  Eleven bright gamma-ray bursts (GRBs) detected by BATSE have also been seen
at much higher energies by EGRET, six at energies above 10 MeV. Such
observations imply that these bursts are optically thin to photon-photon pair
production at all observed energies. For bursts more than about 30pc away,
internal transparency can be achieved only if the source is moving with a
relativistic bulk Lorentz factor $\Gamma\gg 1$, or if the radiation is highly
beamed. Early calculations of $\gamma\gamma\to e^+e^-$ considerations for GRBs
were limited to cases of a beam with opening half-angle $\Thetab\sim 1/\Gamma$,
or expansions of infinitely thin spherical shells. This paper presents our
extension of pair production optical depth calculations in relativistically
expanding sources to more general geometries, including shells of finite
thickness and arbitrary opening angle. The problem is reduced analytically to a
single integral in the broadly applicable case of observing photons along the
axis of the expansion. We find that the minimum bulk Lorentz factor for the
EGRET sources to be optically thin is only moderately dependent on the shell
thickness and virtually independent of its opening solid angle if
$\Thetab\gtrsim 1/\Gamma$. This insensitivity to $\Thetab$ relieves the
commonly-perceived number problem for non-repeating sources at cosmological
distances, i.e. it is not necessary to invoke small $\Thetab$ to effect photon
escape.
