tract 30.

CONOIDAL SECTIONS.

135

always in a. constant ratio; consequently afc is a conic sec-tion, and every section parallel to afc will be of the samekind with it, and similar to it. q. e. d.

Carol. 1. The above constant ratio, in which ae x ec isto ef ! , is that of ki 1 to im% the squares of the diameters ofthe generating section respectively parallel to Ac, gh ; thatis, the ratio of the square of the diameter parallel to thesection, to the square of the revolving axe of the generat-ing plane.—This will appear by conceiving ac and gh to bemoved into the positions kl, mn, intersecting in i, thecentre of the generating section.

Corol. 2. And hence it appears, that the axes ac and 2 efof the section, supposing e now to be the middle of AC,will be to each other, as the diameter kl is to the diametermn of the generating section.

Corol. 3. If the section of the solid be made so as to re-turn into itself, it will evidently be an ellipse. Which al-ways happens in the spheroid, except when it is perpendi-cular to the axe; which position is also to be excepted inthe other solids, the section being always then a circle: inthe paraboloid the section is always an ellipse, exceptingwhen it is parallel to the axe: and in the hyperboloid thesection is always an ellipse, when its axe makes with theaxe of the solid, an angle greater than that made by thesaid axe of the solid and the asymptote of the generatinghyperbola; the section being an hypeibola in all othercases, but when those angles are equal, and then it is a.parabola.

Corol. 4 . But if the section be parallel to the fixed axebd, it will be of the same kind with, and similar to, thegenerating plane abc ; that is, the section parallel to theaxe, in a spheroid, is an ellipse similar to the generatingellipse; in the paraboloid, the section is a parabola similarto the generating parabola; and in an hyperboloid, it is anhyperbola similar to the generating hyperbola of the solid.

Corol. s. In the spheroid, the section through the axe is