Curve

Curve (Lat. curvus, bent), a word commonly meaning a shape represented by a line bending continuously out of the straight without making an angle, but only properly to be defined in its geometrical sense in the terms set out below. This subject is treated here from an historical point of view, for the purpose of showing how the different leading ideas were successively arrived at and developed.

1. A curve is a line, or continuous singly infinite system of points. We consider in the first instance, and chiefly, a plane curve described according to a law. Such a curve may be regarded geometrically as actually described, or kinematically as in the course of description by the motion of a point; in the former point of view, it is the locus of all the points which satisfy a given condition; in the latter, it is the locus of a point moving subject to a given condition. Thus the most simple and earliest known curve, the circle, is the locus of all the points at a given distance from a fixed centre, or else the locus of a point moving so as to be always at a given distance from a fixed centre. (The straight line and the point are not for the moment regarded as curves.)

Next to the circle we have the conic sections, the invention of them attributed to Plato (who lived 430-347 B.C.); the original definition of them as the sections of a cone was by the Greek geometers who studied them soon replaced by a proper definition in plano like that for the circle, viz. a conic section (or as we now say a “conic”) is the locus of a point such that its distance from a given point, the focus, is in a given ratio to its (perpendicular) distance from a given line, the directrix; or it is the locus of a point which moves so as always to satisfy the foregoing condition. Similarly any other property might be used as a definition; an ellipse is the locus of a point such that the sum of its distances from two fixed points (the foci) is constant, &c., &c.

The Greek geometers invented other curves; in particular, the conchoid (q.v.), which is the locus of a point such that its distance from a given line, measured along the line drawn through it to a fixed point, is constant; and the cissoid (q.v.), which is the locus of a point such that its distance from a fixed point is always equal to the intercept (on the line through the fixed point) between a circle passing through the fixed point and the tangent to the circle at the point opposite to the fixed point. Obviously the number of such geometrical or kinematical definitions is infinite. In a machine of any kind, each point describes a curve; a simple but important instance is the “three-bar curve,” or locus of a point in or rigidly connected with a bar pivoted on to two other bars which rotate about fixed centres respectively. Every curve thus arbitrarily defined has its own properties; and there was not any principle of classification.

2. Cartesian Co-ordinates.—The principle of classification first presented itself in the Géometrie of Descartes (1637). The idea was to represent any curve whatever by means of a relation between the co-ordinates (x, y) of a point of the curve, or say to represent the curve by means of its equation. (See Geometry: Analytical.)

Any relation whatever between (x, y) determines a curve, and conversely every curve whatever is determined by a relation between (x, y).

Observe that the distinctive feature is in the exclusive use of such determination of a curve by means of its equation. The Greek geometers were perfectly familiar with the property of an ellipse which in the Cartesian notation is x²/a² + y²/b² = 1, the equation of the curve; but it was as one of a number of properties, and in no wise selected out of the others for the characteristic property of the curve.

3. Order of a Curve.—We obtain from the equation the notion of an algebraical as opposed to a transcendental curve, viz. an algebraical curve is a curve having an equation F(x, y) = 0 where F(x, y) is a rational and integral function of the co-ordinates (x, y); and in what follows we attend throughout (unless the contrary is stated) only to such curves. The equation is sometimes given, and may conveniently be used, in an irrational form, but we always imagine it reduced to the foregoing rational and integral form, and regard this as the equation of the curve. And we have hence the notion of a curve of a given order, viz. the order of the curve is equal to that of the term or terms of highest order in the co-ordinates (x, y) conjointly in the equation of the curve; for instance, xy − 1 = 0 is a curve of the second order.

It is to be noticed here that the axes of co-ordinates may be any two lines at right angles to each other whatever; and that the equation of a curve will be different according to the selection of the axes of co-ordinates; but the order is independent of the axes, and has a determinate value for any given curve.

