1867.] 



Prof. A. Cayley's Eighth Memoir on Quantics. 



331 



of the degree 6. The determination of the two co variants (Tables 83 and 

 84 post.), and of the syzygies of the degree 6, occupies the commence- 

 ment of the present memoir. The remainder of the memoir is in a great 

 measure a reproduction (with various additions and developments) of re- 

 searches contained in Prof. Sylvester's Trilogy, and in a recent memoir by 

 M. Hermite*. In particular, I establish in a more general form (de- 

 fining for that purpose the functions which I call " Auxiliars ") the theory 

 which is the basis of Prof. Sylvester's criteria for the reality of the roots 

 of a quintic equation, or, say, the theory of the determination of the cha- 

 racter of an equation of any order. By way of illustration, I first apply 

 this to the quartic equation ; and I then apply it to the quintic equation, 

 following Prof. Sylvester's track, but so as to dispense altogether with his 

 amphigenous surface, making the investigation to depend solely on the 

 discussion of the bicorn curve, which is a principal section of this surface. 

 I explain the new form which M. Hermite has given to the Tschirnhauseri 

 transformation, leading to a transformed equation, the coefficients whereof 

 are all invariants; and, in the case of the quintic, I identify with my 

 Tables his cubicovariants (.r, y) and (p., (a?, y). And in the two new 

 Tables, 85 and 86, I give the leading coefficients of the other two cubi- 

 covariants ^3 (x, y) and (.r, y). In the transformed equation the second 

 term (or that in z^) vanishes, and the coefficient ^ of is obtained as a 

 quadric function of four indeterminates. The discussion of this form led to 

 criteria for the character of a quintic equation, expressed like those of 

 Prof. Sylvester in terms of invariants, but of a different and less simple 

 form ; two such sets of criteria are obtained, and the identification of these 

 and of a third set resulting from a separate investigation, with the criteria 

 of Prof. Sylvester, is a point made out in the present memoir. The theory 

 is also given of the canonical forms, which is the mechanism by which 

 M. Hermite's investigations were carried on. The memoir contains other in- 

 vestigations and formulae in relation to the binary quintic ; and as part of 

 the foregoing theory of the determination of the character of an equation, 

 I was led to consider the question of the imaginary linear transformations 

 which give rise to a real equation : this is discussed in the concluding arti- 

 cles of the memoir, and in an annex I have given a somewhat singular 

 analytical theorem arising thereout. 



^ Sylvester "On the Eeal and Imaginary Eoots of Algebraical Equations ; a Trilogy," 

 Phil. Trans, t. dir. (1864) pp. 579-666; Hermite, "Sur 1' Equation du 5^ Degre,' 

 Comptes Rendus, t. Ixi. (1866), and in a separate form, Paris, 1866. 



