There is a theory of diameters; these are the polars of points at infinity. The polar line of a point at infinity is the locus of the mean centre of the in intersections of the curve with chords through the point (proved by Newton for cubics) ; this property of a diametral line is analogous the bisection property of a diam eter of a conic. The other polars (curvilinear diameters) also can be explained in terms of the segments of the chords. (Salmon, H. P. C., chap. 4).
The first enumeration of varieties of a curve of any order beyond the second is' Newton's linearum tertii ordinis' (1706). He proves that all the 72 varieties (it should be 78) can be obtained by projection from the five types of cubic with an inflexional tangent at infinity (divergent parabolas), bipartite, uni partite, crunodal, acnodal and cuspidal. Simi larly when once the distinct types of a curve of any order have been enumerated, the varie ties can be obtained at once. There are 144 types of quartic (R. Gentry the forms of plane quartic curves,' 1896) ;. the number for higher curves, even for quintics, must be very great. It does not appear that any special purpose would be served directly by the enumera tion of these, though there are matters of in terest on which this might throw light. For exan'ple, the theory of the inflexional tangents of a cubic has been thoroughly worked out; suitably taken in threes they determine three inflexions on a straight line; the three tangents and this line form a framework for the curve. As regards quartics, a closely corresponding theory is that of the bitangents; suitably taken in fours, these determine sets of eight points on the quartic, each set lying on a conk; the curve is conveniently referred to the four bi tangents and the conic. What is the generaliza tion of this? Even for the quintic, this is as yet unknown.
A more profitable classification of curves is according to their genus, and the values of the 3(p-1) characteristic constants, or moduli. Rational (p—O), elliptic (p-1), and hyper elliptic curves (which include among others all curves with p=2) have been extensively treated. (Clebsch-Lindemann, pp. 883-903, 903-915, 915 923 ; also 711-712, 717-720; Loria, (II passato ed it presente delle principali teorie geome triche,) 2d edition, 1896, pp. 76-79 for refer ences).
Investigations on special classes of algebraic curves are too numerous to mention; in par ticular, bicircular quartics and cartesians (with nodes, or cusps, at the circular points) have a literature to themselves. The Steiner curve, or deltoid (hypocycloid with three cusps), is perhaps the most interesting individual among algebraic curves, on account of its geometrical properties. (Luria, (II passato,' etc., pp. 61-76).
If a curve has an equation that cannot be expressed in finite algebraic form it is said to be transcendental. Algebraic and transcend ental curves, however, are by no means as widely separated as this would suggest; e.g., the equation r sin ba represents curves which are algebraic when b is rational, transcendental for all other values of b. Thus the algebraic curves of the series bear to the whole series the relation that is borne by rational numbers to all numbers ; they are isolated members, whose number is insignificant. The same is probably true of all algebraic curves ; they are isolated members of transcendental families. It is not surprising, therefore, that there is as yet no general theory of transcendental curves. Results proved for algebraic curves by means of the whole equation (e.g., Pliidcer's equations) are not applicable to transcendental curves, which from one point of view are of infinite order; while results that depend only on a small arc are in general applicable. Such knowledge as we have of transcendental curves is obtained from metric investigations of special curves.
Among these special curves, one of the most important divisions is that of roulettes. A roulette is traced by a point attached to a curve, which itself rolls without sliding on a fixed curve. A point on the circumference of a circle which rolls on a straight line traces a cycloid; if the circle rolls in or on a circle the roulette is an epicycloid or hypocycloid. If the point is not on the circumference of the rolling circle, the curve is a trochoid, epitrochoid or hypo. trochoid. The epicycloid or hypocycloid is algebraic if the radii of the two circles are com mensurable. An important theorem, due to Descartes, is that the normal to a roulette at any point passes through the corresponding point of contact of the rolling curve with the fixed curve.