Unequal representation in the sub-classes of a classification is not an uncommon situation in the biological sciences where data must often be taken as they are found or where death and disease can upset the most carefully contrived design. Thus, in a survey recently undertaken for the purpose of estimating the mean levels of concentration of the major elements in apple leaves (Mason and Whitfield [1959]) it was required also to apportion the variation among its several sources (year-to-year, orchardto-orchard, tree-to-tree) with a view to making recommendations as to subsequent sampling procedures. The design of the survey, however, was such that in each of three years the same four trees were sampled at each of 30 orchards of which 21 were under a permanent grass cover while the remaining nine were cultivated. For any particular element, then, the data formed a two-way classification, Years X Systems of Management, with an hierarchy of sub-samples within each sub-class. Furthermore, while it was convenient, and not unreasonable in the context, to regard the years as being a random sample, the two systems of management represented a broad classification covering all possible systems and could in no way be thought of as a random selection. The problem thus involved the estimation of variance components in a two-way classification with unequal representation in the sub-classes and when not all effects could be treated as random. Ganguli [1941], and more recently in this journal, Gates and Shiue [1962] and Gower [1962] have considered a similar situation involving a hierarchal classification, while Fairfield-Smith [1951] has considered the two-way classification when all effects are random effects. When one dimension of a two-way classification involves the complete enumeration of a small population of classes, as in the problem described above or, for example, when plants or animals are classified according to sex or by species, such a fully random model is no longer applicable, and one embracing both fixed and random effects must be assumed.