1. Introduction. This note is concerned with testing the hypothesis that a sample from a normal n-variate population is in fact from a population for which the variances are all equal and the correlations are all zero. A population having this symmetry will be called" spherical." Under a linear orthogonal transformation of variates, a spherical population remains spherical, and consequently the features of a sample which furnish information relevant to this hypothesis must be invariant under such transformations. A situation for which this test is indicated arises when the sample consists of N n-dimensional vectors, for which the variates are the n components along coordinate axes known to be mutually perpendicular, but having an orientation which is, a priori at least, quite arbitrary. A specific application for two dimensions, treated elsewhere [1], may be mentioned. Each of N days furnishes a sine and a cosine Fourier coefficient for a given periodicity, and these, when plotted as ordinate and abcissa, yield a somewhat elliptical cloud of N points. The sine and cosine functions are orthogonal, and their variances have