Invariant tests for uniformity on the vertices of a regular polygon

Beran (1968) and Watson (1974) indicated how to construct optimal invariant tests for local and distant alternatives for distributions on the vertices of regular polygons. Here we give the details of tests which are locally optimal against a linear trend, a single and a double peak in the alternative probabilities. These tests are useful for testing the equality of multinomial probabilities when the labels on the cells can be cyclically permuted without changing the problem e.g. the number of births in the 24 hours of the day since the time origin is biologically arbitrary.