arising as follows:—1. The cusp: the point as it travels along the line may come to rest, and then reverse the direction of its motion.3. The stationary tangent: the line may in the course of its rotation come to rest, and then reverse the direction of its rotation.2. The node: the point may in the course of its motion come to coincide with a former position of the point, the two positions of the line not in general coinciding.4. The double tangent: the line may in the course of its motion come to coincide with a former position of the line, the two positions of the point not in general coinciding.
arising as follows:—
1. The cusp: the point as it travels along the line may come to rest, and then reverse the direction of its motion.3. The stationary tangent: the line may in the course of its rotation come to rest, and then reverse the direction of its rotation.2. The node: the point may in the course of its motion come to coincide with a former position of the point, the two positions of the line not in general coinciding.4. The double tangent: the line may in the course of its motion come to coincide with a former position of the line, the two positions of the point not in general coinciding.
1. The cusp: the point as it travels along the line may come to rest, and then reverse the direction of its motion.
3. The stationary tangent: the line may in the course of its rotation come to rest, and then reverse the direction of its rotation.
2. The node: the point may in the course of its motion come to coincide with a former position of the point, the two positions of the line not in general coinciding.
4. The double tangent: the line may in the course of its motion come to coincide with a former position of the line, the two positions of the point not in general coinciding.
It may be remarked that we cannot with a real point and line obtain the node with two imaginary tangents (conjugate or isolated point or acnode), nor again the real double tangent with two imaginary points of contact; but this is of little consequence, since in the general theory the distinction between real and imaginary is not attended to.
The singularities (1) and (3) have been termed proper singularities, and (2) and (4) improper; in each of the first-mentioned cases there is a real singularity, or peculiarity in the motion; in the other two cases there is not; in (2) there is not when the point is first at the node, or when it is secondly at the node, any peculiarity in the motion; the singularity consists in the point coming twice into the same position; and so in (4) the singularity is in the line coming twice into the same position. Moreover (1) and (2) are, the former a proper singularity, and the latter an improper singularity,as regards the motion of the point; and similarly (3) and (4) are, the former a proper singularity, and the latter an improper singularity,as regards the motion of the line.
But as regards the representation of a curve by an equation, the case is very different.
First, if the equation be in point-co-ordinates, (3) and (4) are in a sense not singularities at all. The curve (*x, y, z)m= 0, or general curve of the order m, has double tangents and inflections; (2) presents itself as a singularity, for the equations dx(*x, y, z)m= 0, dy(*x, y, z)m= 0, dz(*x, y, z)m= 0, implying (*x, y, z)m= 0, are not in general satisfied by any values (a, b, c) whatever of (x, y, z), but if such values exist, then the point (a, b, c) is a node or double point; and (1) presents itself as a further singularity or sub-case of (2), a cusp being a double point for which the two tangents becomes coincident.
In line-co-ordinates all is reversed:—(1) and (2) are not singularities; (3) presents itself as a sub-case of (4).
The theory of compound singularities will be referred to farther on.
In regard to the ordinary singularities, we have
and this being so, Plücker’s “six equations” are
It is easy to derive the further forms—
the whole system being equivalent to three equations only; and it may be added that using a to denote the equal quantities 3m + ι and 3n + κ everything may be expressed in terms of m, n, a. We have
κ = a − 3n,ι = a − 3m,2δ = m² − m + 8n − 3a.2τ = n² − n + 8m − 3a.
κ = a − 3n,
ι = a − 3m,
2δ = m² − m + 8n − 3a.
2τ = n² − n + 8m − 3a.
