Assume that a particle is moving under an arbitrary central force F 1 (r), and let its radius r and azimuthal angle φ be denoted as r (t) and φ 1 (t) as a function of time t. Now consider a second particle with the same mass m that shares the same radial motion r (t), but one whose angular speed is k times faster than that of the first particle. In other words, the azimuthal angles of the two particles are related by the equation φ 2 (t) = k φ 1 (t). Newton showed that the force acting on the second particle equals the force F 1 (r) acting on the first particle, plus an inverse-cube central force F 2 ( r ) = F 1 ( r ) + L 1 2 m r 3 ( 1 − − k 2 ) where L 1 is the magnitude of the first particle's angular momentum.