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Certain exact relationships between the radiation pattern and impedance characteristics of a radiating rectangular waveguide with vanishingly thin walls are developed. In particular, it is shown that the ratio of radiation intensities in certain preferred directions and the power gain in these directions are related in a simple manner to the reflection coefficient inside the waveguide. Although the information obtained is restricted in the sense that it applies to only a few discrete points on the radiation pattern, it is quite general as regards the manner in which the waveguide is broken or perforated to allow radiation. The results are shown to apply equally well if conducting sheets, having arbitrary shapes but lying on specified planes, are present in the vicinity of the radiating waveguide.