CONIC SECTIONS, three curves, the hyperbola, the parabola and the ellipse, are called the conic sections, because these curves are formed by the intersection of the surface of a cone with planes that cut the cone in various directions. If the cutting-plane be parallel to the axis of the cone (fig. 1), the curve formed is the hyperbola, which has two branches, as shown in the figure. If the cut ting-plane be parallel to a straight line on the surface of the cone (fig. 2), the curve formed is a parabola. Any other section is an ellipse (fig. 3). It must be noticed, however, that this general description includes three peculiar cases. In the case of a plane parallel to the axis of the cone, when that plane contains the axis, the seetitm, instead of being a hyperbola, is in this limiting case a pair of straight lines meeting each other at an angle equal to that of the angle of the cone so as to form a triangle (fig. 4). When a plane, which would other wise form a parabolic section, is a tangent plane to the cone, thc parabola degenerates into a • straight line passing through the vertex of the cone. Lastly, when a plane that would other wise form an ellipse is perpendicular to the axis of the cone, the ellipse becomes a circle (fig. 5). Lastly, when a plane cuts the cone
only in the vertex, the conic section degenerates into a single point (fig. 6). A pair of points cannot be produced by any plane section of a done, but on account of its analytical proper ties, this figure is also considered as a degen erate conic section. The properties of these• curves are discussed under their several names. It will there be seen that other definitions may be given of the curves; and that from these their properties are •more conveniently derived' than from the consideration that they are formed by the sections of a conical surface. The most important of these properties are that of being represehted in a system of Cartesian co-ordinates by an equation of the second de-. gree, and that of being the locus of, the corre *ending lines of two projective pencils in the plane. The properties of these curves are of the greatest physical interest; and the geometry of the conic sections has, ever since the time of. the Greek mathematicians, been considered as the best of the more advanced geometrical studies.