  The critical properties of the phase transition from a normal gas to a BEC
(superfluid) of a harmonically confined Bose gas are addressed with the
knowledge of an equation of state of the underlying homogeneous Bose fluid. It
is shown that while the presence of the confinement trap arrests the usual
divergences of the isothermal compressibility and heat capacities, the critical
behavior manifests itself now in the divergence of derivatives of the mentioned
susceptibilities. This result is illustrated with a mean-field like model of an
equation of state for the homogeneous particle density as a function of the
chemical potential and temperature of the gas. The model assumes the form of an
ideal Bose gas in the normal fluid while in the superfluid state a function is
proposed such that, both, asymptotically reaches the Thomas-Fermi solution of a
weakly interacting Bose gas at large densities and low temperatures and, at the
transition, matches the critical properties of the ideal Bose gas. With this
model we obtain the {\it global} thermodynamics of the harmonically confined
gas, from which we analyze its critical properties. We discuss how these
properties can be experimentally tested.
