Corresponding to each relative equilibrium in the n-body problem, there is a family of periodic solutions for which each body traces out an ellipse with eccentricity e. Over the course of the orbit, the configuration of the bodies maintains its shape, but alters its size (homographic motion). While there has been some progress made in understanding the linear stability of the circular periodic solutions (e=0), little is known about solutions which are genuinely elliptic. Using e as a parameter, we investigate the case e \neq 0 and reveal that in certain examples the elliptic orbits can be linearly stable even though the circular ones are not.