  In 1970 Don Saari conjectured that the only solutions of the Newtonian
$n$-body problem that have constant moment of inertia are the relative
equilibria. We prove this conjecture in the collinear case for any potential
that involves only the mutual distances. Furthermore, in the case of
homogeneous potentials, we show that the only collinear and non-zero angular
momentum solutions are homographic motions with central configurations.
