We apply the potential distribution theorems for the cavity distribution functions to the development of thermodynamic formulas for fused-sphere chain molecules. Alternative forms of the potential distribution theorems are derived : in terms of the cavity functions, and in terms of the singlet direct correlation functions. We point out the connections to integral equation theories. To determine the behavior of fused dispheres, we examine the successful Wertheim thermodynamic perturbation theory (TPT) at different bond lengths l in light of the cavity functions. For ternary mixtures of spheres S and B, and fused dispheres (SB), we discover a confluence point when all cavity functions at different mixture compositions converge. This takes place at the tangent disphere limit l=d (l being the bond length, and d, the hard sphere diameter). This point is also in common with the excess Helmholtz free energy from the TPT theory for tangent dumbbells. The cavity functions are obtained from the accurate equation of state of Boublik. To verify the chemical potentials calculated, we compare with new Monte Carlo simulations for mixtures of hard spheres and dumbbells. TPT does not hold for l

Keywords:DIRECTIONAL ATTRACTIVE FORCES;INTEGRAL-EQUATION THEORY;THERMODYNAMIC PERTURBATION-THEORY;ASSOCIATING LIQUID-MIXTURES;DENSITY-FUNCTIONAL THEORY;MULTIPLE BONDING SITES;LENNARD-JONES FLUIDS;COMPUTER-SIMULATION;PHASE-EQUILIBRIA;CHAIN MOLECULES