  For $\Gamma$ a cofinite Fuchsian group, and $l$ a fixed closed geodesic, we
study the asymptotics of the number of those images of $l$ that have a
prescribed orientation and distance from $l$ less than or equal to $X$. We give
a new concrete proof of the error bound $O(X^{2/3})$ that appears in the works
of Good and Hejhal. Furthermore, we prove a new bound $O(X^{1/2}\log{X})$ for
the mean square of the error. For particular arithmetic groups, we provide
interpretations in terms of correlation sums of the number of ideals of norm at
most $X$ in associated number fields, generalizing previous examples due to
Hejhal.
