Directrices.--The distance from each point of the ellipse to one of the foci, and to the directrix adjacent to that focus, are to each other as the eccentricity is to the major axis. Equations of the tangent and of the normal at any point of the ellipse.[6]--The point in which the tangent meets one of the axes prolonged is independent of the length of the other axis.--Construction of the tangent at any point of the ellipse by means of this property. [Footnote 6: They will be deduced from the property, previously demonstrated, of the derivative of the ordinate with respect to the abscissa.] The radii vectores, drawn from the foci to any point of the ellipse, make equal angles with the tangent at that point or the same side of it.--The normal bisects the angle made by the radii vectores with each other.--This property may serve to draw a tangent to the ellipse through a point on the curve, or through a point exterior to it. The diameters of the ellipse are right lines passing through the centre of the curve.--The chords which a diameter bisects are parallel to the tangent drawn through the extremity of that diameter.-- Supplementary chords. By means of them a tangent to the ellipse can be drawn through a given point on that curve or parallel to a given right line. Conjugate diameters.--Two conjugate diameters are always parallel to supplementary chords, and reciprocally.--Limit of the angle of two conjugate diameters.--An ellipse always contains two equal conjugate diameters.--The sum of the squares of two conjugate diameters is constant.--The area of the parallelogram constructed on two conjugate diameters is constant.--To construct an ellipse, knowing two conjugate diameters and the angle which they make with each other. Expression of the area of an ellipse in function of its axes. _Of the hyperbola._ Centre and axes.--Ratio of the squares of the ordinates perpendicular to the transverse axes. Of foci and of directrices; of the tangent and of the normal; of diameters and of supplementary chords.--Properties of these points and of these lines, analogous to those which they possess in the ellipse. Asymptotes of the hyperbola.--The asymptotes coincide with the diagonals of the parallelogram formed on any two conjugate diameters.--The portions of a secant comprised between the hyperbola and its asymptotes are equal.--Application to the tangent and to its construction. The rectangle of the parts of a secant, comprised between a point of the curve and the asymptotes, is equal to the square of half of the diameter to which the secant is parallel. Form of the equation of the hyperbola referred to its asymptotes. _Of the parabola._ Axis of the parabola.--Ratio of the squares of the ordinates perpendicular to the axis. Focus and directrix of the parabola.--Every point of the curve is equally distant from the focus and from the directrix.--Construction of the parabola.