  We use the path-integral Monte Carlo (PIMC) method and state-of-the-art
two-body and three-body potentials to calculate the fourth virial coefficients
$D(T)$ of $^4$He and $^3$He as functions of temperature from 2.6K to 2000K. We
derive expressions for the contributions of exchange effects due to the bosonic
or fermionic nature of the helium isotope; these effects have been omitted from
previous calculations. The exchange effects are relatively insignificant for
$^4$He at the temperatures considered, but for $^3$He they are necessary for
quantitative accuracy below about 4K. Our results are consistent with previous
theoretical work (and with some of the limited and scattered experimental data)
for $^4$He; for $^3$He there are no experimental values and this work provides
the first values of $D(T)$ calculated at this level. The uncertainty of the
results depends on the statistical uncertainty of the PIMC calculation, the
estimated effect of omitting four-body and higher terms in the potential
energy, and the uncertainty contribution propagated from the uncertainty of the
potentials. At low temperatures, the uncertainty is dominated by the
statistical uncertainty of the PIMC calculations, while at high temperatures
the uncertainties related to the three-body potential and to omitted
higher-order contributions become dominant.