We hence divide curves according to their order, viz. a curve is of the first order, second order, third order, &c., according as it is represented by an equation of the first order, ax + by + c = 0, or say (* x, y, 1) = 0; or by an equation of the second order, ax² + 2hxy + by² + 2fy + 2gx + c = 0, say (* x, y, 1)² = 0; or by an equation of the third order, &c.; or what is the same thing, according as the equation is linear, quadric, cubic, &c.

A curve of the first order is a right line; and conversely every right line is a curve of the first order. A curve of the second order is a conic, and is also called a quadric curve; and conversely every conic is a curve of the second order or quadric curve. A curve of the third order is called a cubic; one of the fourth order a quartic; and so on.

A curve of the order m has for its equation (* x, y, 1)m = 0; and when the coefficients of the function are arbitrary, the curve is said to be the general curve of the order m. The number of coefficients is ½(m + 1)(m + 2); but there is no loss of generality if the equation be divided by one coefficient so as to reduce the coefficient of the corresponding term to unity, hence the number of coefficients may be reckoned as ½(m + 1)(m + 2) − 1, that is, ½m(m + 3); and a curve of the order m may be made to satisfy this number of conditions; for example, to pass through ½m(m + 3) points.

It is to be remarked that an equation may break up; thus a quadric equation may be (ax + by + c)(a′x + b′y + c′) = 0, breaking up into the two equations ax + by + c = 0, a′x + b′y + c′ = 0, viz. the original equation is satisfied if either of these is satisfied. Each of these last equations represents a curve of the first order, or right line; and the original equation represents this pair of lines, viz. the pair of lines is considered as a quadric curve. But it is an improper quadric curve; and in speaking of curves of the second or any other given order, we frequently imply that the curve is a proper curve represented by an equation which does not break up.

4. Intersections of Curves.—The intersections of two curves are obtained by combining their equations; viz. the elimination from the two equations of y (or x) gives for x (or y) an equation of a certain order, say the resultant equation; and then to each value of x (or y) satisfying this equation there corresponds in general a single value of y (or x), and consequently a single point of intersection; the number of intersections is thus equal to the order of the resultant equation in x (or y).

Supposing that the two curves are of the orders m, n, respectively, then the order of the resultant equation is in general and at most = mn; in particular, if the curve of the order n is an arbitrary line (n = 1), then the order of the resultant equation is = m; and the curve of the order m meets therefore the line in m points. But the resultant equation may have all or any of its roots imaginary, and it is thus not always that there are m real intersections.

The notion of imaginary intersections, thus presenting itself, through algebra, in geometry, must be accepted in geometry—and it in fact plays an all-important part in modern geometry. As in algebra we say that an equation of the mth order has m roots, viz. we state this generally without in the first instance, or it may be without ever, distinguishing whether these are real or imaginary; so in geometry we say that a curve of the mth order is met by an arbitrary line in m points, or rather we thus, through algebra, obtain the proper geometrical definition of a curve of the mth order, as a curve which is met by an arbitrary line in m points (that is, of course, in m, and not more than m, points).

The theorem of the m intersections has been stated in regard to an arbitrary line; in fact, for particular lines the resultant equation may be or appear to be of an order less than m; for instance, taking m = 2, if the hyperbola xy − 1 = 0 be cut by the line y = β, the resultant equation in x is βx − 1 = 0, and there is apparently only the intersection (x = 1/β, y = β); but the theorem is, in fact, true for every line whatever: a curve of the order m meets every line whatever in precisely m points. We have, in the case just referred to, to take account of a point at infinity on the line y = β; the two intersections are the point (x = 1/β, y = β), and the point at infinity on the line y = β.

It is, moreover, to be noticed that the points at infinity may be all or any of them imaginary, and that the points of intersection, whether finite or at infinity, real or imaginary, may coincide two or more of them together, and have to be counted accordingly; to support the theorem in its universality, it is necessary to take account of these various circumstances.