It is implied in Plücker’s theorem that, m, n, δ, κ, τ, ι signifying as above in regard to any curve, then in regard to the reciprocal curve, n, m, τ, ι, δ, κ will have the same significations, viz. for the reciprocal curve these letters denote respectively the order, class, number of nodes, cusps, double tangent and inflections.The expression ½m(m + 3) − δ − 2κ is that of the number of the disposable constants in a curve of the order m with δ nodes and κ cusps (in fact that there shall be a node is 1 condition, a cusp 2 conditions) and the equation (9) thus expresses that the curve and its reciprocal contain each of them the same number of disposable constants.For a curve of the order m, the expression ½m(m − 1) − δ − κ is termed the “deficiency” (as to this more hereafter); the equation (10) expresses therefore that the curve and its reciprocal have each of them the same deficiency.The relations m² − 2δ − 3κ = n² − 2τ − 3ι = m + n, present themselves in the theory of envelopes, as will appear farther on.
It is implied in Plücker’s theorem that, m, n, δ, κ, τ, ι signifying as above in regard to any curve, then in regard to the reciprocal curve, n, m, τ, ι, δ, κ will have the same significations, viz. for the reciprocal curve these letters denote respectively the order, class, number of nodes, cusps, double tangent and inflections.
The expression ½m(m + 3) − δ − 2κ is that of the number of the disposable constants in a curve of the order m with δ nodes and κ cusps (in fact that there shall be a node is 1 condition, a cusp 2 conditions) and the equation (9) thus expresses that the curve and its reciprocal contain each of them the same number of disposable constants.
For a curve of the order m, the expression ½m(m − 1) − δ − κ is termed the “deficiency” (as to this more hereafter); the equation (10) expresses therefore that the curve and its reciprocal have each of them the same deficiency.
The relations m² − 2δ − 3κ = n² − 2τ − 3ι = m + n, present themselves in the theory of envelopes, as will appear farther on.
With regard to the demonstration of Plücker’s equations it is to be remarked that we are not able to write down the equation in point-co-ordinates of a curve of the order m, having the given numbers δ and κ of nodes and cusps. We can only use the general equation (*x, y, z)m= 0, say for shortness u = 0, of a curve of themth order, which equation, so long as the coefficients remain arbitrary, represents a curve without nodes or cusps. Seeking then, for this curve, the values, n, ι, τ of the class, number of inflections, and number of double tangents,—first, as regards the class, this is equal to the number of tangents which can be drawn to the curve from an arbitrary point, or what is the same thing, it is equal to the number of the points of contact of these tangents. The points of contact are found as the intersections of the curve u = 0 by a curve depending on the position of the arbitrary point, and called the “first polar” of this point; the order of the first polar is = m − 1, and the number of intersections is thus = m(m − 1). But it can be shown, analytically or geometrically, that if the given curve has a node, the first polar passes through this node, which therefore counts as two intersections, and that if the curve has a cusp, the first polar passes through the cusp, touching the curve there, and hence the cusp counts as three intersections. But, as is evident, the node or cusp is not a point of contact of a proper tangent from the arbitrary point; we have, therefore, for a node a diminution 2, and for a cusp a diminution 3, in the number of the intersections; and thus, for a curve with δ nodes and κ cusps, there is a diminution 2δ + 3κ, and the value of n is n = m(m − 1) − 2δ − 3κ.
Secondly, as to the inflections, the process is a similar one; it can be shown that the inflections are the intersections of the curve by a derivative curve called (after Ludwig Otto Hesse who first considered it) the Hessian, defined geometrically as the locus of a point such that its conic polar (§8 below) in regard to the curve breaks up into a pair of lines, and which has an equation H = 0, where H is the determinant formed with the second differential coefficients of u in regard to the variables (x, y, z); H = 0 is thus a curve of the order 3(m − 2), and the number of inflections is = 3m(m − 2). But if the given curve has a node, then not only the Hessian passes through the node, but it has there a node the two branches at which touch respectively the two branches of the curve; and the node thus counts as six intersections; so if the curve has a cusp, then the Hessian not only passes through the cusp, but it has there a cusp through which it again passes, that is, there is a cuspidal branch touching the cuspidal branch of the curve, and besides a simple branch passing through the cusp, and hence the cusp counts as eight intersections. The node or cusp is not an inflection, and we have thus for a node a diminution 6, and for a cusp a diminution 8, in the number of the intersections; hence for a curve with δ nodes and κ cusps, the diminution is = 6δ + 8κ, and the number of inflections is ι = 3m(m − 2) − 6δ − 8κ.
