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Waterman's -matrix formulation of classical electromagnetic scattering from a single, homogeneous scatterer has been extended by Peterson and Ström to include scattering from multilayered scatterers. The -matrix refers to expansions in spherical wave solutions of the vector Helmholtz equation. This theory and the computational method for calculating backscattering cross sections for axisymmetric scatterers are developed. This approach, using the equivalence principle, yields a more concise and systematic development to the -matrix formulation than that of Peterson and Ström which relies upon the conceptually similar Poincare-Huygens principle. The method is realistically applicable to scatterers of sizes up to and for a wide range of dielectric constants. Sample computations are compared with extended Mie theory calculations of scattering by concentric shells of varying size and with measured backscattering cross section obtained from a displaced spherical perfect conductor imbedded in a lossy, spherical dielectric.