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A novel procedure for the analytic evaluation of cluster integrals is given. By means of a result of Silverstone and Moats which transforms the spherical harmonic expansion of a function around a given point into a new spherical harmonic expansion around a displaced point, a 3 N -dimensional cluster integral for N point particles ( N > 2) may be reduced to 2N+1 trivial integrals and N− 1 interesting integrals, an improvement over the usual reduction to six trivial integrals and 3N −6 nontrivial integrals. For hard spheres, the N −1 integrals involve only a series of simple polynomials taken between linear algebraic bounds

Upon the utility of spherical harmonic expansions in the evaluation of cluster integrals