On the distribution of mass in collinear central configurations

Moulton’s Theorem says that given an ordering of masses, $m_1,m_2, \dotsc ,m_n$, there exists a unique collinear central configuration with center of mass at the origin and moment of inertia equal to 1. This theorem allows us to ask the questions: What is the distribution of mass in this standardized collinear central configuration? What is the behavior of the distribution as $n\to \infty$? In this paper, we define continuous configurations, prove a continuous version of Moulton’s Theorem, and then, in the spirit of limit theorems in probability theory, prove that as $n\to \infty$, under rather general conditions, the discrete mass distributions of the standardized collinear central configurations have distribution functions which converge uniformly to the distribution function of the unique continuous standardized collinear central configuration which we determine.