  After a brief review of the kinematics of deep inelastic scattering (DIS)
within the so-called $\Sigma$ method, we derive the necessary formulae for the
treatment of QED radiative corrections to DIS originating from hard photon
radiation. The results are applied to a calculation of the corrections to DIS
with a tagged photon with next-to-leading logarithmic accuracy under HERA
conditions. It turns out that the next-to-leading order corrections are quite
important for the $\Sigma$ method. We also discuss the dependence of the
corrections on the longitudinal structure function of the proton, $F_L$, in the
region of low $Q^2$ and moderate $x$.
