  We complete the existing literature on the structure and stability of
polytropic gas spheres reported in the classical monograph of Chandrasekhar
(1942). For isolated polytropes with index $1<n<5$, we provide a new,
alternative, proof that the onset of instability occurs for $n=3$ and we
express the perturbation profiles of density and velocity at the point of
marginal stability in terms of the Milne variables. Then, we consider the case
of polytropes confined within a box of radius $R$ (an extension of the Antonov
problem for isothermal gas spheres). For $n\ge 3$, the mass-density relation
presents some damped oscillations and there exists a limiting mass above which
no hydrostatic equilibrium is possible. Like for isothermal gas spheres, the
onset of instability occurs precisely at the point of maximum mass. Analytical
results are obtained for the particular index $n=5$. We also discuss the
relation of our study with extended thermodynamics (Tsallis entropy) recently
investigated by Taruya & Sakagami (cond-mat/0107494).
