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CONOIDAL SECTIONS.

TRACT 30.

PROPOSITION II.

If si be the semi-diameter belonging to the double ordinateaec of the generating plane, aec being the diameter of thesection afc, conceived to be moved continually parallel to it-self; and if x denote any part of the diameter si, interceptedby e the middle of ac, and any given fixed point taken in si;then will the section afc be always as a b.v 4- cxx ; a, b, c,being constant quantities ; b in some cases affirmative, and inothers negative ; c being affirmative in the hyperbola, and ne-gative in the ellipse, and nothing in the parabola ; and a mayalways be supposed to denote the distance of the given fixedpoint from the vertex s.

Demon. In any conic section, ac 2, is as a + bx 4- cxx ;but all the parallel sections are like and similar figures, andsimilar plane figures are as the squares of their like dimen-sions ; therefore the section afc is as ac 2 , that is, asa bx + cxx. q. e. d.

Corollary. If the given fixed point, where x begins, co-incide with the vertex s, then will a be equal to nothing,and the section will be as bx ± cxx, or as x ± dxx, in thehyperbola and ellipse, and as bx, or as x, in the parabola.