  We discuss the statistical mechanics of rotating self-gravitating systems by
allowing properly for the conservation of angular momentum. We study
analytically the case of slowly rotating isothermal spheres by expanding the
solutions of the Boltzmann-Poisson equation in a series of Legendre
polynomials, adapting the procedure introduced by Chandrasekhar (1933) for
distorted polytropes. We show how the classical spiral of Lynden-Bell & Wood
(1967) in the temperature-energy plane is deformed by rotation. We find that
gravitational instability occurs sooner in the microcanonical ensemble and
later in the canonical ensemble. According to standard turning point arguments,
the onset of the collapse coincides with the minimum energy or minimum
temperature state in the series of equilibria. Interestingly, it happens to be
close to the point of maximum flattening. We determine analytically the
generalization of the singular isothermal solution to the case of a slowly
rotating configuration. We also consider slowly rotating configurations of the
self-gravitating Fermi gas at non zero temperature.
