We determine the effective interaction of quasiparticles on the Fermi surface for the electron gas in a rigid positive background. The quasiparticle scattering amplitude is calculated from the two coupled Bethe-Salpeter equations for the two-particle vertex functions in the particle-hole (p-h) channels for densities from r/sub s/ = 1 to r/sub s/ = 37. The density and spin-density mean fields are fitted to the compressibility and spin susceptibility of Green's-function Monte Carlo calculations and the local-field factors G(q) of microscopic models. We find p-wave superconductivity for 1035.