The theory of the preceding papers is generalized and the notation simplified. A cluster of molecules in a stable phase surrounded by an unstable phase is itself unstable until a critical size is reached, though for statistical reasons a distribution of such clusters may exist. Beyond the critical size, the cluster tends to grow steadily. The designation ``nuclei'' or ``grains'' is used according as the clusters are below or above the critical size. It is shown that a comprehensive description of the phenomena of phase change may be summarized in Phase Change, Grain Number and Microstructure Formulas or Diagrams, giving, respectively, the transformed volume, grain, and microstructure densities as a function of time, temperature, and other variables. To facilitate the deduction of formulas for these densities the related densities of the ``extended'' grain population are introduced. The extended population is that system of interpenetrating volumes that would obtain if the grains granulated and grew through each other without mutual interference. The extended densities are much more readily derivable from an analysis of the fundamental processes of granulation and growth. It is shown that, under very general circumstances, the densities of the actual grain population may be expressed simply in terms of the extended population.