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A circular cylindrical dielectric layer is an idealized but rigorously analyzable model for radome covers. With a line source located on the concave side, an exact integral formulation is derived for the field transmitted to the convex side. Alternative representations are developed therefrom in terms of discrete guided modes and continuous spectra, and of ray integrals which, asymptotically at high frequencies, yield geometric optical fields that experience multiple internal reflections between the layer boundaries and also multiple reflections on the concave side. It is then shown that the higher order multiple reflected contributions can be expressed collectively as a ray field with a weighted transmission coefficient that is equivalent to the plane wave transmission coefficient for a plane parallel layer but includes a simple curvature correction. When source and observer are close to the inner and outer layer boundaries, respectively, and are also separated by a large angular interval, guided mode effects may have to be included as well. The result is a general and novel representation of the transmitted field in terms of a certain number of ordinary multiple reflected geometric optical ray fields, a single "collective" ray field, which includes in a composite manner all of the remaining internal reflections, and, possibly, the guided modes along the layer.