We consider thin spherical shells of matter in both Newtonian gravity and general relativity and examine their equilibrium configurations and dynamical stability. Thin-shell models are admittedly a poor substitute for realistic stellar models. But the simplicity of the equations that govern their dynamics, compared to the much more complicated mechanics of a self-gravitating fluid, allows us to deliver, in a very direct and easy manner, powerful insights into their equilibria and stability. We explore, in particular, the link between the existence of a maximum mass along a sequence of equilibrium configurations and the onset of dynamical instability. Such a link is well-established in the case of fluid bodies in both Newtonian gravity and general relativity, but the demonstration of this link is both subtle and difficult. The proof is very simple, however, in the case of thin shells, and it is constructed with nothing more than straightforward algebra and a little calculus.

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