  For some time, the theoretical result for the transition temperature of a
dilute three-dimensional Bose gas in an arbitrarily wide harmonic trap has been
known to first order in the interaction strength. We extend that result to
second order. The first-order result for a gas trapped in a harmonic potential
can be computed in mean field theory (in contrast to the first order result for
a uniform gas, which cannot). We show that, at second order, perturbation
theory suffices for relating the transition temperature to the chemical
potential at the transition, but the chemical potential is non-perturbative at
the desired order. The necessary information about the chemical potential can
be extracted, however, from recent lattice simulations of uniform Bose gases.
