**HODOGRAPH** (Gr.

This will be evident if we consider that, since radii vectores of the **hodograph** represent velocities in the orbit, the elementary arc between two consecutive radii vectores of the **hodograph** represents the velocity which must be compounded with the velocity of the moving point at the beginning of any short interval of time to get the velocity at the end of that interval, that is to say, represents the change of velocity for that interval.

Hence the elementary arc divided by the element of time is the rate of change of velocity of the moving-point, or in other words, the velocity in the **hodograph** is the acceleration in the orbit.

Every orbit must clearly have a **hodograph**, and, conversely, every **hodograph** a corresponding orbit; and, theoretically speaking, it is possible to deduce the one from the other, having given the other circumstances of the motion.

For applications of the **hodograph** to the solution of kinematical problems see Mechanics.

**Hodograph**.The motion of a particle subject to a force which passes always through a fixed point 0 is necessarily in a plane orbit.

The locus of the point V is called the hodograp/z (q.v.); and it appears that the velocity of the point V along the **hodograph** represents in magnitude and in directon tbt acceleration in the original orbit.

In the motion of a projectile under gravity the **hodograph** is a vertical line described with constant velocity.

In elliptic harmonic motion the velocity of P is parallel and proportional to the semi-diameter CD which is conjugate to the radius CP; the **hodograph** is therefore an ellipse similar to the actual orbit.

Hence the **hodograph** is similar and similarly situated to the locus of Z (the Flo.

70 exhibits the various cases, with the **hodograph** in its proper orientation.

The pole 0 of the **hodograph** is inside on or outside the circle, according as the orbit is an ellipse, parabola or hyperbola.

In any case of a central orbit the **hodograph** (when turned through a right angle) is similar and similarly situated to the reciprocal polar of the orbit with respect to the centre of force.

Thus for a circular orbit with the centre of force at an excentric point, the **hodograph** is a conic with the pole as focus.