The table is arranged according to the value of m; and we have
m = 0, n = 1, the point; m = 1, n = 0, the line; m = 2, n = 2, the
conic; of m = 3, the cubic, there are three cases, the class being
6, 4 or 3, according as the curve is without singularities, or as it
has 1 node or 1 cusp; and so of m = 4, the quartic, there are ten
cases, where observe that in two of them the class is = 6,—the
reduction of class arising from two cusps or else from three nodes.
The ten cases may be also grouped together into four, according
as the number of nodes and cusps (δ + κ) is = 0, 1, 2 or 3.
The cases may be divided into sub-cases, by the consideration of compound singularities; thus when m = 4, n = 6, δ = 3, the three nodes may be all distinct, which is the general case, or two of them may unite together into the singularity called a tacnode, or all three may unite together into a triple point or else into an oscnode.
We may further consider the inflections and double tangents, as well in general as in regard to cubic and quartic curves.
The expression for the number of inflections 3m(m − 2) for a curve of the order m was obtained analytically by Plücker, but the theory was first given in a complete form by Hesse in the two papers “Über die Elimination, u.s.w.,” and “Über die Wendepuncte der Curven dritter Ordnung” (Crelle, t. xxviii., 1844); in the latter of these the points of inflection are obtained as the intersections of the curve u = 0 with the Hessian, or curve Δ = 0, where Δ is the determinant formed with the second derived functions of u. We have in the Hessian the first instance of a covariant of a ternary form. The whole theory of the inflections of a cubic curve is discussed in a very interesting manner by means of the canonical form of the equation x3 + y3 + z3 + 6lxyz = 0; and in particular a proof is given of Plücker’s theorem that the nine points of inflection of a cubic curve lie by threes in twelve lines.
It may be noticed that the nine inflections of a cubic curve represented by an equation with real coefficients are three real, six imaginary; the three real inflections lie in a line, as was known to Newton and Maclaurin. For an acnodal cubic the six imaginary inflections disappear, and there remain three real inflections lying in a line. For a crunodal cubic the six inflections which disappear are two of them real, the other four imaginary, and there remain two imaginary inflections and one real inflection. For a cuspidal cubic the six imaginary inflections and two of the real inflections disappear, and there remains one real inflection.
A quartic curve has 24 inflections; it was conjectured by George Salmon, and has been verified by H. G. Zeuthen that at most eight of these are real.
The expression ½m(m − 2)(m2 − 9) for the number of double tangents of a curve of the order m was obtained by Plücker only as a consequence of his first, second, fourth and fifth equations. An investigation by means of the curve Π = 0, which by its intersections with the given curve determines the points of contact of the double tangents, is indicated by Cayley, “Recherches sur l’élimination et la théorie des courbes” (Crelle, t. xxxiv., 1847; Collected Works, vol. i. p. 337), and in part carried out by Hesse in the memoir “Über Curven dritter Ordnung” (Crelle, t. xxxvi., 1848). A better process was indicated by Salmon in the “Note on the Double Tangents to Plane Curves,” Phil. Mag., 1858; considering the m − 2 points in which any tangent to the curve again meets the curve, he showed how to form the equation of a curve of the order (m − 2), giving by its intersection with the tangent the points in question; making the tangent touch this curve of the order (m − 2), it will be a double tangent of the original curve. See Cayley, “On the Double Tangents of a Plane Curve” (Phil. Trans. t. cxlviii., 1859; Collected Works, iv. 186), and O. Dersch (Math. Ann. t. vii., 1874). The solution is still in so far incomplete that we have no properties of the curve Π = 0, to distinguish one such curve from the several other curves which pass through the points of contact of the double tangents.
A quartic curve has 28 double tangents, their points of contact determined as the intersections of the curve by a curve Π = 0 of the order 14, the equation of which in a very elegant form was first obtained by Hesse (1849). Investigations in regard to them are given by Plücker in the Theorie der algebraischen Curven, and in two memoirs by Hesse and Jacob Steiner (Crelle, t. xlv., 1855), in respect to the triads of double tangents which have their points of contact on a conic and other like relations. It was assumed by Plücker that the number of real double tangents might be 28, 16, 8, 4 or 0, but Zeuthen has found that the last case does not exist.
8. Invariants and Covariants. Polar Curves.—The Hessian Δ has just been spoken of as a covariant of the form u; the notion of invariants and covariants belongs rather to the form u than to the curve u = 0 represented by means of this form; and the theory may be very briefly referred to. A curve u = 0 may have some invariantive property, viz. a property independent of the particular axes of co-ordinates used in the representation of the curve by its equation; for instance, the curve may have a node, and in order to this, a relation, say A = 0, must exist between the coefficients of the equation; supposing the axes of co-ordinates altered, so that the equation becomes u′ = 0, and writing A′ = 0 for the relation between the new coefficients, then the relations A = 0, A′ = 0, as two different expressions of the same geometrical property, must each of them imply the other; this can only be the case when A, A′ are functions differing only by a constant factor, or say, when A is an invariant of u. If, however, the geometrical property requires two or more relations between the coefficients, say A = 0, B = 0, &c., then we must have between the new coefficients the like relations, A′ = 0, B′ = 0, &c., and the two systems of equations must each of them imply the other; when this is so, the system of equations, A = 0, B = 0, &c., is said to be invariantive, but it does not follow that A, B, &c., are of necessity invariants of u. Similarly, if we have a curve U = 0 derived from the curve u = 0 in a manner independent of the particular axes of co-ordinates, then from the transformed equation u’ = 0 deriving in like manner the curve U′ = 0, the two equations U = 0, U′ = 0 must each of them imply the other; and when this is so, U will be a covariant of u. The case is less frequent, but it may arise, that there are covariant systems U = 0, V = 0, &c., and U′ = 0, V′ = 0, &c., each implying the other, but where the functions U, V, &c., are not of necessity covariants of u.
If we take a fixed point (x′, y′, z′) and a curve u = 0 of order m, and suppose the axes of reference altered, so that x′, y′, z′ are linearly transformed in the same way as the current x, y, z, the curves [x′(∂/∂x) + y′(∂/∂y) + z′(∂/∂z)]ru = 0, (r = 1, 2, ... m − 1) have the covariant property. They are the polar curves of the point with regard to u = 0.
The theory of the invariants and covariants of a ternary cubic function u has been studied in detail, and brought into connexion with the cubic curve u = 0; but the theory of the invariants and covariants for the next succeeding case, the ternary quartic function, is still very incomplete.
9. Envelope of a Curve.—In further illustration of the Plückerian dual generation of a curve, we may consider the question of the envelope of a variable curve. The notion is very probably older, but it is at any rate to be found in Lagrange’s Théorie des fonctions analytiques (1798); it is there remarked that the equation obtained by the elimination of the parameter a from an equation ƒ(x, y, a) = 0 and the derived equation in respect to a is a curve, the envelope of the series of curves represented by the equation ƒ(x, y, a) = 0 in question. To develop the theory, consider the curve corresponding to any particular value of the parameter; this has with the consecutive curve (or curve belonging to the consecutive value of the parameter) a certain number of intersections and of common tangents, which may be considered as the tangents at the intersections; and the so-called envelope is the curve which is at the same time generated by the points of intersection and enveloped by the common tangents; we have thus a dual generation. But the question needs to be further examined. Suppose that in general the variable curve is of the order m with δ nodes and κ cusps, and therefore of the class n with τ double tangents and ι inflections, m, n, δ, κ, τ, ι being connected by the Plückerian equations,—the number of nodes or cusps may be
