Exact and approximate tests for directions. I

where 01 is the angle between an observed vector and a fixed polar vector. K is a parameter whose value determines the spread of the vectors: when K = 0 the distribution is uniform round the circle, and when K is large the distribution has very small dispersion. IO(K) is the imaginary Bessel function of order zero. The distribution has been tabulated by Gumbel, Greenwood & Durand (1953). When K = 0, the problem has been examined by many writers, including Kluyver (1906), Rayleigh (1919), and, more recently, Greenwood & Durand (1955) and Durand & Greenwood (1957), who give important results. The corresponding probability density in three dimensions is given by