  Every closed hyperbolic geodesic $\gamma$ on the triply--punctured sphere $M
=\widehat{{\mathbb C}} - \{0,1,\infty\}$ has a self--intersection number
$I(\gamma) \ge 1$ and a combinatorial length $L(\gamma) \ge 2$, the latter
defined by the number of times $\gamma$ passes through the upper halfplane.
  In this paper we show that $\delta(\gamma) = I(\gamma) - L(\gamma) \ge -1$
for all closed geodesics; and that for each fixed $\delta$, the number of
geodesics with invariants $(\delta,L)$ is given exactly by a quadratic
polynomial $p_\delta(L)$ for all $L \ge 4 + \delta$.
