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The backscattering solutions for a metallic cylinder and a metallic sphere coated with single and double layers of absorbing material are investigated. In each case, attention is confined to the high-frequency specular return. The formal series solution obeying exact boundary conditions is expanded asymptotically into a Luneberg-Kline series whose first term, geometric optics, would be obtained by using an impedance boundary condition. The second term may be viewed as a correction term which depends upon the coating characteristics and radii of curvature of the body; in the case of the cylinder, its form for the TE polarization component differs from that for the TM polarization component. No practical conditions can be imposed upon the absorbing layers to cause cancellation of second order terms simultaneously for both polarization components. However, it is possible to adjust the flat-plate reflection coefficient of the absorber coat to cancel second order terms for one polarization component. The reflection coefficient will then depend ,upon curvature; for example, the voltage reflection coefficient must be i/(SkQ) for polarization parallel to a cylinder with a thin magnetic absorber coat. Comparison of the sphere solution with the cylinder solution yields information concerning absorbers on a general convex body.