5. Line at Infinity.—The foregoing notion of a point at infinity is a very important one in modern geometry; and we have also to consider the paradoxical statement that in plane geometry, or say as regards the plane, infinity is a right line. This admits of an easy illustration in solid geometry. If with a given centre of projection, by drawing from it lines to every point of a given line, we project the given line on a given plane, the projection is a line, i.e. this projection is the intersection of the given plane with the plane through the centre and the given line. Say the projection is always a line, then if the figure is such that the two planes are parallel, the projection is the intersection of the given plane by a parallel plane, or it is the system of points at infinity on the given plane, that is, these points at infinity are regarded as situate on a given line, the line infinity of the given plane.1

Reverting to the purely plane theory, infinity is a line, related like any other right line to the curve, and thus intersecting it in m points, real or imaginary, distinct or coincident.

Descartes in the Géométrie defined and considered the remarkable curves called after him the ovals of Descartes, or simply Cartesians, which will be again referred to. The next important work, founded on the Géométrie, was Sir Isaac Newton’s Enumeratio linearum tertii ordinis (1706), establishing a classification of cubic curves founded chiefly on the nature of their infinite branches, which was in some details completed by James Stirling (1692-1770), Patrick Murdoch (d. 1774) and Gabriel Cramer; the work also contains the remarkable theorem (to be again referred to), that there are five kinds of cubic curves giving by their projections every cubic curve whatever. Various properties of curves in general, and of cubic curves, are established in Colin Maclaurin’s memoir, “De linearum geometricarum proprietatibus generalibus Tractatus” (posthumous, say 1746, published in the 6th edition of his Algebra). We have in it a particular kind of correspondence of two points on a cubic curve, viz. two points correspond to each other when the tangents at the two points again meet the cubic in the same point.

6. Reciprocal Polars. Intersections of Circles. Duality. Trilinear and Tangential Co-ordinates.—The Géométrie descriptive, by Gaspard Monge, was written in the year 1794 or 1795 (7th edition, Paris, 1847), and in it we have stated, in plano with regard to the circle, and in three dimensions with regard to a surface of the second order, the fundamental theorem of reciprocal polars, viz. “Given a surface of the second order and a circumscribed conic surface which touches it ... then if the conic surface moves so that its summit is always in the same plane, the plane of the curve of contact passes always through the same point.” The theorem is here referred to partly on account of its bearing on the theory of imaginaries in geometry. It is in Charles Julian Brianchon’s memoir “Sur les surfaces du second degré” (Jour. Polyt. t. vi. 1806) shown how for any given position of the summit the plane of contact is determined, or reciprocally; say the plane XY is determined when the point P is given, or reciprocally; and it is noticed that when P is situate in the interior of the surface the plane XY does not cut the surface; that is, we have a real plane XY intersecting the surface in the imaginary curve of contact of the imaginary circumscribed cone having for its summit a given real point P inside the surface.

Stating the theorem in regard to a conic, we have a real point P (called the pole) and a real line XY (called the polar), the line joining the two (real or imaginary) points of contact of the (real or imaginary) tangents drawn from the point to the conic; and the theorem is that when the point describes a line the line passes through a point, this line and point being polar and pole to each other. The term “pole” was first used by François Joseph Servois, and “polar” by Joseph Diez Gergonne (Gerg. t. i. and iii., 1810-1813); and from the theorem we have the method of reciprocal polars for the transformation of geometrical theorems, used already by Brianchon (in the memoir above referred to) for the demonstration of the theorem called by his name, and in a similar manner by various writers in the earlier volumes of Gergonne. We are here concerned with the method less in itself than as leading to the general notion of duality.

Bearing in a somewhat similar manner also on the theory of imaginaries in geometry (but the notion presents itself in a more explicit form), there is the memoir by L. Gaultier, on the graphical construction of circles and spheres (Jour. Polyt. t. ix., 1813). The well-known theorem as to radical axes may be stated as follows. Consider two circles partially drawn so that it does not appear whether the circles, if completed, would or would not intersect in real points, say two arcs of circles; then we can, by means of a third circle drawn so as to intersect in two real points each of the two arcs, determine a right line, which, if the complete circles intersect in two real points, passes through the points, and which is on this account regarded as a line passing through two (real or imaginary) points of intersection of the two circles. The construction in fact is, join the two points in which the third circle meets the first arc, and join also the two points in which the third circle meets the second arc, and from the point of intersection of the two joining lines, let fall a perpendicular on the line joining the centre of the two circles; this perpendicular (considered as an indefinite line) is what Gaultier terms the “radical axis of the two circles”; it is a line determined by a real construction and itself always real; and by what precedes it is the line joining two (real or imaginary, as the case may be) intersections of the given circles.

