  Macbeath gave a formula for the number of fixed points for each non-identity
element of a cyclic group of automorphisms of a compact Riemann surface in
terms of the universal covering transformation group of the cyclic group. We
observe that this formula generalizes to determine the fixed-point set of each
non-identity element of a cyclic group of automorphisms acting on a closed
non-orientable surface with one exception; namely, when this element has order
2. In this case the fixed-point set may have simple closed curves (called
ovals) as well as fixed points. In this note we extend Macbeath's results to
include the number of ovals and also determine whether they are twisted or not.
