Charm Production in Deep-Inelastic $e\gamma$ Scattering To Next-To-Leading Order in QCD

We discuss the calculation of $F_2^{\gamma}({\rm charm})$ to next-to-leading order (NLO) in QCD, including contributions from both hadronlike and pointlike photons. We show that the former dominates strongly below $x\simeq 0.01$, and the latter above this value. This fact makes $F_2^{\gamma}({\rm charm})$ for $x \geq 0.01$ calculable, whereas for $x \leq 0.01$ it serves to constrain the small-$x$ gluon density in the photon. Both ranges in $x$ are accessible at LEP2. Theoretical uncertainties are well under control. We present rates for single-tag events for the process for $e^+e^- \rightarrow e^+e^- c X$ for LEP2. Although these event rates are small, we believe a measurement of $F_2^{\gamma}({\rm charm})$ is feasible.