## Abstract

It is shown how to evaluate the two-body, and three-body cluster integrals, <latex>$\eta _{3}$</latex>, <latex>$\eta _{3}^{\ast}$</latex>, <latex>$\beta _{3}$</latex>, <latex>$\beta _{3}^{\ast}$</latex> (equations (1.1) to (1.4)) for the hard-sphere, square-well and Lennard-Jones (<latex>$\nu $</latex>: <latex>$\frac{1}{2}$</latex><latex>$\nu $</latex>) potentials; the three-body potential used is the dipole-dipole-dipole potential of Axilrod & Teller. Explicit expressions are presented for the integrals <latex>$\eta _{3}^{\ast}$</latex>, <latex>$\beta _{3}^{\ast}$</latex> using the above potentials; in the case of the first integral, its values for both small and large values of the separation distance are also given, for the Lennard-Jones (<latex>$\nu $</latex>: <latex>$\frac{1}{2}$</latex><latex>$\nu $</latex>) potential. Similar considerations have been carried out for <latex>$\eta _{3}$</latex>, and <latex>$\beta _{3}$</latex>, except that explicit expressions for the hard-sphere, and square-well potentials are not given, since these had been done before by other authors. The intermediate expressions for the four cluster integrals, are in terms of single integrals, and such expressions are valid for any continuous potential. Numerical results based on some of the expressions in this paper are compared with the results of numerical evaluation of the above integrals by other authors, and the agreement is seen to be good. Making use of the Mikolaj-Pings relation, the above results are used to obtain relationships between the second virial coefficient, and X-ray scattering data, as well as a means of deducing the pair potential at large separations, directly from a knowledge of X-ray scattering data, and the second virial coefficient.

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