  The transition temperature for a dilute, homogeneous, three-dimensional Bose
gas has the expansion T_c = T_0 {1 + c_1 a n^(1/3) + [c_2' ln(a n^(1/3)) +
c_2''] a^2 n^(2/3) + O(a^3 n)}, where a is the scattering length, n the number
density, and T_0 the ideal gas result. The first-order coefficient c_1 depends
on non-perturbative physics. In this paper, we show that the coefficient c_2'
can be computed perturbatively. We also show that the remaining second-order
coefficient c_2'' depends on non-perturbative physics but can be related, by a
perturbative calculation, to quantities that have previously been measured
using lattice simulations of three-dimensional O(2) scalar field theory. Making
use of those simulation results, we find T_c = T_0 {1 + (1.32+-0.02) a n^(1/3)
+ [19.7518 ln(a n^(1/3)) + (75.7+-0.4)] a^2 n^(2/3) + O(a^3 n)}.
