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 \geq 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 in the series of equilibrium .
Analytical results are obtained for the particular index n = 5 .
We also discuss the relation of our study with generalized thermodynamics ( Tsallis entropy ) recently investigated by Taruya & Sakagami ( cond-mat/0107494 ) .