Onsager made corrections to the Poisson-Boltzmann equation, to more correctly describe interactions in asymmetric electrolytes (those which have different charges, in absolute value, for the cation and the anion). The equations established by Onsager allow the calculation of symmetric total correlation functions, such that h ij (r) = h ji (r). In this article it has been shown that these correlation functions, describing in particular asymmetric electrolytes, can also be deduced from the Ornstein Zernicke integral equations by considering a nonlinear closure relation of the HNC type. The excess free energy deduced from these correlation functions has been found similar to that obtained in Mayer's theory. Relations with the electrostatic association models on the one hand, and with the dressed ions theory on the other hand, are recovered.