Correlation effects in the quantum crystals ${\mathrm{He}}^{3}$ and ${\mathrm{He}}^{4}$ are studied in detail. The single-particle wave function is obtained in the harmonic effective-field approximation; the parameters of the harmonic-oscillator potential are determined self-consistently from the two-body correlation function and the bare interatomic potential. We determine the two-body correlation function by solving numerically an equation derived by decoupling the three-body correlation function into a product of two-body correlation functions. This equation is similar to one derived on the basis of the $T$-matrix approximation, but yields the correct behavior of the correlation function as the distance between the particles tends to zero. The ground-state energy, pressure, and compressibility are computed over a wide range of molar volume for ${\mathrm{He}}^{3}$ in the bcc and hcp structures, and for hcp ${\mathrm{He}}^{4}$. In bcc ${\mathrm{He}}^{3}$ we employ two different interatomic potentials, the Lennard-Jones 6-12 potential generally used in calculations of this type and also the Yntema-Schneider potential. We find a ground-state energy about 1\textonehalf{}\ifmmode^\circ\else\textdegree\fi{}K too low with the former and about 5\ifmmode^\circ\else\textdegree\fi{}K too high with the latter, demonstrating that the choice of potential is of considerable importance for detailed comparison of theory and experiment. We also examine the effect of three-body correlations on the equation for the two-body correlation function; the result suggests that the effect is to force the two particles toward one another, which also seems physically reasonable, as the atoms surrounding a particular pair of particles should tend to push them together. We incorporate this effect into the two-particle equation by including a potential linear in the interparticle distance; for a given molar volume, its magnitude is chosen by imposing a constraint on the average interparticle distance. The calculated correlation functions for neighboring particles have peaks at about the corresponding lattice distance, and the numerical results for energy, pressure, and compressibility are in reasonable agreement with experiment. Our approach is compared with other recent theories of quantum crystals.