Imagine a second particle with the same mass m and with 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 θ 2 (t) = k θ 1 (t). Newton showed that the motion of the second particle can be produced by adding an inverse-cube central force to whatever force F 1 (r) acts on the first particle