The cubic period-distance relation for the Kater reversible pendulum

We describe the correct cubic relation between the mass configuration of a Kater reversible pendulum and its period of oscillation. From an analysis of its solutions we conclude that there could be as many as three distinct mass configurations for which the periods of small oscillations about the two pivots of the pendulum have the same value. We also discuss a real compound Kater pendulum that realizes this property.