Polarography with the electrolyte dropping electrode is used to study the transfer of several non-adsorbing ions including Na+, H+, Cl-, tetrabutylammonium(+) (TBA(+)) and tetraethylammonium(+) (TEA(+)) across the water/1,2-dichloroethane interface. The transfer of Na+, H+, Cl- and TBA(+) in the thermodynamically favored direction is accompanied by the interfacial instability, which is manifested by the substantial faradaic current amplification (polarographic current maximum). No such amplification is observed in case of the transfer of TEA+ in either direction. The potential range of the interfacial instability, as well as of its absence, is correctly predicted by the linear instability analysis developed by Aogaki et al. (Electrochim. Acta 23 (1978) 867). The analysis also provides an explanation for the experimentally observed decrease of the interfacial tension in the potential range of the instability. Two remarkable effects of the interfacial instability are demonstrated, specifically the irregularly repeated swapping of the faradaic current and the interfacial tension occurring close to the electrocapillary maximum between the amplified current/low interfacial tension and diffusion current/high interfacial tension levels, and the mechanical oscillation of the electrolyte drop in the former state.