  It has been recently realized (Rephaeli 1995) that the relativistic
corrections to the spectral distortions of the cosmic microwave background
(CMB) measured in the direction of clusters of galaxies containing hot gas are
significant and should be detectable with the forthcoming experiments. In the
present paper we calculate the correction terms that are proportional to
$V_{r}/c\times kT_e/m_ec^2$ and $(V/c)^2$ to the standard formulae describing
the spectral distortions caused by the bulk motion of the free electrons
(kinematic effect) and due to the presence of the hot gas (thermal effect) for
the case of a cluster having a peculiar velocity $V$ ($V_{r}$ is its radial
component). The results of our analytical calculations are confirmed by
Monte-Carlo simulations (Sazonov & Sunyaev 1998).
