Multivariate analysis of variance tests are developed for situations where the underlying covariance structure is uniform (equal variances and covariances) in terms of statistics analogous to Hotelling'sT2andT20. Extensions are made to several populations as well as to blocks of uniform covariance matrices. A special case, which is typical of the test procedures given here, is the problem of testing whether the mean vector of a bivariate normal distribution is equal to some specified vector based onnobservations. The uniform structure assumes that the two unknown variances are equal though the correlation is arbitrary. The testing procedure leads to a statisticUwhich is distributed as the sum of two independentF1,n–1ratios which may be contrasted with theT2statistic proportional toF2,n–2used in the situation where the variances are not assumed equal.