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XVI. An Eighth Memoir on Quantics. By A. Cayley, F.B.S. 
Beceived January 8, — Bead January 17, 1867. 
The present Memoir relates mainly to the binary quintic, continuing the investigations 
in relation to this form contained in my Second, Third, and Fifth Memoirs on Quantics ; 
the investigations which it contains in relation to a quantic of any order are given with 
a view to their application to the quintic. All the invariants of a binary quintic (viz. 
those of the degrees 4, 8, 12, and 18) are given in the Memoirs above referred to, and 
also the covariants up to the degree 5 ; it was interesting to proceed one step further, 
viz. to the covariants of the degree 6 ; in fact, while for the degree 5 we obtain 3 cova- 
riants and a single syzygy, for the degree 6 we obtain only 2 covariants, but as many as 
7 syzygies; one of these is, however, the syzygy of the degree 5 multiplied into the 
quintic itself, so that, excluding this derived syzygy, there remain (7 — 1 = ) 6 syzygies of 
the degree 6. The determination of the two covariants (Tables 83 and 84 post) and of 
the syzygies of the degree 6, occupies the commencement of the present Memoir. 
The remainder of the Memoir is in a great measure a reproduction (with various 
additions and developments) of researches contained in Professor Sylvester’s Trilogy, 
and in a recent memoir by M. Hermite*. In particular, I establish in a more general 
form (defining for that purpose the functions which I call “ Auxiliars”) the theory which 
is the basis of Professor Sylvester’s criteria for the reality of the roots of a quintic 
equation, or, say, the theory of the determination of the character of an equation of any 
order. By way of illustration, 1 first apply this to the quartic equation ; and I then apply 
it to the quintic equation, following Professor 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 Tschirnhausen 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 p,(# , y) and <p 2 (#, y). 
And in the two new Tables, 85 and 86, I give the leading coefficients of the other two 
cubicovariants <p 3 (tv, y) and <p t (w, y). In the transformed equation the second term (or 
that in z 4 ) vanishes, and the coefficient 91 of z 3 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 Professor Sylvester in terms of invariants, but of a 
* Sylvester “On the Beal and Imaginary Boots of Algebraical Equations ; a Trilogy,” Phil. Trans, vol. 154 
(1864), pp. 579-666. Hermite, “ Sur l’Equation du 5 e degre,” Comptes Bendus, t. 61 (1866), and in a 
separate form, Paris, 1866. 
4 A 
MDCCCLXVII. 
