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A study is made of transmission by radomes, with particular reference to cross polarization, and of the consequent radiation patterns of the radome with its aerial for both linear and circular polarizations. The work is embodied in a computer program which considers a scanner of specifiable size, position, and power distribution at different orientations inside a multilayer radome of given dimensions, class of shape, and construction in terms of the number and properties of its layers. Initially, rays are traced from points on the scanner, and details of their paths and propagation are presented in tables of preliminary results; later, diffraction theory is used. Polar diagrams of cross polarization, of main beam, etc., are presented in final tables for the system. Nearly all the parameters are specifiable, and so the program has reasonably general applicability, and it can also assess the effects of bandwidth, tapering of layer thickness, complex permittivity, and other parameters of the system.