We consider the conditions on the triplet-distribution function g (3), and the pair-distribution g(2), sufficient that thermodynamic quantities calculated from g(2) be state functions. We show that g(3) cannot be, even to first order in density, equivalent to the Kirkwood superposition approximation. We obtain a g(3) that does yield a thermodynamically self-consistent g(2), to second order in density. We show that this g(2) differs little from the exact g(2), at that order. We also examine the low-density behavior of a partially self-consistent closure in which angular correlations appear explicitly. The latter closure is found to describe three-particle correlations well when two of the particles are in contact. A principal conclusion which follows from these studies is that the criterion of thermodynamic self-consistency is useful as a constraint in constructing closures, although it does not in itself determine a closure.