Within an assumed pair potential framework, it has been generally accepted for a long time that far from the critical point the asymptotic form of the direct correlation function c(r) at large r is given by [-phi(r)/k(B)T]. Here phi(r) is the pair potential and k(B)T the thermal energy. Subsequently, Kumar, March, and Wasserman [Phys. Chem. Liquids 11, 271 (1982)] examined the condition for thermodynamic consistency between virial and compressibility equations of state. Their study, together with later work by Senatore, Rashid, and March [Phys. Chem. Liquids 16, 1 (1986)], resulted in a decomposition of c(r) into a potential part c(p)(r) given by Kumar for all r and involving the pair function g(r) and its density derivative, plus a "collective" part c(c)(r), which must obey a simple sum rule to satisfy thermodynamic consistency. The more recent study of B. C. Eu and K. Rah [J. Chem. Phys. 3, 3327 (1999)] prompts us to bring their results into direct contact with the study of Kumar The work of Eu and Rah gives a prominent place to the Mayer function f(r)=e(B)((-[phi(r)/k)T]-1 which tends to -[phi(r)/k(B)T] as r-->infinity for potentials tending to zero at infinity.