148 
AN EIGHTH MEMOIR ON QUANTICS. 
[405 
I then apply it to the quintic equation, following Professor Sylvester’s track, but so 
as to dispense altogether with his amphigenous surface, and 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 Tschirn- 
hausen 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 cubi- 
covariants </>j (x, y) and <p. 2 (x, y). And in the two new Tables, 85 and 86, I give the 
leading coefficients of the other two cubicovariants <£ :; (x, y) and </> 4 (x, y), [these are 
now also identified with my Tables], In the transformed equation the second term (or 
that in z 4 ) vanishes, and the coefficient 21 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 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 Professor Sylvester, is a point made out in the present memoir. 
The theory is also given of the canonical form which is the mechanism by which 
M. Hermite’s investigations were carried on. The Memoir contains other investigations 
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 articles of the memoir, and in an Annex I have given a 
somewhat singular analytical theorem arising thereout. 
The paragraphs and Tables are numbered consecutively with those of my former 
Memoirs on Quantics. I notice that in the Second Memoir, p. 126, we should have 
No. 26 = (No. 19) 2 - 128 (No. 25), viz. the coefficient of the last term is 128 instead 
of 1152. [This correction is made in the present reprint, 141, where the equation 
is given in the form Q' = G 2 — 128Q.] 
Article Nos. 251 to 254.—The Binary Quintic, Covariants and Syzygies of the degree 6. 
251. The number of asyzygetic covariants of any degree is obtained as in my 
Second Memoir on Quantics, Philosophical Transactions, vol. cxlvi. (1856), pp. 101—126, 
[141], viz. by developing the function 
1 
(1 — z) (1 — xz) (1 — afz) (1 — a?z) (1 — Pz) (1 — x’z) ’ 
as shown p. 114, and then subtracting from each coefficient that which immediately 
precedes it; or, what is the same thing, by developing the function 
1 — x 
(1 — z) (1 — xz) (1 — x~z) (1 — a?z) (1 — xtz) (1 — xPz) ’ 
which would lead directly to the second of the two Tables which are there given; 
the Table is there calculated only up to z*, but I have since continued it up to z 18 , 
so as to show the number of the asyzygetic covariants of every order in the variables 
up to the degree 18 in the coefficients, being the degree of the skew invariant, the 
highest of the irreducible invariants of the quintic. The Table is, for greater 
convenience, arranged in a different form, as follows: