The extension of the Teorell-Meyer-Sievers theory of the dialysis potential to a general theory of capillary systems with narrow pores outlined in Part I of this series of publications has been applied to electroosmotic phenomena in Part II. In this Part, the electrical conductivity, including the electrical convection conductivity, will be treated in terms of the new theory. The corresponding equations already referred to in Part I are derived. In addition, results of measurements of the electrical conductivity of collodion membranes with graded porosity and graded electrochemical activity in aqueous KCl solutions of different concentrations are reported. They are used to test the new theory. It will be shown that it is possible to determine the fixed ion concentration A of the membranes by using electrical conductivity data. The theory predicts that the value of A should be identical with the ＇selectivity constant＇ of the Meyer-Sievers theory of the dialysis potential. This prediction will be checked in Part IV of this series of contributions.