Thirdly, for the double tangents; the points of contact of these are obtained as the intersections of the curve by a curve Π = 0, which has not as yet been geometrically defined, but which is found analytically to be of the order (m − 2)(m² − 9); the number of intersections is thus = m(m − 2)(m² − 9); but if the given curve has a node then there is a diminution = 4(m² − m − 6), and if it has a cusp then there is a diminution = 6(m² − m − 6), where, however, it is to be noticed that the factor (m² − m − 6) is in the case of a curve having only a node or only a cusp the number of the tangents which can be drawn from the node or cusp to the curve, and is used as denoting the number of these tangents, and ceases to be the correct expression if the number of nodes and cusps is greater than unity. Hence, in the case of a curve which has δ nodes and κ cusps, the apparent diminution 2(m² − m − 6)(2δ + 3κ) is too great, and it has in fact to be diminished by 2{2δ(δ − 1) + 6δκ +9⁄2κ(κ − 1)}, or the half thereof is 4 for each pair of nodes, 6 for each combination of a node and cusp, and 9 for each pair of cusps. We have thus finally an expression for 2τ, = m(m − 2)(m² − 9) − &c.; or dividing the whole by 2, we have the expression for τ given by the third of Plücker’s equations.
It is obvious that we cannot by consideration of the equation u = 0 in point-co-ordinates obtain the remaining three of Plücker’s equations; they might be obtained in a precisely analogous manner by means of the equation v = 0 in line-co-ordinates, but they follow at once from the principle of duality, viz. they are obtained by the mere interchange of m, δ, κ, with n, τ, ι respectively.
To complete Plücker’s theory it is necessary to take account of compound singularities; it might be possible, but it is at any rate difficult, to effect this by considering the curve as in course of description by the point moving along the rotating line; and it seems easier to consider the compound singularity as arising from the variation of an actually described curve with ordinary singularities. The most simple case is when three double points come into coincidence, thereby giving rise to a triple point; and a somewhat more complicated one is when we have a cusp of the second kind, or node-cusp arising from the coincidence of a node, a cusp, an inflection, and a double tangent, as shown in the annexed figure, which represents the singularities as on the point of coalescing. The general conclusion (see Cayley,Quart. Math. Jour.t. vii., 1866, “On the higher singularities of plane curves”;Collected Works, v. 520) is that every singularity whatever may be considered as compounded of ordinary singularities, say we have a singularity = δ′ nodes, κ′ cusps, τ′ double tangents and ι′ inflections. So that, in fact, Plücker’s equations properly understood apply to a curve with any singularities whatever.
By means of Plücker’s equations we may form a table—
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 siximaginaryinflections 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 theTheorie 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 theenvelopeof a variable curve. The notion is very probably older, but it is at any rate to be found in Lagrange’sThéorie 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 begreater 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 them-th order, as one which is met by every right line inmpoints, 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.
The two legs of a hyperbolic branch may belong to different asymptotes, and in this case we have the forms which Newtoncalls inscribed, circumscribed, ambigene, &c.; or they may belong to the same asymptote, and in this case we have the serpentine form, where the branch cuts the asymptote, so as to touch it at its two extremities on opposite sides, or the conchoidal form, where it touches the asymptote on the same side. The two legs of a parabolic branch may converge to ultimate parallelism, as in the conic parabola, or diverge to ultimate parallelism, as in the semi-cubical parabola y² = x³, and the branch is said to be convergent, or divergent, accordingly; or they may tend to parallelism in opposite senses, as in the cubical parabola y = x³. As mentioned with regard to a branch generally, an infinite branch of any kind may have cusps, or, by cutting itself or another branch, may have or give rise to a crunode, &c.
