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Scattering of a plane electromagnetic wave by an infinite circular dielectric cylinder coating eccentrically an elliptic dielectric one, is under consideration. Both E and H polarizations are treated for normal incidence. The electromagnetic field is expressed in terms of both circular and elliptical-cylindrical wave functions. Using proper transformation theorems between the field expressions in different coordinate systems, for the satisfaction of the boundary conditions, we obtain two infinite sets of linear nonhomogeneous equations for the expansion coefficients of the field. In case of small values of h=k2c/2, where c is the interfocal distance of the elliptic cylinder and k2 the wavenumber of the dielectric coating, the former sets of equations provide, by truncation, semianalytical expressions of the form S(h)=S(0)[1+gh2+O(h4)] for the scattered field and the various scattering cross sections. The coefficients g are independent of h. Graphical results for the scattering cross sections are given for various values of the parameters.