  We consider a dilute Bose gas confined by a harmonic potential. We define an
appropriate thermodynamic limit and analyze the properties of the phases and
phase transition in this limit. Critical properties in the presence of the
potential are found to be different from, though simply related, to those in
the usual translationally invariant case. We argue that the properties of
magnetically trapped rubidium\cite{Wieman} and sodium\cite{ketterle} gases (in
which Bose-Einstein condensation has been recently observed) are well
approximated by our thermodynamic limit except in a narrow window of
temperature around the critical temperature. We also consider the effect of the
confining potential on the non-equilibrium dynamics following a rapid quench to
the ordered side and give a scaling description of the late time universal
dynamics.
