We have revisited the problem of the beating mercury systems for acidic and basic medium. The results are analyzed in terms of electrochemical process and direct conduction of electrons between nanocontacts (contacts of few atoms size) formed by Fe and W with Hg in the acidic and basic medium, respectively. It is observed that, in some cases, the voltage(current) does not drop abruptly during the contact process (direct conduction), but in the steps that are related to the size of the nanocontacts. The phenomenon is similar to the one observed in the nanocontacts formed by metallic wires in which the conductance is quantized. Also, the analysis takes into account the completion electrochemical formation, of the film on the mercury ball in the acidic medium during the electrochemical process. The agreement between theory and experiment is very good when the nanocontacts conductance and completion of the film is added to previous existing theory.