  The density matrix for a system of particles interacting via the Coulomb
potential is obtained in the high--temperature limit following almost entirely
the original work by Kelbg. For this purpose the Bl\"och equation is solved in
the first order of perturbation theory. We tried to explain all the
transformations in the derivation in order to simplify the understanding of
this non-trivial theory. The solution of Kelbg is widely used in path integral
simulations of Coulomb systems.
