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A method of calculating the electromagnetic scattering from and internal field distribution of inhomogeneous dielectric bodies of revolution (BOR) is presented. The method uses a typical mode-by-mode solution scheme. The electric flux density is chosen as the unknown quantity, which, together with the special construction of basis and testing functions, enables considerable reduction of the number of unknowns. A key element in this technique is expressing of the azimuthal field components of basis functions in terms of transverse components. A Galerkin testing procedure is used, with special attention put on the efficiency of calculating scalar potential term. Results of calculation for a few classes of dielectric bodies are given and compared with calculations done by other authors.