CONOID, (from the Greek, icor/v/4(5in partaking of the figure (.,f a cone), a figure generated by the revolution of a conic section round one of its axes. There are three kinds of colloids, viz., the elliptical, the hyperbolical, and the pa•a bolical ; which are sometimes otherwise denominated, ellip soid, or spheroid, hyperholoid, and paraboloid.
Now because the solid is generated by the revolution Of the section of a cone upon its axis, the axis will then also be that of the solid. In this case, since, in the generation of the solid, every point of the curve will describe a circle, every section of the solid parallel to the base will he a circle.
11' a conoid be cut by a plane meeting the base, or the plane of the base produced. the section will be either an ellipsis, or an hyperbola, or a parabola.
Every section of an ellip%oid oblique to its axis, is an ellipsis; and it' a paraboloid or hyperboloid be cult by a plane meeting the plane of the base, produced on the outside of the figure, the section will also be an ellipsis. In the paraboloid, it' the cutting plane be parallel to the axis. the section will be an equal parabola. In the hyperboloid, it the solid be cut by a plane parallel to a section of the cone. made by a plane passing; through the point where the asymptote of the generating section meets the axis of the solid produced, the section will be an hyperbola; but it the plane he parallel to the plane in which is the asymptote, and at right angles- with the genorating section, the section will be a parabola.
Thus the ellipsoid has only two sections. viz. the circle and the ellipsis : the paiaboloid. three sections, viz. the circle, the ellipsis, and the : the hyperboloid, four sections, viz. the circle. the ellipsis, the hyperbola, and the parabola : and the cone itself has live sections, viz. the triangle, the circle, the ellipsis, the hyperbola, and the parabola. The triangle is a section peculiar to the come alone ; the hyperbola, to the cone and hyperboloid; the parabola, to the cone and parabolical and hyperbolical colloids ; and the eirele and ellipsis are common not only to the cone, but also to each of the three com,ids.
All parallel sections of colloids are (4 srrnilar figures; though it, may seem singular that this should be a general property, when it is considered that, in a cone. a section through the apex, or point, is a triangle. and a section parallel thereto is an hyperbola ; so that if' the property existed generally, the triangular and hyperbolic sections of the cone so ought also to be similar. To reconcile this paradox, let ins
consider, that in all hyperbolical parallel sections of the ass niptotes make equal angles, and the sections o high are nearer to that passing through the apex of the cone, hove a greater degree of curvature at the vertex of these curves, than those which are more remote, though both figures be similar. Farther, if the legs of the hyperbola be infinitely extended, they will be infinitely near a straight line, as they will fall in with the asytnptotes nearly, and the curved por tion o ill bear no sensible tinmitude, compared with the part which is comparatively straight, as the legs of the hyperbola become straighter and straighter as they are more and more produced. Thus the curved portion inay be considered as a inere point to the whole figure, in a section through the vertex, the ideas of the general property seeming to vanish, or not apply ; but it' we allow a parallel section, though ever so little distant, it can very easily he compared with any remote parallel section, and their ditferenee will be this, that, in like portions of' the two curves. the similar figures inscribed in the section nearer to the apex will be incomparably small to those of the sections more remote, and in a parallel section passing through the vertex, the similar figures of comparison will lie lost, as being of infinitely small magnitude.
The section through the axis, which is the generating plane, is, in the spheroid, the greatest of all the paridlel sec tions ; but in the hyperboloid, it is the least ; and in the paraboloid. it is equal to any other parallel section.
If an hyperbola be supposed to revolve with its asymptote upon its axis, the curve will generate a colloid, and the asymptote. a cone ; if these t WO Solids be imagined to be cut by a plane in any position, then the two sections will be similar and concentric tigures, of the same species in each To find the :solidity of r/ CO/Wid.—To the area of the base, add t'our times the area of the middle section. multiply one sixth of the sum by the height, and the product will give the solidity. In the spheroid, one-sixth of fiair times the middle section only, multiplied by the height, gives the solidity ; that is, two-thirds of the circumscribing cylinder.
Other particular rules and properties will be found under ELLIPSOID, PARABOLOID, and 11 YPERBOLOID.