  The simplest bounded potential is that of penetrable spheres, which takes a
positive finite value $\epsilon$ if the two spheres are overlapped, being 0
otherwise. In this paper we derive the cavity function to second order in
density and the fourth virial coefficient as functions of $T^*\equiv
k_BT/\epsilon$ (where $k_B$ is the Boltzmann constant and $T$ is the
temperature) for penetrable sphere fluids. The expressions are exact, except
for the function represented by an elementary diagram inside the core, which is
approximated by a polynomial form in excellent agreement with accurate results
obtained by Monte Carlo integration. Comparison with the hypernetted-chain
(HNC) and Percus-Yevick (PY) theories shows that the latter is better than the
former for $T^*\lesssim 1$ only. However, even at zero temperature (hard sphere
limit), the PY solution is not accurate inside the overlapping region, where no
practical cancelation of the neglected diagrams takes place. The exact fourth
virial coefficient is positive for $T^*\lesssim 0.73$, reaches a minimum
negative value at $T^*\approx 1.1$, and then goes to zero from below as
$1/{T^*}^4$ for high temperatures. These features are captured qualitatively,
but not quantitatively, by the HNC and PY predictions. In addition, in both
theories the compressibility route is the best one for $T^*\lesssim 0.7$, while
the virial route is preferable if $T^*\gtrsim 0.7$.
