(13.) The parabola referred to any diameter. A diameter and the tangent at its vertex being the axes, the equation is If 4m, 4m, are the parameters corresponding to the ordinates Pp, Qq respectively, then The subtangent NT is bisected by the curve at A. If from the several points of any line given in position pairs of tangents be drawn to a parabola, the chords - joining the corresponding points of contact will pass through the same point. The chord of curvature through the focus=4SP. (H. C. S. 71-84; Hust. Prop. 9-15; Biot, 191-223; THE ELLIPSE. (14.) The Ellipse referred to its axes. Let S be the focus, and SP PQ e : 1; e being a constant quantity, and < 1, The equation referred to the axis, and a tangent at the vertex is AN.Na NP2 :: AC2: BC2. The equation referred to the centre and axes is The intersection of an ellipse with the straight line, which has only two roots, hence a straight line cannot cut an ellipse in more than two points. The equation to a tangent at the point (x,y) is Let PT be a tangent at the point P, then If the ordinate NP be produced to meet the circumference of a circle described on Aa in R, the tangents at the points P and R will meet the axis produced in the same point. The equation to the normal PG is To draw a tangent to an ellipse from a given point (~2,y2) without it the point of contact (x,y) must be determined from the equations (H. C. S. 89-111; Hust. Prop. 8-12; Biot, 133—60.) (15.) The ellipse referred to the focus. Let S, H, be the foci, then The polar equation to the ellipse, the focus being the pole, is If SY, HZ be drawn perpendicular to the tangent, the points Y, Z are always in the circumference of a circle described on A a. PI=AC. SY.HZ=BC2. (H. C. S. 112-22; Hust. Prop. 1-7.) (16.) The ellipse referred to any conjugate diameters. If CD is conjugate to CP, conversely CP is conjugate to CD. A tangent at P is parallel to CD. The axes are the only conjugate diameters that are perpendicular to each other. The equation to an ellipse referred to the centre and any conjugate diameters a1, b1 is If the extremity of the diameter is the origin, then If any = chords Qq, Rr, parallel to CD, CP, respectively, intersect each other in O, then Q0.0q RO. Or: CD2: CP2. If a1, b1, are any conjugate diameters, 19 Let Pp be the diameter drawn through any point P, whose co-ordinates are a1, y1, and PQ, Qp the supplemental chords: the equation to PQ being If the chords be drawn from the extremities of the axis major, their equations are |