11.Classification of Cubic Curves.—We may now consider the various forms of cubic curves as appearing by Newton’sEnumeratio, and by the figures belonging thereto. The species are reckoned as 72, which are numbered accordingly 1 to 72; but to these should be added 10a, 13a, 22aand 22b. It is not intended here to consider the division into species, nor even completely that into genera, but only to explain the principle of classification. It may be remarked generally that there are at most three infinite branches, and that there may besides be a re-entrant branch or oval.
The genera may be arranged as follows:—
and thus arranged they correspond to the different relations of the line infinity to the curve. First, if the three intersections by the line infinity are all distinct, we have the hyperbolas; if the points are real, the redundant hyperbolas, with three hyperbolic branches; but if only one of them is real, the defective hyperbolas, with one hyperbolic branch. Secondly, if two of the intersections coincide, say if the line infinity meets the curve in a onefold point and a twofold point, both of them real, then there is always one asymptote: the line infinity may at the twofold point touch the curve, and we have the parabolic hyperbolas; or the twofold point may be a singular point,—viz., a crunode giving the hyperbolisms of the hyperbola; an acnode, giving the hyperbolisms of the ellipse; or a cusp, giving the hyperbolisms of the parabola. As regards the so-called hyperbolisms, observe that (besides the single asymptote) we have in the case of those of the hyperbola two parallel asymptotes; in the case of those of the ellipse the two parallel asymptotes become imaginary, that is, they disappear; and in the case of those of the parabola they become coincident, that is, there is here an ordinary asymptote, and a special asymptote answering to a cusp at infinity. Thirdly, the three intersections by the line infinity may be coincident and real; or say we have a threefold point: this may be an inflection, a crunode or a cusp, that is, the line infinity may be a tangent at an inflection, and we have the divergent parabolas; a tangent at a crunode to one branch, and we have the trident curve; or lastly, a tangent at a cusp, and we have the cubical parabola.
It is to be remarked that the classification mixes together non-singular and singular curves, in fact, the five kinds presently referred to: thus the hyperbolas and the divergent parabolas include curves of every kind, the separation being made in the species; the hyperbolisms of the hyperbola and ellipse, and the trident curve, are nodal; the hyperbolisms of the parabola, and the cubical parabola, are cuspidal. The divergent parabolas are of five species which respectively belong to and determine the five kinds of cubic curves; Newton gives (in two short paragraphs without any development) the remarkable theorem that the five divergent parabolas by their shadows generate and exhibit all the cubic curves.
The five divergent parabolas are curves each of them symmetrical with regard to an axis. There are two non-singular kinds, the one with, the other without, an oval, but each of them has an infinite (as Newton describes it)campaniformbranch; this cuts the axis at right angles, being at first concave, but ultimately convex, towards the axis, the two legs continually tending to become at right angles to the axis. The oval may unite itself with the infinite branch, or it may dwindle into a point, and we have the crunodal and the acnodal forms respectively; or if simultaneously the oval dwindles into a point and unites itself to the infinite branch, we have the cuspidal form. (SeeParabola.) Drawing a line to cut any one of these curves and projecting the line to infinity, it would not be difficult to show how the line should be drawn in order to obtain a curve of any given species. We have herein a better principle of classification; considering cubic curves, in the first instance, according to singularities, the curves are non-singular, nodal (viz. crunodal or acnodal), or cuspidal; and we see further that there are two kinds of non-singular curves, the complex and the simplex. There is thus a complete division into the five kinds, the complex, simplex, crunodal, acnodal and cuspidal. Each singular kind presents itself as a limit separating two kinds of inferior singularity; the cuspidal separates the crunodal and the acnodal, and these last separate from each other the complex and the simplex.
