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 three covariants and a single syzygy, for the degree 6 we obtain only two covariants, but as many as seven 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 = ) six 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 Prof. 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 Prof. 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 I then apply it to the quintic equation, following Prof. Sylvester’s track, hut 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
ϕ
1
(
x
,
y
) and
ϕ
2
(
x
,
y
). And in the two new Tables, 85 and 86, I give the leading coefficients of the other two cubi covariants
ϕ
3
(
x
,
y
) and
ϕ
4
(
x
,
y
). In the transformed equation the second term (or that in
z
4
) vanishes, and the coefficient
A
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 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 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.
Analogues of the binomial coefficient theorems of Gauss and Jacobi