We assume that the neutron-neutron potential is well-behaved and velocity-dependent. We can then apply perturbation theory to find the energy per particle of a neutron gas, in the range of Fermi wave numbers $0.5<\mathrm{}{k}_f<2\mathrm{}$ ${\mathrm{f}}^{-1\mathrm{}}$. The energy through first order is found in closed form, or by a single numerical integration. We use two different velocity-dependent potentials adjusted to fit observed nucleon-nucleon ${}_{}{}^1{}_{}{}^{}S$ and ${}_{}{}^1{}_{}{}^{}D$ phase shifts. In the range of densities $0.5<\mathrm{}{k}_f<1\mathrm{}$ ${\mathrm{f}}^{-1\mathrm{}}$, our two potentials give nearly the same energy/particle (within 0.5 Mev); our values tend to run an Mev below values found by Brueckner et al., for the Gammel-Thaler potential. Wider divergences appear at higher densities. Our values, and Brueckner's are higher than those found by Salpeter by a semiempirical approach. A crude estimate of the second-order energy for our potentials indicates that perturbation theory converges rapidly in the density range considered. Our results suggest that at moderately low densities the energy/particle in a many-body system is insensitive to the shape or nonlocal character of the assumed two-body potential.

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