Page:Encyclopædia Britannica, Ninth Edition, v. 6.djvu/760

724 CURVE the hyperbolisms of the ellipse ; or a cusp, giving the hyper bolisms 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 havs a threefold point : this may be an inflexion, a crunode, or a cusp, that is, the line infinity may be a tangent at an inflexion, 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 end ellipse, and the trident curve, are nodal ; the hyperbol isms of the parabola, and the cubical parabola, are cuspidal. The divergent parabolas are of five species which respec tively 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 diver gent 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 convex, but ultimately concave, 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 simul taneously the oval dwindles into a point and unites itself to the infinite branch, we have the cuspidal form. Draw ing 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 classi fication; 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 crunoial 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 Mobius 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 co.ne 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 inflexion, and which is met by any plane through the vertex in one or in three lines ; the twin-pair sheet has no lines of inflexion, 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 pro jection 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, arid 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 contain ing 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 inflexions, but on the even circuit no inflexion ; 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 the acnodal kind the acnode must be regarded as an even circuit. The analogous question of the classification of quartics (in particular non-singular quartics and nodal quartics) is considered in Zeuthen s memoir " Sur les differentes formes des courbes planes du quatrieme 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 inflexions. 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