Of the properties of a tangent it may be noticed that the tangent at any point is equally inclined to the focal distances of that point; that the feet of the perpendiculars from the foci on any tangent always lie on the auxiliary circle, and the product of these perpendiculars is constant, and equal to the product of the distances of a focus from the two vertices. From any point without the curve two, and only two, tangents can be drawn; if OP, OP' be two tangents from O, and S, S' the foci, then the angles OSP, OSP' are equal and also SOP, S'OP'. If the tangents be at right angles, then the locus of the point is a circle having the same centre as the ellipse; this is named the _director circle_. The middle points of a system of parallel chords is a straight line, and the tangent at the point where this line meets the curve is parallel to the chords. The straight line and the line through the centre parallel to the chords are named _conjugate diameters_; each bisects the chords parallel to the other. An important metrical property of conjugate diameters is the sum of their squares equals the sum of the squares of the major and minor axis. In analytical geometry, the equation ax² + 2hxy + by² + 2gx + 2fy + c = 0 represents an ellipse when ab > h²; if the centre of the curve be the origin, the equation is a¹x² + 2h¹xy + b¹y² = C¹, and if in addition a pair of conjugate diameters are the axes, the equation is further simplified to Ax² + By² = C. The simplest form is x²/a² + y²/b² = 1, in which the centre is the origin and the major and minor axes the axes of co-ordinates. It is obvious that the co-ordinates of any point on an ellipse may be expressed in terms of a single parameter, the abscissa being a cos [phi], and the ordinate b sin [phi], since on eliminating [phi] between x = a cos [phi] and y = b sin [phi] we obtain the equation to the ellipse. The angle [phi] is termed the _eccentric angle_, and is geometrically represented as the angle between the axis of x (the major axis of the ellipse) and the radius of a point on the auxiliary circle which has the same abscissa as the point on the ellipse. The equation to the tangent at [theta] is x cos [theta]/a + y sin [theta]/b = 1, and to the normal ax/cos [theta] - by/sin [theta] = a² - b². The area of the ellipse is [pi]ab, where a, b are the semi-axes; this result may be deduced by regarding the ellipse as the orthogonal projection of a circle, or by means of the calculus. The perimeter can only be expressed as a series, the analytical evaluation leading to an integral termed _elliptic_ (see FUNCTION, ii. _Complex_). There are several approximation formulae:--S = [pi](a + b) makes the perimeter about 1/200th too small; s =[pi][root](a²+b²) about 1/200th too great; 2s = [pi](a + b) + [pi][root](a² + b²) is within 1/30,000 of the truth.