Chapter IX: Part 9

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, ax^2 + 2hxy + by^2 + 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 1/2(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 1/2(m + 1)(m + 2) - 1, that is, 1/2m(m + 3); and a curve of the order m may be made to satisfy this number of conditions; for example, to pass through 1/2m(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 _m_th 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 _m_th 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 _m_th 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 = [beta], the resultant equation in x is [beta]x - 1 = 0, and there is apparently only the intersection (x = 1/[beta], y = [beta]); 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 = [beta]; the two intersections are the point (x = 1/[beta], y = [beta]), and the point at infinity on the line y = [beta].

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 _Geometrie_ 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 _Geometrie_, 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 Geometrie 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 degre" (_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 Francois 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 _Traite des proprietes projectives_ (1822), where we find (p. 49) the statement, "deux circles places arbitrairement sur un plan ... ont idealement deux points imaginaires communs a 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 theorie generale des polaires reciproques_, although presented to the Paris Academy in 1824, were only published (_Crelle_, t. iii. and iv., 1828, 1829) subsequent to the memoir by Gergonne, _Considerations philosophiques sur les elemens de la science de l'etendue_ (_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 geometrie plane, a chaque theoreme il en repond necessairement un autre qui s'en deduit en echangeant simplement entre eux les deux mots _points_ et _droites_; tandis que dans la geometrie de l'espace ce sont les mots _points_ et _plans_ qu'il faut echanger entre eux pour passer d'un theoreme a son correlatif"; 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 theoremes, &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 _m_th degree (order) when it has with a line m real or ideal intersections," and "a plane curve is said to be of the _m_th 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 proprietes de l'etendue_ (_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 _Memoire de geometrie sur deux principes generaux de la science, la dualite et l'homographie_, which forms a sequel to the _Apercu historique sur l'origine et le developpement des methodes en geometrie_ (_Mem. de Brux._ t. xi., 1837).

We now come to Julius Plucker; his "six equations" were given in a short memoir in _Crelle_ (1842) preceding his great work, the _Theorie der algebraischen Curven_ (1844). Plucker 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.

Plucker, 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 ([xi], [eta], [zeta]) 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 [xi]x + [eta]y + [zeta]z = 0 to be the equation of a line, we say that ([xi], [eta], [zeta]) are the line-co-ordinates of this line. A linear relation a[xi] + b[eta] + c[zeta] = 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 a[xi] + b[eta] + c[zeta] = 0 expresses that the equation [xi]x + [eta]y + [zeta]z = 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 ([xi], [eta], [zeta]) are the line-co-ordinates of any line whatever; but when these, instead of being absolutely arbitrary, are subject to the restriction a[xi] + b[eta] + c[zeta] = 0, this obliges the line to pass through a point (a, b, c); and the last-mentioned equation a[xi] + b[eta] + c[zeta] = 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 (*() [xi], [eta], [zeta])^n = 0 is the relation which is satisfied by the line-co-ordinates ([xi], [eta], [zeta]) 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. Plucker'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, 2. The double point
\ cusp or spinode; or node;

Line-singularities-- / 3. The stationary tangent 4. The double
\ or inflection; tangent;

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.

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 d_x(*() x, y, z)^m = 0, d_y(*() x, y, z)^m = 0, d_z(*() 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,
[delta] " number of double points,
[kappa] " " cusps,
[tau] " " double tangents,
[iota] " " inflections;

and this being so, Plucker's "six equations" are

(1) n = m(m - 1) - 2[delta] - 3[kappa],
(2) [iota] = 3m(m - 2) - 6[delta] - 8[kappa],
(3) [tau] = 1/2m(m - 2)(m^2 - 9) - (m^2 - m - 6)(2[delta] + 3[kappa])
+ 2[delta]([delta] - 1)
+ 6[delta][kappa] + (9/2)[kappa]([kappa] - 1),
(4) m = n(n - 1) - 2[tau] - 3[iota],
(5) [kappa] = 3n(n - 2) - 6[tau] - 8[iota],
(6) [delta] = 1/2n(n - 2)(n^2 - 9) - (n^2 - n - 6)(2[tau] + 3[iota])
+ 2[tau]([tau] - 1) + 6[tau][iota]
+ (9/2)[iota]([iota] - i).

It is easy to derive the further forms--

(7) [iota] - [kappa] = 3(n - m),
(8) 2([tau] - [delta]) = (n - m)(n + m - 9),
(9) 1/2m(m + 3) - [delta]-2[kappa] = 1/2n(n + 3) - [tau] - 2[iota],
(10) 1/2(m - 1)(m - 2) - [delta] - [kappa]
= 1/2(n - 1)(n - 2) - [tau] - [iota],
(11, 12) m^2 - 2[delta] - 3[kappa] = n^2 - 2[tau] - 3^[iota],
= 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 + [iota] and 3n + [kappa] everything may be expressed in terms of m, n, a. We have

[kappa] = a - 3n,
[iota] = a - 3m,
2[delta] = m^2 - m + 8n - 3a.
2[tau] = n^2 - n + 8m - 3a.

