SUMMARY Consider a finite set of points, located on the circumference of a circle. Several tests have been proposed of the hypothesis that the points constitute a random sample from a uniform distribution. In this paper we study a test statistic defined as the maximal number of points that can be covered by some semicircle. Exact and asymptotic distributions under the null hypothesis, and under a certain alternative hypothesis, are given together with some tables. A related test statistic is studied briefly. An expression is obtained concerning most powerful invariant tests of the hypothesis of a uniform circular distribution. In 1965, Dr G. Borenius described an unpublished experiment with a bubble chamber, where points representing events were observed through a circular window. A natural hypothesis was that the events occurred at random with a constant probability density within the circle. In one case it was observed that 67 out of 100 events fell within a suitably chosen semicircle. The question then arose whether this asymmetry should be judged inconsistent with the hypothesis of a uniform distribution. More generally, suppose that n points are observed and that each point is moved radially to the circumferences of the circle. We then have a sample of n points on the circumference and want to test the hypothesis that the underlying probability distribution is uniform over the circumference. The test statistic suggested by the foregoing paragraph is the maximal number of points in the sample that can be covered by a suitably chosen semicircle. In the following this test statistic will be denoted by N. We reject the hypothesis if N is too large. Many other tests of the same hypothesis have been proposed. For example, the classical Kolmogorov-Smirnov test has been adapted to circular distributions by Kuiper; see Kuiper (1960) and Stephens (1965). Watson (1961) did the same thing for the Crame6r-von Mises test. A detailed study of the null distribution of Watson's test statistic has been made by Stephens (1963, 1964). For a general review of statistical methods in connexion with circular distributions, see Batschelet (1965). The problem of determining the distribution of N under the hypothesis is purely combinatorial. It was solved by Borenius for sample sizes n up to n = 15 and the general solution was inductively conjectured by him. He thus found that the above-mentioned observation, N = 67 for a sample of size n = 100, corresponds to a level of significance P = 1U6 0/. Dr S. Johansen, University of Copenhagen, has told the author that he, too, has found the null-distribution of N. This was done in connexion with an application to the study of