Analytical expressions for the fourth virial coefficient of a hard-sphere mixture

A method of numerical calculation of the fourth virial coefficients of the mixture of additive hard spheres is proposed. The results are compared with an exact analytical formula for the fourth partial virial coefficient B-4([1]) (i.e., three spheres of diameters sigma(1) and one sphere of diameter sigma(2)) and a semiempirical expression for B-4([2]) (i.e., two spheres of each kind). It is shown that the first formula is nonanalytic and the implication to the equations of state for hard-sphere mixtures is discussed.