The method by which the cycloid is generated shows that it consists of an infinite number of cusps placed along the fixed line and separated by a constant distance equal to the circumference of the rolling circle. The name cycloid is usually restricted to the portion between two consecutive cusps (fig. 1, curve a); the fixed line LM is termed the base, and the line PQ which divides the curve symmetrically is the _axis_. The co-ordinates of any point R on the cycloid are expressible in the form x = a([theta] + sin [theta]); y = a(1 - cos [theta]), where the co-ordinate axes are the tangent at the vertex O and the axis of the curve, a is the radius of the generating circle, and [theta] the angle R´CO, where RR´ is parallel to LM and C is the centre of the circle in its symmetric position. Eliminating [theta] between these two relations the equation is obtained in the form x = (2ay - y²)½ + a vers-¹ y/a. The clumsiness of the relation renders it practically useless, and the two separate relations in terms of a single parameter [theta] suffice for the deduction of most of the properties of the curve. The length of any arc may be determined by geometrical considerations or by the methods of the integral calculus. When measured from the vertex the results may be expressed in the forms s = 4a sin ½[theta] and s = [root](8ay); the total length of the curve is 8a. The intrinsic equation is s = 4a sin [psi], and the equation to the evolute is s = 4a cos [psi], which proves the evolute to be a similar cycloid placed as in fig. 2, in which the curve QOP is the evolute and QPR the original cycloid. The radius of curvature at any point is readily deduced from the intrinsic equation and has the value [rho] = 4 cos ½[theta], and is equal to twice the normal which is 2a cos ½[theta]. Entry: CYCLOID