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The fully stressed column with self-weight and supporting a top load is found. The equilibrium solution is unique, and the cross-sectional area tapers exponentially toward the top. Such an optimum column will be important in column design. The vibration frequency is obtained by first transforming the governing equation and then using an efficient initial-value numerical method. The problem depends on two non-dimensional parameters: One represents the column density, and one represents the product of rigidity and top load. Tables for the free top case and the sliding top case are presented.