TRACT 30 .

CONOIDAL SECTIONS.

1ST

Thus, let an hyperbola and its asymptote be revolved to-gether about the transverse axe, the former describing anhyperboloid, and the latter a cone circumscribing it; nowlet them be supposed to be both cut bv one plane in anyposition, then the two sections will be like, similar, andconcentric figures: that is, if the plane cut both sides ofeach, the sections will be concentric,-similar ellipses ; il thecutting plane be parallel to the asymptote, or to the side ofo the cone, the sections will be parabolas; and in all otherpositions, the sections will be similar and concentric hyper-bolas.

That the sections are like figures, appears from the fore-going corollaries. That they are concentric, will be evidentwhen we consider that c c is = a a, producing ac both waysto meet the asymptotes in a and c. And that they are simi-lar, or have their transverse and conjugate axes proportionalto each other, will appear thus: Produce gh both ways tomeet the asymptotes in g and h- and on the diameters gh,gh, describe the semi-circles gqh, gB.h, meeting eqr, drawnperpendicular to gh, in q and r; eq and er being thenevidently the semi-conjugate axes, and ec, ec, the semi-transverse axes of the sections. Now if gh and ac be con-ceived to be moved parallel to themselves, ae X ec or ce 1 ,will be to ge x eh or eq 1 , in a constant ratio, or ce to eq.will be a constant ratio ; and since ce is as e g, and <7e as e h,ae x ec or ce 1 , will be to gB x e h or er 1 , in a constant ratio,or ce to er will be a constant ratio ; but at an infinite dis-tance from the vertex, c and c coincide, or ec = ec, asalso eg = E;g, consequently eq is then — er, and ce to eq' will be =: ce to er ; but as these ratios are constant, if theybe equal to each other in one place, they must be always so;and consequently ce : ec :: qe : er.

And this analogy of the sections will not seem strange,when we consider that a cone is a species of the hyperbo-loid ; or a triangle a species of the hyperbola, rvhose axesare infinitely small.