. Mathematics, compiled from the best authors and intended to be the text-book of the course of private lectures on these sciences in the University at Cambridge [microform]. .double ordinates are bisected by the axis ; and that the?whole figure, made up of all the double ordinates, is alsobisected by the axis. 2. The two foci are equally distant from the centre, orfrom either vertex. Cor. 3. When the angle, Which the plane of the sec-tion makes with the base of the cone, decreases till it be-come equal to the angle made by the side of the cone and* CONIC SECTIONS. 303 the base, or till the se
. Mathematics, compiled from the best authors and intended to be the text-book of the course of private lectures on these sciences in the University at Cambridge [microform]. .double ordinates are bisected by the axis ; and that the?whole figure, made up of all the double ordinates, is alsobisected by the axis. 2. The two foci are equally distant from the centre, orfrom either vertex. Cor. 3. When the angle, Which the plane of the sec-tion makes with the base of the cone, decreases till it be-come equal to the angle made by the side of the cone and* CONIC SECTIONS. 303 the base, or till the section be parallell to the opposite side ofthe cone j then the axis becomes infinitely long, anddie hyperbola degenerates into a parabola ; and the ab-scisses are to each other as the squares of their ordinates. For then the infinites FB and HB are in a ratio of equality,and the general property, namely, AF-FB : AH-HB :: FG2 : HI*, becomes AF : AH :: FG2 : HI2. PROPOSITION II. As the square of the transverse axis :Is to the square of the conjugate : :So is the rectangle of the abscisses :To the squaie of their ordinate. That is, AB3 : a£3orAC2 : «C2 :: AD-DB : For, by Prop. I. AC-CB : AD-DB :: C«2 : DE2 ;But, if C be the centre, then AC-CB =:AC2, and Ca isthe semiconjugate ; therefore, AC2 : AD-DB :: aC* : DE2 ; or, by permutation, AC2 : aC2 :: AD-DB : DE2 ;or, by doubling, ABa : ah* :: AD-DB : DE2. Q. E. D. J04 MATHEMATICS. Cor. 1. Or, AB8 : ab* :: CD2— CA2 : as the rectangle AT>DB=CD»—CA2,the same property is CA2 : Ca2 :: CD2—CA2 : DES, Cor. 2. Or, as the transverseIs to its parameter,So is the rectangle of the abscissesTo the square of their ordinate. ah*For, : jjt :: CD2—CA2 : DE2, that is, AB : p :: AD-DB or CD2—CA9 : DE* ; wl a** p is the parameter . vt, by the definition of it. Cor. 3. When the axis AB is infinitely long, the curvebecomes a parabola, and the infinites AB, DB, are then in aratio of equality ; and then the last propert
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