Several approximate forms of the three-particle distribution function p (3)(1,2,3) (Kirkwood superposition p K, convolution p c, Abe p A) are studied in relation to the accuracy with which they fulfill the sequential relation connecting two- and three-particle distribution functions. In a generalized Abe form, four conditions are imposed on six parameters by the sequential relation. In the uniform limit the radial distribution function g( r) differs little from its asymptotic value. The parameter α = g( o) − 1 can then be used as an ordering parameter in cluster expansions which represent physical quantities as functionals in the radial distribution function. The Abe-type correction factor to p K is determined by a cluster expansion procedure through terms of order α 3. The corresponding error term in the sequential relation is also of order α 3. Two rules are stated which eliminate infinite classes of diagrams from a diagrammatic representation of the correction factor and, in a particular approximation (in conjunction with the sequential relation), determine three out of four parameters at the values fixed by the cluster-expansion procedure. In the uniform limit, the Abe form for p (3) and the BBGKY relation yield the hypernetted-chain connection between the two-particle correlation function u [Eq. (13)] and the radial distribution function g with an explicit correction term of order α 3. The p (3) function for a pure state is subject to a dynamical consistency condition. The condition is developed in both coordinate and momentum spaces and partially evaluated for both p K and p c using g( r) computed from the observed liquid structure function of liquid 4He. In configuration space the condition singles out certain straight line and equilateral triangular configurations for which p K is apparently most seriously inadequate. An Appendix exhibits a four-point distribution function p c(1,2,3,4) which generates p c(1,2,3) in the sequential relation connecting p (4) and p (3). A second Appendix discusses the application of the hnc connection between g( r) and u( r) to compute the ground state energy of a boson system in the uniform limit.