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We summarize the previous treatments of the question of superfluidity in ${\mathrm{He}}^{3}$, 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 ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{T}}}_{j}$ was carried out in the limiting case of large angular momentum $j$. In estimating ${T}_{c}$ 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 ${T}_{c}$ then yields the unattainably low value ${T}_{c}\ensuremath{\sim}{10}^{\ensuremath{-}100}$ deg. We have projected precisely ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{T}}}_{j}(0\ensuremath{\le}j\ensuremath{\le}10)$ in second-order perturbation theory. We used as the bare vertex function the off-energy-shell $\mathcal{K}$ matrix, the particular form of which was chosen so that $\mathcal{K}$ 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 ${\mathcal{K}}_{j}$, for $j=2$ and $j\ensuremath{\ge}6$, in contrast to Pitaevskii's result that the second-order diagram for ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{T}}}_{j}$ 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 $jl20$). 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 ${T}_{c}$.