  The fourth virial coefficient of additive hard-sphere mixtures, as predicted
by the Percus-Yevick (PY) and hypernetted-chain (HNC) theories, is derived via
the compressibility, virial, and chemical-potential routes, the outcomes being
compared with exact results. Except in the case of the HNC compressibility
route, the other five expressions exhibit a common structure involving the
first three moments of the size distribution. In both theories the
chemical-potential route is slightly better than the virial one and the best
behavior is generally presented by the compressibility route. Moreover, the PY
results with any of the three routes are more accurate than any of the HNC
results.