The intersections which lie on the radical axis are two out of the four intersections of the two circles. The question as to the remaining two intersections did not present itself to Gaultier, but it is answered in Jean Victor Poncelet’s Traité des propriétés projectives (1822), where we find (p. 49) the statement, “deux circles placés arbitrairement sur un plan ... ont idéalement deux points imaginaires communs à l’infini”; that is, a circle qua curve of the second order is met by the line infinity in two points; but, more than this, they are the same two points for any circle whatever. The points in question have since been called (it is believed first by Dr George Salmon) the circular points at infinity, or they may be called the circular points; these are also frequently spoken of as the points I, J; and we have thus the circle characterized as a conic which passes through the two circular points at infinity; the number of conditions thus imposed upon the conic is = 2, and there remain three arbitrary constants, which is the right number for the circle. Poncelet throughout his work makes continual use of the foregoing theories of imaginaries and infinity, and also of the before-mentioned theory of reciprocal polars.

Poncelet’s two memoirs Sur les centres des moyennes harmoniques and Sur la théorie générale des polaires réciproques, although presented to the Paris Academy in 1824, were only published (Crelle, t. iii. and iv., 1828, 1829) subsequent to the memoir by Gergonne, Considérations philosophiques sur les élémens de la science de l’étendue (Gerg. t. xvi., 1825-1826). In this memoir by Gergonne, the theory of duality is very clearly and explicitly stated; for instance, we find “dans la géométrie plane, à chaque théorème il en répond nécessairement un autre qui s’en déduit en échangeant simplement entre eux les deux mots points et droites; tandis que dans la géométrie de l’espace ce sont les mots points et plans qu’il faut échanger entre eux pour passer d’un théorème à son corrélatif”; and the plan is introduced of printing correlative theorems, opposite to each other, in two columns. There was a reclamation as to priority by Poncelet in the Bulletin universel reprinted with remarks by Gergonne (Gerg. t. xix., 1827), and followed by a short paper by Gergonne, Rectifications de quelques théorèmes, &c., which is important as first introducing the word class. We find in it explicitly the two correlative definitions: “a plane curve is said to be of the mth degree (order) when it has with a line m real or ideal intersections,” and “a plane curve is said to be of the mth class when from any point of its plane there can be drawn to it m real or ideal tangents.”

It may be remarked that in Poncelet’s memoir on reciprocal polars, above referred to, we have the theorem that the number of tangents from a point to a curve of the order m, or say the class of the curve, is in general and at most = m(m − 1), and that he mentions that this number is subject to reduction when the curve has double points or cusps.

The theorem of duality as regards plane figures may be thus stated: two figures may correspond to each other in such manner that to each point and line in either figure there correspond in the other figure a line and point respectively. It is to be understood that the theorem extends to all points or lines, drawn or not drawn; thus if in the first figure there are any number of points on a line drawn or not drawn, the corresponding lines in the second figure, produced if necessary, must meet in a point. And we thus see how the theorem extends to curves, their points and tangents; if there is in the first figure a curve of the order m, any line meets it in m points; and hence from the corresponding point in the second figure there must be to the corresponding curve m tangents; that is, the corresponding curve must be of the class m.

Trilinear co-ordinates (see Geometry: Analytical) were first used by E. E. Bobillier in the memoir Essai sur un nouveau mode de recherche des propriétés de l’étendue (Gerg. t. xviii., 1827-1828). It is convenient to use these rather than Cartesian co-ordinates. We represent a curve of the order m by an equation (* x, y, z)m = 0, the function on the left hand being a homogeneous rational and integral function of the order m of the three co-ordinates (x, y, z); clearly the number of constants is the same as for the equation (* x, y, 1)m = 0 in Cartesian co-ordinates.

