results, unlike the compressibility route (PYC). It is worth noting that the results stemming from the $C_{1}$ approximation, although rather close to the MC data in the low-density branch, clearly fail to accurately reproduce them for higher densities, as expected.

In spite of their lack of accuracy, the $C_{0}$ and $C_{1}$ approximations provide however a rather sound basis for getting some insight into phase equilibria of polydisperse SHS fluids, since they allow simple analytical, or semi-analytical, treatments.

A first important feature of the $C_{0}$ and $C_{1}$ approximations for polydisperse SHS is that the corresponding free energy has a *truncatable* structure, that is it depends upon few ($4$ at most) moments of the (discrete) size distribution, $\xi _{\nu
}=(\pi /6)\rho \sum_{j}x_{j}\sigma _{j}^{\nu }$ with $\nu =0,1,2,3$.

A second remarkable fact is that the $C_{0}$ and $C_{1}$ approximations are able to describe the so-called *fractionation* phenomena characteristic in phase equilibria of polydisperse systems. While we refer to a recent review [@Sollich02] for detailed description of the increased complexity in the polydisperse phase diagrams, here we just mention the two important points. First, fractionation means that daughter phases, obtained from demixing of a parent homogeneous phase, need not have the same composition of 