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In general, the beam patterns of a multiple-beam antenna system may overlap. The degree of overlap is usefully defined by a cross-correlation type of integral which includes phase factors as well as polar diagrams in the description of the beams. It is shown that when beam overlap exists, conservation of energy implies the unavoidable existence of cross couplings between the feed lines, and a related limitation on the radiation efficiency (or correspending receiving cross section) of any single beam. Quantitative relationships are derived describing these limitations, and examples given. The implied character of the scattering matrix, comprised of the array of cross-coupling factors, is also discussed with especial emphasis on lossless, reciprocal systems.