The whole question is discussed very fully and ably by A. F. Möbius in the memoir “Ueber die Grundformen der Linien dritter Ordnung” (Abh. der K. Sachs. Ges. zu Leipzig, t. i., 1852). The author considers not only plane curves, but also cones, or, what is almost the same thing, the spherical curves which are their sections by a concentric sphere. Stated in regard to the cone, we have there the fundamental theorem that there are two different kinds of sheets; viz., the single sheet, not separated into two parts by the vertex (an instance is afforded by the plane considered as a cone of the first order generated by the motion of a line about a point), and the double or twin-pair sheet, separated into two parts by the vertex (as in the cone of the second order). And it then appears that there are two kinds of non-singular cubic cones, viz. the simplex, consisting of a single sheet, and the complex, consisting of a single sheet and a twin-pair sheet; and we thence obtain (as for cubic curves) the crunodal, the acnodal and the cuspidal kinds of cubic cones. It may be mentioned that the single sheet is a sort of wavy form, having upon it three lines of inflection, and which is met by any plane through the vertex in one or in three lines; the twin-pair sheet has no lines of inflection, and resembles in its form a cone on an oval base.
In general a cone consists of one or more single or twin-pair sheets, and if we consider the section of the cone by a plane, the curve consists of one or more complete branches, or say circuits, each of them the section of one sheet of the cone; thus, a cone of the second order is one twin-pair sheet, and any section of it is one circuit composed, it may be, of two branches. But although we thus arrive by projection at the notion of a circuit, it is not necessary to go out of the plane, and we may (with Zeuthen, using the shorter termcircuitfor hiscomplete branch) define a circuit as any portion (of a curve) capable of description by the continuous motion of a point, it being understood that a passage through infinity is permitted. And we then say that a curve consists of one or more circuits; thus the right line, or curve of the first order, consists of one circuit; a curve of the second order consists of one circuit; a cubic curve consists of one circuit or else of two circuits.
A circuit is met by any right line always in an even number, or always in an odd number, of points, and it is said to be an even circuit or an odd circuit accordingly; the right line is an odd circuit, the conic an even circuit. And we have then the theorem, two odd circuits intersect in an odd number of points; an odd and an even circuit, or two even circuits, in an even number of points. An even circuit not cutting itself divides the plane into two parts, the one called the internal part, incapable of containing any odd circuit, the other called the external part, capable of containing an odd circuit.
We may now state in a more convenient form the fundamental distinction of the kinds of cubic curve. A non-singular cubic is simplex, consisting of one odd circuit, or it is complex, consisting of one odd circuit and one even circuit. It may be added that there are on the odd circuit three inflections, but on the even circuit no inflection; it hence also appears that from any point of the odd circuit there can be drawn to the odd circuit two tangents, and to the even circuit (if any) two tangents, but that from a point of the even circuit there cannot be drawn (either to the odd or the even circuit) any real tangent; consequently, in a simplex curve the number of tangents from any point is two; but in a complex curve the number is four, or none,—four if the point is on the odd circuit, none if it is on the even circuit. It at once appears from inspection of the figure of a non-singular cubic curve, which is the odd and which the even circuit. The singular kinds arise as before; in the crunodal and the cuspidal kinds the whole curve is an odd circuit, but in an acnodal kind the acnode must be regarded as an even circuit.
12.Quartic Curves.—The analogous question of the classification of quartics (in particular non-singular quartics and nodal quartics) is considered in Zeuthen’s memoir “Sur les différentes formes des courbes planes du quatrième ordre” (Math. Ann.t. vii., 1874). A non-singular quartic has only even circuits; it has at most four circuits external to each other, or two circuits one internal to the other, and in this last case the internal circuit has no double tangents or inflections. A very remarkable theorem is established as to the double tangents of such a quartic: distinguishing as a double tangent of the first kind a real double tangent which either twice touches the same circuit, or else touches the curve in two imaginary points, the number of the double tangents of the first kind of a non-singular quartic is = 4; it follows that the quartic has at most 8 real inflections. The forms of the non-singular quartics are very numerous, but it is not necessary to go further into the question.
