Page:Encyclopædia Britannica, Ninth Edition, v. 6.djvu/759

Rh CURVE 723 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, the 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 conies, 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 ; an acnode is a branch by itself, it may be considered as an indefinitely small re entrant branch. A branch may have inflexions and double tangents, or there may be double tangents which touch two distinct branches ; there are also double tangents with ima ginary points of contact, which are thus lines having no visible connection with the curve. A re-entrant branch not cutting itself may be everywhere convex, and it is then pro perly 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, kc., 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 Iima9on. 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 pro jections, in the same way as we in effect consider the hyperbola and the parabola as projections of the ellipse. If a line O cut an arc aa, so that the two segments ab, ba lie on opposite sides of the line, then projecting the figure so that the line fl 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 la 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 fi touch at 5 the arc aa so that the two portions ab, la lie on the same side of the line O, then projecting the figure as before, the tangent at b, that is, the line 13 itself, is projected to infinity ; the arc ab is projected into a parabolic leg, and at the same time the arc la 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 O is a tangent at an inflexion, passes through a crunode, or touches one of the branches of a crunode, &c. ; thus, if the line O passes through a crunode we have pairs of hyperbolic legs belong ing to two parallel asymptotes. The foregoing considera tions 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 y 2 = # 3, 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 = y?. 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. 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 10&quot;, 13 a, 22 a , and 22&quot;. It is not intended here to consider the division into species, nor even completely that into genera, but only to explain the principle of classifica tion. 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 ,, ,, ellipse 11 ,, ,, parabola 12 trident curve 13 divergent parabolas 1 4 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 maybe a singular point, viz., a crunoue giving the hyperbolisms O f the hyperbola ; an acnode, giving