Starting with the Feynman-Dyson $S$ matrix, we have computed the two leading terms in the high-energy expansion of the cross section for multiple photon production by electron pair annihilation in flight. In the extreme relativistic limit, when $\mathrm{ln}(\frac{{E}_{+}}{m})\ensuremath{\gg}1$, where ${E}_{+}$ is the positron energy in the laboratory system, our result reduces to that previously obtained by Gupta.We also calculated the first four terms in the high-energy expansion of the three-photon production cross section. Gupta obtained the leading term only and found a discrepancy between ${\ensuremath{\sigma}}_{3}$ and ${\ensuremath{\sigma}}_{n}$ for $n=3$ by a factor $\frac{3}{2}$. This discrepancy is removed.For the sake of simplicity, Gupta assumed that the principal contributions to the cross section come from (1) low-energy photons and (2) small scattering angles. We proved that all these assumptions are indeed valid.The possibility of observing multiple photon production is discussed.