  In a general framework that has been labeled the ``gauging of equations
method'', we study the diagrams that contribute to Compton scattering off a
relativistic composite system. These contributions can be derived for
$N$--particle bound states described by the covariant Bethe--Salpeter equation
with a method equivalent to minimal substitution in the one--particle case and
yield the correct contributions (including subtraction terms) in the order
${\cal O}(e^2)$. We give the Ward--Takahashi identities for the general
two--photon vertex as well as the corresponding constraints for the two--photon
irreducible interaction kernel and the Bethe--Salpeter amplitude describing the
bound state. From this we can show that gauge invariance holds for the full
two--photon vertex. We furthermore study in detail the low--energy limit of the
Compton scattering tensor in this approach (including a discussion of the pole
terms) and can prove that the full amplitude yields the correct Born--Thomson
limit as we shall explicitly show for the spin--0 case.
  The calculations are completed by the investigation of certain approximations
that can be formulated for arbitrary $N$--particle bound states. We neglect for
instance contributions from $n$--photon irreducible interaction kernels and
show that in this case gauge invariance is only realized if either the
interaction kernel in the Bethe--Salpeter equation is independent of the total
momentum and additionally is of local type, or if the photon energies vanish;
furthermore, we find the correct low--energy limit in this approximation. To
clarify our approach, we also give the results in the order ${\cal O} (e)$; as
examples, we will quote some resulting lowest order expressions for a $q\bar q$
system explicitly.
