To investigate the form of the curve use may be made of the definition: the ellipse is the locus of a point which moves so that the ratio of its distance from a fixed point (the _focus_) to its distance from a straight line (the _directrix_) is constant and is less than unity. This ratio is termed the _eccentricity_, and will be denoted by e. Let KX (fig. 1) be the directrix, S the focus, and X the foot of the perpendicular from S to KX. If SX be divided at A so that SA/AX = e, then A is a point on the curve. SX may be also divided externally at A', so that SA'/A'X = e, since e is less than unity; the points A and A' are the _vertices_, and the line AA' the _major axis_ of the curve. It is obvious that the curve is symmetrical about AA'. If AA' be bisected at C, and the line BCB' be drawn perpendicular to AA', then it is readily seen that the curve is symmetrical about this line also; since if we take S' on AA' so that S'A' = SA, and a line K'X' parallel to KX such that AX = A'X', then the same curve will be described if we regard K'X' and S' as the given directrix and focus, the eccentricity remaining the same. If B and B' be points on the curve, BB' is the _minor axis_ and C the _centre_ of the curve. [Illustration: FIG. 1.] Metrical relations between the axes, eccentricity, distance between the foci, and between these quantities and the co-ordinates of points on the curve (referred to the axes and the centre), and focal distances are readily obtained by the methods of geometrical conics or analytically. The semi-major axis is generally denoted by a, and the semi-minor axis by b, and we have the relation b² = a²(1 - e²). Also a² = CS·CX, i.e. the square on the semi-major axis equals the rectangle contained by the distances of the focus and directrix from the centre; and 2a = SP + S'P, where P is any point on the curve, i.e. the sum of the focal distances of any point on the curve equals the major axis. The most important relation between the co-ordinates of a point on an ellipse is: if N be the foot of the perpendicular from a point P, then the square on PN bears a constant ratio to the product of the segments AN, NA' of the major axis, this ratio being the square of the ratio of the minor to the major axis; symbolically PN² = AN·NA'(CB/CA)². From this or otherwise it is readily deduced that the ordinates of an ellipse and of the circle described on the major axis are in the ratio of the minor to the major axis. This circle is termed the _auxiliary circle_.