It is implied in Plucker's theorem that, m, n, [delta], [kappa],
[tau], [iota] signifying as above in regard to any curve, then in
regard to the reciprocal curve, n, m, [tau], [iota], [delta], [kappa]
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 1/2m(m + 3) - [delta] - 2[kappa] is that of the number
of the disposable constants in a curve of the order m with [delta]
nodes and [kappa] 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 1/2m(m - 1) - [delta] -
[kappa] 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 m^2 - 2[delta] - 3[kappa] = n^2 - 2[tau] - 3[iota] = m +
n, present themselves in the theory of envelopes, as will appear
farther on.

With regard to the demonstration of Plucker'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 [delta] and [kappa] 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 _m_th 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, [iota], [tau] 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 [delta] nodes and [kappa] cusps, there is a diminution 2[delta] + 3[kappa], and the value of n is n = m(m - 1) - 2[delta] -3[kappa].

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 (S8 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 [delta] nodes and [kappa] cusps, the diminution is = 6[delta] + 8[kappa], and the number of inflections is [iota] = 3m(m - 2) - 6[delta] - 8[kappa].

Thirdly, for the double tangents; the points of contact of these are obtained as the intersections of the curve by a curve [Pi] = 0, which has not as yet been geometrically defined, but which is found analytically to be of the order (m - 2)(m^2 - 9); the number of intersections is thus = m(m - 2)(m^2 - 9); but if the given curve has a node then there is a diminution = 4(m^2 - m - 6), and if it has a cusp then there is a diminution = 6(m^2 - m - 6), where, however, it is to be noticed that the factor (m^2 - 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 [delta] nodes and [kappa] cusps, the apparent diminution 2(m^2 - m - 6)(2[delta] + 3[kappa]) is too great, and it has in fact to be diminished by 2{2[delta]([delta] - 1) + 6[delta][kappa] + (9/2)[kappa]([kappa] - 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 2[tau], = m(m - 2)(m^2 - 9) - &c.; or dividing the whole by 2, we have the expression for [tau] given by the third of Plucker's equations.

It is obvious that we cannot by consideration of the equation u = 0 in point-co-ordinates obtain the remaining three of Plucker'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, [delta], [kappa], with n, [tau], [iota] respectively.

To complete Plucker'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 = [delta]' nodes, [kappa]' cusps, [tau]' double tangents and [iota]' inflections. So that, in fact, Plucker's equations properly understood apply to a curve with any singularities whatever.

By means of Plucker's equations we may form a table--

+-------+-------+-------+-------+-------+-------+
| m | n |[delta]|[kappa]| [tau] | [iota]|
+-------+-------+-------+-------+-------+-------+
| 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 ([delta] + [kappa]) 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, [delta] = 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 Plucker, but the theory was first given in a complete form by Hesse in the two papers "Uber die Elimination, u.s.w.," and "Uber 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 [Delta] = 0, where [Delta] 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 x^3 + y^3 + z^3 + 6lxyz = 0; and in particular a proof is given of Plucker'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 1/2m(m - 2)(m^2 - 9) for the number of double tangents of a curve of the order m was obtained by Plucker only as a consequence of his first, second, fourth and fifth equations. An investigation by means of the curve [Pi] = 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'elimination et la theorie des courbes" (_Crelle_, t. xxxiv., 1847; _Collected Works_, vol. i. p. 337), and in part carried out by Hesse in the memoir "Uber 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 [Pi] = 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 [Pi] = 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 Plucker 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 Plucker 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 [Delta] 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'([dP]/[dP]x) + y'([dP]/[dP]y) + z'([dP]/[dP]z)]^(r)u = 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 Pluckerian 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 _Theorie des fonctions analytiques_ (1798); it is there remarked that the equation obtained by the elimination of the parameter a from an equation f(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 f(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 [delta] nodes and [kappa] cusps, and therefore of the class n with [tau] double tangents and [iota] inflections, m, n, [delta], [kappa], [tau], [iota] being connected by the Pluckerian 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 [delta] and [kappa] 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 = m^2 - 2[delta] - 3[kappa]. 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 = n^2 -2[tau] - 3[iota] (= m^2 - 2[delta] - 3[kappa], inasmuch as each of these numbers is as was seen = m + n). At any one of the m^2 - 2[delta] -3[kappa] 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 n^2 -2[tau] - 3[iota] 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 n^2 - 2[tau] - 3[iota] common tangents are the tangents to the variable curve at the m^2 - 2[delta] -3[kappa] points respectively, and the envelope is at the same time generated by the m^2 - 2[delta] - 3[kappa] points, and enveloped by the n^2 - 2[tau] - 3[iota] tangents; we have thus a dual generation of the envelope, which only differs from Plucker's dual generation, in that in place of a single point and tangent we have the group of m^2 - 2[delta] -3[kappa] points and n^2 - 2[tau] - 3[iota] 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.