Singular continuation of planar central configurations with clusters of bodies

Planar central configurations of point masses that have one or more clusters of bodies are created by analytic continuation. The singularity of the gravitational interaction is removed from the continuation equations by algebraic means. The continuation splits a single body into several, and the initial mass of the single body can be nonzero. Necessary conditions are derived for these continuations that may be useful in addressing the question of finiteness of central configurations.