  In Newton's and in Einstein's theory we give criteria on the equation of
state of a barotropic perfect fluid which guarantee that the corresponding
one-parameter family of static, spherically symmetric solutions has finite
extent. These criteria are closely related to ones which are known to ensure
finite or infinite extent of the fluid region if the assumption of spherical
symmetry is replaced by certain asymptotic falloff conditions on the solutions.
We improve this result by relaxing the asymptotic asumptions. Our conditions on
the equation of state are also related to (but less restrictive than) ones
under which it has been shown in Relativity that static, asymptotically flat
fluid solutions are spherically symmetric. We present all these results in a
unified way.
