The Number of Planar Central Configurations Is Finite When N − 1 Mass Positions Are Fixed

In this paper, it is proved that for n > 2 and n 6= 4, if n−1 masses are located at fixed points in a plane, then there are only a finite number of n-point central configurations that can be generated by positioning a given additional nth mass in the same plane. The result is established by proving an equivalent isolation result for planar central configurations of five or more points. Other general properties of central configurations are established in the process. These relate to the amount of centrality lost when a point mass is perturbed and to derivatives associated with central configurations.