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:—
Taking u, X, Y, Z to be rational and integral functions (X, Y, Z all of the same order) of the co-ordinates (x, y, z), and u′, X′, Y′, Z′ rational and integral functions (X′, Y′, Z′, all of the same order) of the co-ordinates ( x′, y′, z′), we transform a given curve u = 0, by the equations of x′ : y′ : z′ = X : Y : Z, thereby obtaining a transformed curve u′ = 0, and a converse set of equations x : y : z = X′ : Y′ : Z′; viz. assuming that this is so, the point (x, y, z) on the curve u = 0 and the point ( x′, y′, z′) on the curve u′ = 0 will be points having a (1, 1) correspondence. To show how this is, observe that to a given point (x, y, z) on the curve u = 0 there corresponds a single point ( x′, y′, z′) determined by the equations x′ : y′ : z′ = X : Y : Z; from these equations and the equation u = 0 eliminating x, y, z, we obtain the equation u′ = 0 of the transformed curve. To a given point ( x′, y′, z′) not on the curve u’ = 0 there corresponds, not a single point, but the system of points (x, y, z) given by the equations x′ : y′ : z′ = X : Y : Z, viz., regarding x′, y′, z′ as constants (and to fix the ideas, assuming that the curves X = 0, Y = 0, Z = 0, have no common intersections), these are the points of intersection of the curves X : Y : Z, = x′ : y′ : z′, but no one of these points is situate on the curve u = 0. If, however, the point ( x′, y′, z′) is situate on the curve u′ = 0, then one point of the system of points in question is situate on the curve u = 0, that is, to a given point of the curve u′ = 0 there corresponds a single point of the curve u = 0; and hence also this point must be given by a system of equations such as x : y : z = X′ : Y′ : Z′.
It is an old and easily proved theorem that, for a curve of
