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Tests of randomness of directions in three-dimensional space or equivalently tests of uni-form distribution of points on the unit sphere are treated. One test is against alternatives, which concentrate probability density near an equator, and the other is against alterna-tives which concentrate probability density near opposite poles; in each case the poles are unspecified. The tests are based on the latent roots of the matrix of sums of squares and cross-products of the co-ordinates of the observed points on the unit sphere. Against equa-torial alternatives the null hypothesis is rejected if the smallest root is less than the appropri-ate significance point, and against bimodal alternatives the null hypothesis is rejected if the largest root is greater than the appropriate significance point. Tables of significance points are given, based on Monte-Carlo studies and the asymptotic distributions which are derived. The two-dimensional problem is also discussed