The theorem of duality is considered and developed, but chiefly in regard to its metrical applications, by Michel Chasles in the Mémoire de géométrie sur deux principes généraux de la science, la dualité et l’homographie, which forms a sequel to the Aperçu historique sur l’origine et le développement des méthodes en géométrie (Mém. de Brux. t. xi., 1837).

We now come to Julius Plücker; his “six equations” were given in a short memoir in Crelle (1842) preceding his great work, the Theorie der algebraischen Curven (1844). Plücker first gave a scientific dual definition of a curve, viz.; “A curve is a locus generated by a point, and enveloped by a line—the point moving continuously along the line, while the line rotates continuously about the point”; the point is a point (ineunt.) of the curve, the line is a tangent of the curve. And, assuming the above theory of geometrical imaginaries, a curve such that m of its points are situate in an arbitrary line is said to be of the order m; a curve such that n of its tangents pass through an arbitrary point is said to be of the class n; as already appearing, this notion of the order and class of a curve is, however, due to Gergonne. Thus the line is a curve of the order 1 and class 0; and corresponding dually thereto, we have the point as a curve of the order 0 and class 1.

Plücker, moreover, imagined a system of line-co-ordinates (tangential co-ordinates). (See Geometry: Analytical.) The Cartesian co-ordinates (x, y) and trilinear co-ordinates (x, y, z) are point-co-ordinates for determining the position of a point; the new co-ordinates, say (ξ, η, ζ) are line-co-ordinates for determining the position of a line. It is possible, and (not so much for any application thereof as in order to more fully establish the analogy between the two kinds of co-ordinates) important, to give independent quantitative definitions of the two kinds of co-ordinates; but we may also derive the notion of line-co-ordinates from that of point-co-ordinates; viz. taking ξx + ηy + ζz = 0 to be the equation of a line, we say that (ξ, η, ζ) are the line-co-ordinates of this line. A linear relation aξ + bη + cζ = 0 between these co-ordinate determines a point, viz. the point whose point-co-ordinates are (a, b, c); in fact, the equation in question aξ + bη + cζ = 0 expresses that the equation ξx + ηy + ζz = 0, where (x, y, z) are current point-co-ordinates, is satisfied on writing therein x, y, z = a, b, c; or that the line in question passes through the point (a, b, c). Thus (ξ, η, ζ) are the line-co-ordinates of any line whatever; but when these, instead of being absolutely arbitrary, are subject to the restriction aξ + bη + cζ = 0, this obliges the line to pass through a point (a, b, c); and the last-mentioned equation aξ + bη + cζ = 0 is considered as the line-equation of this point.

A line has only a point-equation, and a point has only a line-equation; but any other curve has a point-equation and also a line-equation; the point-equation (* x, y, z)m = 0 is the relation which is satisfied by the point-co-ordinates (x, y, z) of each point of the curve; and similarly the line-equation (* ξ, η, ζ)n = 0 is the relation which is satisfied by the line-co-ordinates (ξ, η, ζ) of each line (tangent) of the curve.

There is in analytical geometry little occasion for any explicit use of line-co-ordinates; but the theory is very important; it serves to show that in demonstrating by point-co-ordinates any purely descriptive theorem whatever, we demonstrate the correlative theorem; that is, we do not demonstrate the one theorem, and then (as by the method of reciprocal polars) deduce from it the other, but we do at one and the same time demonstrate the two theorems; our (x, y, z.) instead of meaning point-co-ordinates may mean line-co-ordinates, and the demonstration is then in every step of it a demonstration of the correlative theorem.

