  We examine the thermodynamical properties of a family of partially relaxed,
anisotropic stellar systems, derived earlier from the Boltzmann entropy under
the assumption that a third quantity Q is conserved in addition to the total
energy and the total number of stars. We now show that the family of models
conforms to the paradigm of the gravothermal catastrophe, which is expected to
occur (in the presence of adequate energy transport mechanisms) when the
one-parameter equilibrium sequence attains sufficiently high values of the
concentration parameter; these are the values for which the models are well
fitted by the R^(1/4) law. In the intermediate concentration regime the models
belonging to the sequence exhibit significant deviations from the R^(1/4) law.
Curiously, in the low-concentration regime, the global thermodynamical
temperature associated with the models becomes negative when the models become
too anisotropic so that they are unstable against the radial orbit instability;
this latter behavior, while offering a new clue to the physical interpretation
of the radial orbit instability, is at variance with respect to the
low-concentration limit of the classical case of the isotropic, isothermal
sphere investigated by Bonnor (1956) and Lynden-Bell & Wood (1968).
