HODOGRAPH. If from any point 0 a vector OP be drawn representing at any instant in magnitude and direction the velocity of a particle P', which is moving in any manner whatever, the locus of P is the hodograph of the path of P'. (Cf. the definition of radial curve under CURVES, SPECIAL.) A fundamental property of this curve is that the velocity of any point P of the hodograph is equal to the acceleration of the corresponding point P'; also that the direction of motion of P is that of the direction of accel eration of P'. The hodograph of the orbit of a planet or a comet, considered as in a Newtonian field, is always a circle, whatever may be the form and dimensions of the orbit. The pole 0 is inside, on, or outside the circle, according as the orbit is an ellipse, a para bola or a hyperbola. The idea of the hodograph originated with Mobius (1843) and, independently, with Sir William Rowan Hamilton (1846) to whom the name and certain original develop ments of the theory are due. If a particle describes a logarithmic spiral about the pole as a centre of force, the hodograph is also a logarithmic spiral; the same result holds true for sinusoidal spirals (Schouten) . If a particle starts from rest at the vertex of a perfectly smooth inverted cycloid and oscillates under the action of gravity, the hodograph of the motion is a circle through the pole, described with constant velocity. An important improve ment in nautical charts by A. Smith (Proc. Roy. Soc., vol. xv., 1867) introduced a curve which has been called a tidal hodograph (Thomson and Tait).