CONOID

, is a figure resembling a cone, except that the slant sides from the base to the vertex are not straight lines as in the cone, but curved. It is generated by the revolution of a conic section about its axis; and it is therefore threefold, answering to the three sections of the cone, viz, the Elliptical Conoid, or spheroid, the Hyperbolic Conoid, and the Parabolic Conoid.

If a conoid be cut by a plane in any position, the section will be of the figure of some one of the conic sections; and all parallel sections, of the same conoid, are like and similar figures. When the section of the solid returns into itself, it is an ellipse; which is always the case in the sections of the spheroid, except when it is perpendicular to the axis; which position is also to be excepted in the other solids, the section being always a circle in that position. In the parabolic conoid, the section is always an ellipse, except when it is parallel to the axis. And in the hyperbolic conoid, the section is an ellipse, when its axis makes with the axis of the solid, an angle greater than that made by the said axe of the solid and the asymptote of the generating hyperbola; the section being an hyperbola in all other cases, but when those angles are equal, and then it is a parabola.

But when the section is parallel to the fixed axis, it is of the same kind with, and similar to the generating plane itself; that is, the section parallel to the axis, in the spheroid, is an ellipse similar to the generating ellipse; in the parabolic conoid it is a parabola, sunilar to the generating one; and in the hyperbolic conoid, it is an hyperbola similar to the generating one.

The section through the axis, which is the generating plane, is, in the spheroid the greatest of the parallel sections, but in the hyperboloid it is the least, and in the paraboloid those parallel sections are all equal.

The analogy of the sections of the hyperboloid to those of the cone, are very remarkable, all the three conic sections being formed by cutting an hyperboloid in the same positions as the cone is cut. Thus, let an hyperbola and its asymptote be revolved together about the transverse axis, the sormer describing an hyperboloid, and the latter a cone circumscribing it: then let it be supposed that they are both cut by one plane in any position; so shall the two sections be like, similar, and concentric figures: that is, if the plane cut both the sides of each, the sections will be concentric and similar ellipses; but if the cutting plane be parallel to the asymptote, or to the side of the cone, the sections will be parabolas; and in all other positions, the sections will be similar and concentric hyperbolas.

And this analogy of the sections will not seem strange, when it is considered that a cone is a species of the hyperboloid; or a triangle a species of the hyperbola, the axes being infinitely small. See my Mensuration, prop. 1, part 3, sect. 4, pag. 265 edit. 8vo.