. 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]. n ordinate to it; as AK or BK, orDN or EN. Hence in the ellipse and hyperbola, every ordinate hastwo abscisses ; bin in the parabola, only one ; the other ver-tex of the diameter being infinitely distant. CONIC SECTIONS, j«.j 16. The Parameter of any diameter is a third propor-tional to that diameter and its conjugate. 17. The Focus is the point in the axis, where the ordin-ate is equal to half the parameter ; as K and L,
. 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]. n ordinate to it; as AK or BK, orDN or EN. Hence in the ellipse and hyperbola, every ordinate hastwo abscisses ; bin in the parabola, only one ; the other ver-tex of the diameter being infinitely distant. CONIC SECTIONS, j«.j 16. The Parameter of any diameter is a third propor-tional to that diameter and its conjugate. 17. The Focus is the point in the axis, where the ordin-ate is equal to half the parameter ; as K and L, where DKor EL is equal to the semiparameter. Hence, the ellipse and hyperbola have each two foci; butthe parabola only one. 18. If DAE, FBG be two opposite hyperbolas, havingAB for their first or transverse axis, and ab for their secondor conjugate axis ; and if due, jbg be two other oppositehyperbolas, having the same axis, but in a contrary order,namely, ab their first axis, and AB their second ; then thesetwo latter curves dae, fbg, are called the conjugate hyper-bolas to the two former DAE, FBG ; and each pair of op-posite curves mutually conjugate to the 19. And if tangents be drawn to the four vertices of thecurves, or extremities of the axis, forming the inscribedrectangle HIKL ; the diagonals HCK, ICL of this rectan-gle are called the asymptotes of the curves. And if theseasymptotes intersect at right angles, or the inscribed rectan-gle be a square, or the two axes AB and ab be equal, thenthe hyperbolas are said to be right-angled, or equilateral. 176 MATHEMATICS, Scholium. The rectangle, inscribed between the fourconjugate hyperbolas, is similar to a rectangle, circumscrib-ed about an ellipse by drawing tangents, in like manner, tothe four extremities of the two axes ; and the asymptotesor diagonals, in the hyperbola, are analogous to those in theellipse, cutting this curve in similar points and making thepair of equal conjugate diameters. Moreover, the whole
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