7. Singularities of a Curve. Plücker’s Equations.—The above dual generation explains the nature of the singularities of a plane curve. The ordinary singularities, arranged according to a cross division, are

	Proper.	Improper.
Point-singularities—	1. The stationary point, cusp or spinode;	2. The double point or node;
Line-singularities—	3. The stationary tangent or inflection;	4. The double tangent;

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

m,	the	order,
n	”	class,
δ	”	number of double points,
κ	”	”   cusps,
τ	”	”   double tangents,
ι	”	”   inflections;

and this being so, Plücker’s “six equations” are

(1)	n = m (m − 1) − 2 δ − 3κ,
(2)	ι = 3m (m − 2) − 6δ − 8κ,
(3)	τ = ½m (m − 2) (m² − 9) − (m² − m − 6) (2δ + 3κ) + 2δ (δ − 1) + 6δκ + 9⁄2κ (κ − 1),
(4)	m = n (n − 1) − 2τ − 3ι,
(5)	κ = 3n (n − 2) − 6τ − 8ι,
(6)	δ = ½n (n − 2) (n² − 9) − (n² − n − 6) (2τ + 3ι) + 2τ (τ − 1) + 6τι + 9⁄2ι (ι − i).

It is easy to derive the further forms—

(7)	ι − κ	= 3 (n − m),
(8)	2 (τ − δ)	= (n − m) (n + m − 9),
(9)	½m (m + 3) − δ − 2κ	= ½n (n + 3) − τ − 2ι,
(10)	½ (m − 1) (m − 2) − δ − κ	= ½ (n − 1) (n − 2) − τ − ι,
(11, 12)	m² − 2δ − 3κ	= n² − 2τ − 3ι, = m + n,—

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.

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 the mth 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—

m	n	δ	κ	τ	ι
0	1	−	−	0	0
1	0	0	0	−	−
2	2	0	0	0	0
3	6	0	0	0	9
”	4	1	0	0	3
”	3	0	1	0	1
4	12	0	0	28	24
”	10	1	0	16	18
”	9	0	1	10	16
”	8	2	0	8	12
”	7	1	1	4	10
”	6	0	2	1	8
”	6	3	0	4	6
”	5	2	1	2	4
”	4	1	2	1	2
”	3	0	3	1	0

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 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.

The two legs of a hyperbolic branch may belong to different asymptotes, and in this case we have the forms which Newton calls 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’s Enumeratio, 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, 22a and 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:—

1,2,3,4	redundant hyperbolas
5,6	defective hyperbolas
7,8	parabolic hyperbolas
9	hyperbolisms of hyperbola
10	hyperbolisms of ellipse
11	hyperbolisms of parabola
12	trident curve
13	divergent parabolas
14	cubic parabola;

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) campaniform branch; 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. (See Parabola.) 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 term circuit for his complete 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 their focal properties.

13. Foci.—The points called foci presented 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 of distance and of lines at right angles are 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, qua line 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 of correspondence: 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,” or deficiency, 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:—

It is an old and easily proved theorem that, for a curve of the order m, the number δ + κ of nodes and cusps is at most = ½(m − 1)(m − 2); for a given curve the deficiency of the actual number of nodes and cusps below this maximum number, viz. ½(m − 1)(m − 2) − δ − κ, is the “Geschlecht” or “deficiency,” of the curve, say this is = D. When D = 0, the curve is said to be unicursal, when = 1, bicursal, and so on.

The general theorem is that two curves corresponding rationally to each other have the same deficiency. [In particular a curve and its reciprocal have this rational or (1, 1) correspondence, and it has been already seen that a curve and its reciprocal have the same deficiency.]

A curve of a given order can in general be rationally transformed into a curve of a lower order; thus a curve of any order for which D = 0, that is, a unicursal curve, can be transformed into a line; a curve of any order having the deficiency 1 or 2 can be rationally transformed into a curve of the order D + 2, deficiency D; and a curve of any order deficiency = or > 3 can be rationally transformed into a curve of the order D + 3, deficiency D.

Observe that the radical, square root of a quartic function, is connected with the theory of elliptic functions, and the radical, square root of a sextic function, with that of the first kind of Abelian functions, but that the next kind of Abelian functions does not depend on the radical, square root of an octic function.

