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In this work, the scattering of a plane electromagnetic wave by an infinite-circular dielectric cylinder, containing eccentrically an elliptic metallic one is considered. The electromagnetic field is expressed in terms of both circular and elliptical-cylindrical wave functions, connected with one another by well-known expansion formulas. Translational addition theorems for circular cylindrical wave functions are also used for the satisfaction of the boundary conditions in the dielectric cylinder. When the solution is specialized to small values of h = k1c/2 , with k1 the wavenumber of the dielectric cylinder and c the interfocal distance of the elliptic conductor, semianalytical expressions of the form S(h) = S(0)[1 + gh2 + O(h4)] are obtained for the scattered field and the various scattering cross sections of this configuration. Both polarizations are considered for normal incidence. Numerical results are given for various values of the parameters.