We may consider in relation to a curve, not only the line infinity, but also the circular points at infinity; assuming the curve to be real, these present themselves always conjointly; thus a circle is a conic passing through the two circular points, and is thereby distinguished from other conics. Similarly a cubic through the two circular points is termed a circular cubic; a quartic through the two points is termed a circular quartic, and if it passes twice through each of them, that is, has each of them for a node, it is termed a bicircular quartic. Such a quartic is of course binodal (m = 4, δ = 2, κ = 0); it has not in general, but it may have, a third node or a cusp. Or again, we may have a quartic curve having a cusp at each of the circular points: such a curve is a “Cartesian,” it being a complete definition of the Cartesian to say that it is a bicuspidal quartic curve (m = 4, δ = 0, κ = 2), having a cusp at each of the circular points. The circular cubic and the bicircular quartic, together with the Cartesian (being in one point of view a particular case thereof), are interesting curves which have been much studied, generally, and in reference to theirfocalproperties.
13.Foci.—The points calledfocipresented themselves in the theory of the conic, and were well known to the Greek geometers, but the general notion of a focus was first established by Plücker (in the memoir “Über solche Puncte die bei Curven einer höheren Ordnung den Brennpuncten der Kegelschnitte entsprechen” (Crelle, t. x., 1833). We may from each of the circular points draw tangents to a given curve; the intersection of two such tangents (belonging of course to the two circular points respectively) is a focus. There will be from each circular point λ tangents (λ, a number depending on the class of the curve and its relation to the line infinity and the circular points, = 2 for the general conic, 1 for the parabola, 2 for a circular cubic, or bicircular quartic, &c.); the λ tangents from the one circular point and those from the other circular point intersect in λ real foci (viz. each of these is the only real point on each of the tangents through it), and in λ² − λ imaginary foci; each pair of real foci determines a pair of imaginary foci (the so-called antipoints of the two real foci), and the ½λ(λ − 1) pairs of real foci thus determine the λ² − λ imaginary foci. There are in some cases points termed centres, or singular or multiple foci (the nomenclature is unsettled), which are the intersections of improper tangents from the two circular points respectively; thus, in the circular cubic, the tangents to the curve at the two circular points respectively (or two imaginary asymptotes of the curve) meet in a centre.
14.Distance and Angle.Curves described mechanically.—The notions ofdistanceand of linesat right anglesare connected with the circular points; and almost every construction of a curve by means of lines of a determinate length, or at right angles to each other, and (as such) mechanical constructions by means of linkwork, give rise to curves passing the same definite number of times through the two circular points respectively, or say to circular curves, and in which the fixed centres of the construction present themselves as ordinary, or as singular, foci. Thus the general curve of three bar-motion (or locus of the vertex of a triangle, the other two vertices whereof move on fixed circles) is a tricircular sextic, having besides three nodes (m = 6, δ = 3 + 3 + 3 = 9), and having the centres of the fixed circles each for a singular focus; there is a third singular focus, and we have thus the remarkable theorem (due to S. Roberts) of the triple generation of the curve by means of the three several pairs of singular foci.
Again, the normal,qualine at right angles to the tangent, is connected with the circular points, and these accordingly present themselves in the before-mentioned theories of evolutes and parallel curves.
15.Theories of Correspondence.—We have several recent theories which depend on the notion ofcorrespondence: two points whether in the same plane or in different planes, or on the same curve or in different curves, may determine each other in such wise that to any given position of the first point there correspond α′ positions of the second point, and to any given position of the second point a positions of the first point; the two points have then an (α, α) correspondence; and if α, α are each = 1, then the two points have α (1, 1) or rational correspondence. Connecting with each theory the author’s name, the theories in question are G. F. B. Riemann, the rational transformation of a plane curve; Luigi Cremona, the rational transformation of a plane; and Chasles, correspondence of points on the same curve, and united points. The theory first referred to, with the resulting notion of “Geschlecht,” ordeficiency, is more than the other two an essential part of the theory of curves, but they will all be considered.
Riemann’s results are contained in the memoirs on “Abelian Integrals,” &c. (Crelle, t. liv., 1857), and we have next R. F. A. Clebsch, “Über die Singularitäten algebraischer Curven” (Crelle, t. lxv., 1865), and Cayley, “On the Transformation of Plane Curves” (Proc. Lond. Math. Soc.t. i., 1865;Collected Works, vol. vi. p. 1). The fundamental notion of the rational transformation is as follows:—