It is a form of the theorem for the case D = 1, that the co-ordinates x, y, z of a point of the bicursal curve, or in particular the co-ordinates of a point of the cubic, can be expressed as proportional to rational and integral functions of the elliptic functions snu, cnu, dnu; in fact, taking the radical to be √1 − θ²·1 − k²θ², and writing θ = snu, the radical becomes = cnu, dnu; and we have expressions of the form in question.

It will be observed that the equations x′ : y′ : z′ = X : Y : Z before mentioned do not of themselves lead to the other system of equations x : y : z = X′ : Y′ : Z′, and thus that the theory does not in anywise establish a (1, 1) correspondence between the points (x, y, z) and (x′, y′, z′) of two planes or of the same plane; this is the correspondence of Cremona’s theory.

Coming next to Chasles, the principle of correspondence is established and used by him in a series of memoirs relating to the conics which satisfy given conditions, and to other geometrical questions, contained in the Comptes rendus, t. lviii. (1864) et seq. The theorem of united points in regard to points in a right line was given in a paper, June-July 1864, and it was extended to unicursal curves in a paper of the same series (March 1866), “Sur les courbes planes ou à double courbure dont les points peuvent se déterminer individuellement—application du principe de correspondance dans la théorie de ces courbes.”

The principle of correspondence, or say rather the theorem of united points, is a most powerful instrument of investigation, which may be used in place of analysis for the determination of the number of solutions of almost every geometrical problem. We can by means of it investigate the class of a curve, number of inflections, &c.—in fact, Plücker’s equations; but it is necessary to take account of special solutions: thus, in one of the most simple instances, in finding the class of a curve, the cusps present themselves as special solutions.

The principle in its original form as applying to a right line was used throughout by Chasles in the investigations on the number of the conics which satisfy given conditions, and on the number of solutions of very many other geometrical problems.

There is one application of the theory of the (α, α′) correspondence between two planes which it is proper to notice.

16. Systems of Curves satisfying Conditions.—The researches of Chasles (Comptes Rendus, t. lviii., 1864, et seq.) refer to the conics which satisfy given conditions. There is an earlier paper by J. P. E. Fauque de Jonquières, “Théorèmes généraux concernant les courbes géométriques planes d’un ordre quelconque,” Liouv. t. vi. (1861), which establishes the notion of a system of curves (of any order) of the index N, viz. considering the curves of the order n which satisfy ½n(n + 3) − 1 conditions, then the index N is the number of these curves which pass through a given arbitrary point. But Chasles in the first of his papers (February 1864), considering the conics which satisfy four conditions, establishes the notion of the two characteristics (μ, ν) of such a system of conics, viz. μ is the number of the conics which pass through a given arbitrary point, and ν is the number of the conics which touch a given arbitrary line. And he gives the theorem, a system of conics satisfying four conditions, and having the characteristics (μ, ν) contains 2ν − μ line-pairs (that is, conics, each of them a pair of lines), and 2μ − ν point-pairs (that is, conics, each of them a pair of points,—coniques infiniment aplaties), which is a fundamental one in the theory. The characteristics of the system can be determined when it is known how many there are of these two kinds of degenerate conics in the system, and how often each is to be counted. It was thus that Zeuthen (in the paper Nyt Bydrag, “Contribution to the Theory of Systems of Conics which satisfy four Conditions” (Copenhagen, 1865), translated with an addition in the Nouvelles Annales) solved the question of finding the characteristics of the systems of conics which satisfy four conditions of contact with a given curve or curves; and this led to the solution of the further problem of finding the number of the conics which satisfy five conditions of contact with a given curve or curves (Cayley, Comptes Rendus, t. lxiii., 1866; Collected Works, vol. v. p. 542), and “On the Curves which satisfy given Conditions” (Phil. Trans. t. clviii., 1868; Collected Works, vol. vi. p. 191).

