The paper discusses the U-shaped relationship between the equivalence scale ne and the Gini index instead of considering the equivalence scale's relationship to the generalised entropy measures, which was studied by Coulter, et al. (1992). An end-point condition is given for the U-shaped relationship, which corresponds to a condition for that of the generalised entropy measures. Additionally, using a mixture of log-normal distributions approach, five factors are shown to be required for a convex relationship between size elasticity e and the Gini index. Empirically, income distributions satisfy those factors. Thus, the end-point condition essentially determines the shape of the relationship.