  As is well known, approximate integral equations for liquids, such as the
hypernetted chain (HNC) and Percus--Yevick (PY) theories, are in general
thermodynamically inconsistent in the sense that the macroscopic properties
obtained from the spatial correlation functions depend on the route followed.
In particular, the values of the fourth virial coefficient $B_4$ predicted by
the HNC and PY approximations via the virial route differ from those obtained
via the compressibility route. Despite this, it is shown in this paper that the
value of $B_4$ obtained from the virial route in the HNC theory is exactly
three halves the value obtained from the compressibility route in the PY
theory, irrespective of the interaction potential (whether isotropic or not),
the number of components, and the dimensionality of the system. This simple
relationship is confirmed in one-component systems by analytical results for
the one-dimensional penetrable-square-well model and the three-dimensional
penetrable-sphere model, as well as by numerical results for the
one-dimensional Lennard--Jones model, the one-dimensional Gaussian core model,
and the three-dimensional square-well model.