It may be remarked that although, as a process of investigation, it is very convenient to seek for the characteristics of a system of conics satisfying 4 conditions, yet what is really determined is in every case the number of the conics which satisfy 5 conditions; the characteristics of the system (4p) of the conics which pass through 4p points are (5p), (4p, 1l), the number of the conics which pass through 5 points, and which pass through 4 points and touch 1 line: and so in other cases. Similarly as regards cubics, or curves of any other order: a cubic depends on 9 constants, and the elementary problems are to find the number of the cubics (9p), (8p, 1l), &c., which pass through 9 points, pass through 8 points and touch 1 line, &c.; but it is in the investigation convenient to seek for the characteristics of the systems of cubics (8p), &c., which satisfy 8 instead of 9 conditions.

The elementary problems in regard to cubics are solved very completely by S. Maillard in his Thèse, Recherche des caractéristiques des systèmes élémentaires des courbes planes du troisième ordre (Paris, 1871). Thus, considering the several cases of a cubic

	No. of consts.
1. With a given cusp	5
2.  ”  cusp on a given line	6
3.  ”  cusp	7
4.  ”  a given node	6
5.  ”  node on given line	7
6.  ”  node	8
7. non-singular	9

he determines in every case the characteristics (μ, ν) of the corresponding systems of cubics (4p), (3p, 1l), &c. The same problems, or most of them, and also the elementary problems in regard to quartics are solved by Zeuthen, who in the elaborate memoir “Almindelige Egenskaber, &c.,” Danish Academy, t. x. (1873), considers the problem in reference to curves of any order, and applies his results to cubic and quartic curves.

The methods of Maillard and Zeuthen are substantially identical; in each case the question considered is that of finding the characteristics (μ, ν) of a system of curves by consideration of the special or degenerate forms of the curves included in the system. The quantities which have to be considered are very numerous. Zeuthen in the case of curves of any given order establishes between the characteristics μ, ν, and 18 other quantities, in all 20 quantities, a set of 24 equations (equivalent to 23 independent equations), involving (besides the 20 quantities) other quantities relating to the various forms of the degenerate curves, which supplementary terms he determines, partially for curves of any order, but completely only for quartic curves. It is the discussion and complete enumeration of the special or degenerate forms of the curves, and of the supplementary terms to which they give rise, that the great difficulty of the question seems to consist; it would appear that the 24 equations are a complete system, and that (subject to a proper determination of the supplementary terms) they contain the solution of the general problem.

17. Degeneration of Curves.—The remarks which follow have reference to the analytical theory of the degenerate curves which present themselves in the foregoing problem of the curves which satisfy given conditions.

18. Twisted Curves.—In conclusion a little may be said as to curves of double curvature, otherwise twisted curves or curves in space. The analytical theory by Cartesian co-ordinates was first considered by Alexis Claude Clairaut, Recherches sur les courbes à double courbure (Paris, 1731). Such a curve may be considered as described by a point, moving in a line which at the same time rotates about the point in a plane which at the same time rotates about the line; the point is a point, the line a tangent, and the plane an osculating plane, of the curve; moreover the line is a generating line, and the plane a tangent plane, of a developable surface or torse, having the curve for its edge of regression. Analogous to the order and class of a plane curve we have the order, rank and class of the system (assumed to be a geometrical one), viz. if an arbitrary plane contains m points, an arbitrary line meets r lines, and an arbitrary point lies in n planes, of the system, then m, r, n are the order, rank and class respectively. The system has singularities, and there exist between m, r, n and the numbers of the several singularities equations analogous to Plücker’s equations for a plane curve.

It is a leading point in the theory that a curve in space cannot in general be represented by means of two equations U = 0, V = 0; the two equations represent surfaces, intersecting in a curve; but there are curves which are not the complete intersection of any two surfaces; thus we have the cubic in space, or skew cubic, which is the residual intersection of two quadric surfaces which have a line in common; the equations U = 0, V = 0 of the two quadric surfaces represent the cubic curve, not by itself, but together with the line.

---

---