  We show that observational limits on the possible time variation of constants
of Nature are significantly affected by allowing for both space and time
variation. Bekenstein's generalisation of Maxwell's equations to allow for
cosmological variation of $alpha$ is investigated in a universe containing
spherically symmetric inhomogeneities. The time variation of $alpha$ is
determined by the local matter density and hence limits obtained in
high-density geophysical enviroments are far more constraining than those
obtained at high redshift. This new feature is expected to be a property of a
wide class of theories for the variation of constants.
