A variation principle is introduced involving then-particle molecular distribution function (where 1 n N) for a fluid containingN molecules. An integral involving any approximaten-particle distribution function proves to define aleast upper bound to the true system free energy. This integral can, therefore, be minimized with respect to the form of a trial distribution function to provide a best estimate to the exact distribution function. When no other constraints, save the requirement of normalization, are applied to the trial function, the extremum corresponds to the exact function. Using this variation principle, it is possible to demonstrate that the optimum triplet superposition approximation is the Krikwood approximation, and that the optimum quadruplet approximation is the form suggested by Fisher and Kopeliovich. Furthermore, all higher-order optimum superposition approximations are specified.