  This paper is an investigation of the stability of some ideal stars. It is
in- tended as a study in General Relativity, with emphasis on the coupling to
matter, eventually aimed at a better understanding of very strong gravitational
fields and Black Holes. The work is based on an action principle for the
complete system of metric and matter fields. We propose a complete revision of
the treatment of boundary conditions. An ideal star in our terminology has
spherical symmetry and an isentropic equation of state. In our first work on
this subject it was assumed that the density vanishes beyond a finite distance
from the origin. But it is difficult to decide what the proper boundary
conditions should be and we are consequently skeptical of the concept of a
fixed boundary. In this paper we investigate the double polytrope,
characterized by a polytropic index n less than 5 in the bulk of the star and a
value larger than 5 in an outer atmosphere that extends to infinity. It has no
fixed boundary but a region of critical density where the polytropic index
changes from a value that is appropriate for the bulk of the star to a value
that provides a crude model for the atmosphere. The existence of a relation
between mass and radius is confirmed, as well as an upper limit on the mass.
The principal conclusion is that all the static configurations are stable.
