If each of k samples is drawn from some bivariate normal population, criteria are derived for testing the 3 hypotheses: H (that the samples all come from the same population) ; H 1 (that the samples come from populations with the same set of variances and correlations but having means with any value whatever); H 2 (that, assuming the population variances and correlations are the same, the population means are the same). The moments and distributions of these criteria are obtained. In many problems, hypotheses of the H 1 type need to be tested, as well as those of the 2 type. The technique of Analysis of Variance seems unsuited to deal with the former for k > 2. In the multivariate problem one could deal with the variation of each character and the correlation of each pair of characters separately but the application in the first instance of a single comprehensive test has several advantages.