It has been suggested occasionally that the fine structure "constant" may not be a fixed number, but that it may be determined, in a manner not yet understood, by the distribution of mass in the universe. A simple point-particle picture is used to indicate the significance of this idea for the motion of free test particles. The possible generally covariant equations of motion for a point particle are considered, and it is shown that one can find a suitable model which is consistent with the observed structure independence of gravitational acceleration. However, it is indicated that such a model is not consistent with the precise observations, made by Hughes et al. and Drever, of the local isotropy of space. That is, with the observed structure independence of gravitational acceleration and local isotropy of space, and assuming general covariance, it seems to be necessary to rule out any appreciable variation with position in the value of the fine structure constant.