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This article extends a method of antenna design described in an earlier article by the same author. A family of continuous circular aperture distributions is developed in such a way as to involve only two independent parameters, , a quantity uniquely related to the design sidelobe level and a number controlling the degree of uniformity of the sidelobes. An asymptotic approach to the condition of uniform sidelobes thus becomes possible. A companion article by Robert Hansen contains aperture distribution tables and examples.