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 greater for particular values of the parameter, but this is a speciality which may be here disregarded. Considering the variable curve corresponding to a given value of the parameter, or say simply the variable curve, the consecutive curve has then also δ and κ nodes and cusps, consecutive to those of the variable curve; and it is easy to see that among the intersections of the two curves we have the nodes each counting twice, and the cusps each counting three times; the number of the remaining intersections is = m² − 2δ − 3κ. Similarly among the common tangents of the two curves we have the double tangents each counting twice, and the stationary tangents each counting three times, and the number of the remaining common tangents is = n² − 2τ − 3ι (= m² − 2δ − 3κ, inasmuch as each of these numbers is as was seen = m + n). At any one of the m² − 2δ − 3κ points the variable curve and the consecutive curve have tangents distinct from yet infinitesimally near to each other, and each of these two tangents is also infinitesimally near to one of the n² − 2τ − 3ι common tangents of the two curves; whence, attending only to the variable curve, and considering the consecutive curve as coming into actual coincidence with it, the n² − 2τ − 3ι common tangents are the tangents to the variable curve at the m² − 2δ − 3κ points respectively, and the envelope is at the same time generated by the m² − 2δ − 3κ points, and enveloped by the n² − 2τ − 3ι tangents; we have thus a dual generation of the envelope, which only differs from Plücker’s dual generation, in that in place of a single point and tangent we have the group of m² − 2δ − 3κ points and n² − 2τ − 3ι tangents.
The parameter which determines the variable curve may be given as a point upon a given curve, or say as a parametric point; that is, to the different positions of the parametric point on the given curve correspond the different variable curves, and the nature of the envelope will thus depend on that of the given curve; we have thus the envelope as a derivative curve of the given curve. Many well-known derivative curves present themselves in this manner; thus the variable curve may be the normal (or line at right angles to the tangent) at any point of the given curve; the intersection of the consecutive normals is the centre of curvature; and we have the evolute as at once the locus of the centre of curvature and the envelope of the normal. It may be added that the given curve is one of a series of curves, each cutting the several normals at right angles. Any one of these is a “parallel” of the given curve; and it can be obtained as the envelope of a circle of constant radius having its centre on the given curve. We have in like manner, as derivatives of a given curve, the caustic, catacaustic or diacaustic as the case may be, and the secondary caustic, or curve cutting at right angles the reflected or refracted rays.
10. Forms of Real Curves.—We have in much that precedes disregarded, or at least been indifferent to, reality; it is only thus that the conception of a curve of the m-th order, as one which is met by every right line in m points, is arrived at; and the curve itself, and the line which cuts it, although both are tacitly assumed to be real, may perfectly well be imaginary. For real figures we have the general theorem that imaginary intersections, &c., present themselves in conjugate pairs; hence, in particular, that a curve of an even order is met by a line in an even number (which may be = 0) of points; a curve of an odd order in an odd number of points, hence in one point at least; it will be seen further on that the theorem may be generalized in a remarkable manner. Again, when there is in question only one pair of points or lines, these, if coincident, must be real; thus, a line meets a cubic curve in three points, one of them real, and other two real or imaginary; but if two of the intersections coincide they must be real, and we have a line cutting a cubic in one real point and touching it in another real point. It may be remarked that this is a limit separating the two cases where the intersections are all real, and where they are one real, two imaginary.
Considering always real curves, we obtain the notion of a branch; any portion capable of description by the continuous motion of a point is a branch; and a curve consists of one or more branches. Thus the curve of the first order or right line consists of one branch; but in curves of the second order, or conics, the ellipse and the parabola consist each of one branch, the hyperbola of two branches. A branch is either re-entrant, or it extends both ways to infinity, and in this case, we may regard it as consisting of two legs (crura, Newton), each extending one way to infinity, but without any definite separation. The branch, whether re-entrant or infinite, may have a cusp or cusps, or it may cut itself or another branch, thus having or giving rise to crunodes or double points with distinct real tangents; an acnode, or double point with imaginary tangents, is a branch by itself,—it may be considered as an indefinitely small re-entrant branch. a branch may have inflections and double tangents, or there may be double tangents which touch two distinct branches; there are also double tangents with imaginary points of contact, which are thus lines having no visible connexion with the curve. A re-entrant branch not cutting itself may be everywhere convex, and it is then properly said to be an oval; but the term oval may be used more generally for any re-entrant branch not cutting itself; and we may thus speak of a once indented, twice indented oval, &c., or even of a cuspidate oval. Other descriptive names for ovals and re-entrant branches cutting themselves may be used when required; thus, in the last-mentioned case a simple form is that of a figure of eight; such a form may break up into two ovals or into a doubly indented oval or hour-glass. A form which presents itself is when two ovals, one inside the other, unite, so as to give rise to a crunode—in default of a better name this may be called, after the curve of that name, a limaçon (q.v.). Names may also be used for the different forms of infinite branches, but we have first to consider the distinction of hyperbolic and parabolic. The leg of an infinite branch may have at the extremity a tangent; this is an asymptote of the curve, and the leg is then hyperbolic; or the leg may tend to a fixed direction, but so that the tangent goes further and further off to infinity, and the leg is then parabolic; a branch may thus be hyperbolic or parabolic as to its two legs; or it may be hyperbolic as to one leg and parabolic as to the other. The epithets hyperbolic and parabolic are of course derived from the conic hyperbola and parabola respectively. The nature of the two kinds of branches is best understood by considering them as projections, in the same way as we in effect consider the hyperbola and the parabola as projections of the ellipse. If a line Ω cut an arc aa′ at b, so that the two segments ab, ba′ lie on opposite sides of the line, then projecting the figure so that the line Ω goes off to infinity, the tangent at b is projected into the asymptote, and the arc ab is projected into a hyperbolic leg touching the asymptote at one extremity; the arc ba′ will at the same time be projected into a hyperbolic leg touching the same asymptote at the other extremity (and on the opposite side), but so that the two hyperbolic legs may or may not belong to one and the same branch. And we thus see that the two hyperbolic legs belong to a simple intersection of the curve by the line infinity. Next, if the line Ω touch at b the arc aa’ so that the two portions ab, ba’ lie on the same side of the line Ω, then projecting the figure as before, the tangent at b, that is, the line Ω itself, is projected to infinity; the arc ab is projected into a parabolic leg, and at the same time the arc ba′ is projected into a parabolic leg, having at infinity the same direction as the other leg, but so that the two legs may or may not belong to the same branch. And we thus see that the two parabolic legs represent a contact of the line infinity with the curve,—the point of contact being of course the point at infinity determined by the common direction of the two legs. It will readily be understood how the like considerations apply to other cases,—for instance, if the line Ω is a tangent at an inflection, passes through a crunode, or touches one of the branches of a crunode, &c.; thus, if the line Ω passes through a crunode we have pairs of hyperbolic legs belonging to two parallel asymptotes. The foregoing considerations also show (what is very important) how different branches are connected together at infinity, and lead to the notion of a complete branch or circuit.