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A general expression for the cross sections of inelastic collisions of fast(including relativistic) multicharged ions with atoms which is based on thegenelazition of the eikonal approximation is derived. This expression isapplicable for wide range of collision energy and has the standardnonrelativistic limit and in the ultrarelativistic limit coincides with theBaltz's exact solution ~\cite{art13} of the Dirac equation. As an applicationof the obtained result the following processes are calculated: the excitationand ionization cross sections of hydrogenlike atom; the single and doubleexcitation and ionization of heliumlike atom; the multiply ionization of neonand argon atoms; the probability and cross section of K-vacancy production inthe relativistic $U^{92+} - U^{91+}$ collision. The simple analytic formulaefor the cross sections of inelastic collisions and the recurrence relationsbetween the ionization cross sections of different multiplicities are alsoobtained. Comparison of our results with the experimental data and the resultsof other calculations are given.Comment: 25 pages, latex, 7 figures avialable upon request,submitted to PR

Inelastic collisions of relativistic highly charged ions with atoms