A thermodynamically consistent representation, in terms of the equation of state, is obtained for the coefficients of the Wertheim-Till direct correlation functions for a system of solid spheres modeling the microstructure of a simple liquid. It is shown that use of the equation of state in Karnahan - Sterling form significantly improves the results for the radial function, giving good agreement with the results of the semiempirical Verlet-Weis approximation for the data of a numerical experiment on a system of solid spheres. Optimization of the sphere diameter using the entropic method proposed earlier is shown to ensure agreement of the nodes of the radial function of the liquid with experimental data. The agreement of the method proposed with other known methods for the calculation and optimization of the radial function of solid spheres is discussed, and its advantages are noted. © 1978 Plenum Publishing Corporation.