In a previous paper (1), ~ reasons were given why the usual graphical analysis of the distribution of particle sizes should be replaced by an algebraic analysis. In the present paper, this method is developed and explained. Most of the mathematical argument has been removed from the text and is giveri in several appendices. The method of analysis described is comparatively new in that it uses the Chi-Square Minimum Principle instead of the more familiar principle of least squares.
By the results of this analysis, it is possible to characterize the size distribution in a photographic emulsion, in the simpler cases, by two numbers (parameters): The first, denoted by a, which is related to the time of growth of the crystals, and the second, denoted by b, which is inversely proportional to the velocity constant of the growth. The first parameter is subject to statistical fluctuations within an individual emulsion; the second should be constant for one type of emulsion as long as the physicochemical conditions of growth are not altered.
This theory is confirmed by careful analysis of one sample of an emulsion, eight subsamples of which were measured by the observer. The results of the statistical analysis show that the parameter, b, is the same for the eight subsamples within the limits of the experimental error. The values of the parameter, a, show, however, differences in excess of those limits. This is due to the local fluctuations of the time of growth throughout the emulsion, mentioned above. Statistically, there is, therefore, heterogeneity in one of the population parameters. Therefore, the results from the eight subsamples should not be added together. This is usually done with the intention to increase the goodness of the fit, but actually decreases it.