This report is a technical supplement to the University of North Carolina, Institute of Statistics, Mimeo Series Report No. 345, which dealt with the specification of indifference distributions for a wide range of statistical models and within the context of a Bayesian logical probability model. In this paper a more refined approach to indifference procedures is presented and applied to Bernoulli and Poisson models. It is also shown that the proposed procedure yields invariant specifications under certain transformations of the scale of measurement of the random variable. The null distribution of the correlation coefficient corresponding to the structuring of a regression model is presented.