The increasing application of statistical methods in the social sciences has resulted in the consideration of hypotheses more complicated that those for which the theory of univariate statistical processes and the elementary theory of multivariate statistical processes suffice. One cause of the complications may be found in the fact that in the social sciences the results of an experiment are frequently vectors of several components rather than of one component. As examples we may note that the prices and quantities of a given commodity in p localities are vectors of p components, and that mental traits are generally tested by batteries of tests and not by a single test. Although the replacement of these vectors by some function, say an average, of their components is adequate for some purposes, yet in certain situations, any such function may be shown to be unsatisfactory. The fact that vector variates must then be analyzed requires the construction of a statistical theory which will facilitate that analysis. For the normal multivariate distribution, the only distribution for which results have been obtained, the beginnings of that theory may be found in the fairly intensive study, since Gauss, of least squares and multiple correlation, which culminated in the derivation by R. A. Fisher in 1928 of the distribution of the multiple correlation coefficient when the correlation parameter is not zero.t The general theory of vector variates has been developed since 1931. It has included the analysis of variance of a vector variate and the theory of relations between sets of variables. Hotelling derived the distribution of the generalization of Student's ratio? in 1931. Distributions of statistics occurring in the generalized analysis of variance were derived by Wilksll in 1932. In 1936 Hotelling stated and solved a general problem in the theory of relations between sets of variables.? We continue this analysis.