405] 
14 7 
405. 
AN EIGHTH MEMOIR ON QUANTICS. 
[From the Philosophical Transactions of the Royal Society of London, vol. clvii. 
(for the year 1867). Received January 8,—Read January 17, 1867.] 
I HE present Memoir relates mainly to the binary quintic, continuing the investi 
gations in relation to this form contained in my Second, Third, and Fifth Memoirs on 
Quantics, [141], [144], [156]; 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 covariants 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. [These covariants 
83, 84 are the covariants M and N of the paper 143, “ Tables of the covariants M to 
W of the binary quintic ”, and they are accordingly not here reproduced.] 
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( 1 ). 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, I first apply this to the quartic equation ; and 
1 Sylvester “On the Real and Imaginary Roots of Algebraical Equations; a Trilogy,” Phil. Trans, vol. cliv. 
(1864), pp. 579—666. Hermite, “Sur l’Équation du 5° degré,” Comptes Pendus, t. lxi. (1866), and in a separate 
form, Paris, 1866. 
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