We summarize the previous treatments of the question of superfluidity in He3, which have not shown conclusively whether this phase should or should not occur. The most promising calculation was by Pitaevskii, whose determination of the screened quasiparticle transition amplitude T~j was carried out in the limiting case of large angular momentum j. In estimating Tc this was extrapolated to j=2 by Gor'kov and Pitaevskii (GP). The work of GP was found to contain an inadequate approximation (the D-wave Born approximation) plus also an algebraic error. When these corrections are made, their formula for Tc then yields the unattainably low value Tc~10-100 deg. We have projected precisely T~j(0<=j<=10) in second-order perturbation theory. We used as the bare vertex function the off-energy-shell K matrix, the particular form of which was chosen so that K would satisfy two exact identities (Ward identities). It was found that GP's extrapolation underestimates the effective D-wave interaction. In fact, the polarization (particle-hole) diagram is an attractive correction to Kj, for j=2 and j>=6, in contrast to Pitaevskii's result that the second-order diagram for T~j is repulsive. We study carefully the three approximations involved in Pitaevskii's large-j calculation. Theoretical arguments are given, and numerical evidence is furnished, which indicate that as j increases, Pitaevskii's limit is being approached asymptotically, but so slowly that his analytical result does not hold for angular momenta of interest (i.e., for j<20). The possibility of D-state superfluidity is discussed. The transition temperature is such a sensitive function of the calculated theoretical parameters that it is difficult to furnish a precise estimate for Tc.