several quantics are considered 

 as expressed not only in terms of the coefficients, but also in terms 

 of the roots, and I consider the question of the determination of 

 their linear factors, a question, in effect, identical with that of the 

 solution of a quadric, cubic, or biquadratic equation. The expression 

 for the linear factor of a quadric is deduced from a well-known for- 

 mula ; those for the linear factors of a cubic and a quartic were first 

 given in my "Note sur les Covariants d'une fonction quadratique, 

 cubique ou biquadratique a deux indeterminees," Crelle, vol. 1. pp. 

 285 to 287, 1855. It is remarkable that they are in one point of 

 view more simple than the expression for the linear factor of a 



