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Antennas composed of discrete elements equally spaced in angle around a circle or circular cylinder are studied with the objective of designing such antennas to produce required azimuthal radiation patterns. Much has already been written upon this subject under the assumption that a continuous distribution of elementary sources will be an acceptable solution to the design problem or at least will form a step in the attainment of an acceptable solution. In the present writing, however, it is felt that something may be gained by analyzing the problem from the beginning on the basis of discrete elements. The question of how many elements are needed is discussed in detail and it is shown that the envelope of the excitation coefficients is not necessarily equivalent to the continuous solution available by other methods. Practical procedures for finding the envelope of the excitation coefficients, and hence the coefficients themselves, are outlined.