  The vectorial velocity is given as a function of the position of a particle
in orbit when a Newtonian central force is supplemented by an inverse cubic
force as in Newton's theorem on revolving orbits. Such expressions are useful
in fitting orbits to radial velocities of orbital streams. The
Hamilton-Laplace-Runge-Lenz eccentricity vector is generalised to give a
constant of the motion for these systems and an approximate constant for orbits
in general central potentials. A related vector is found for Hooke's centred
ellipse.
