The external wave drag of bodies of revolution moving at supersonic speeds can be expressed either in terms of the geometry of the body, or in terms of the body-simulating axial source distribution. For purposes of deriving optimum bodies under various given condtions, it is found that the second of the methods mentioned is the more tractable. By use of a quasi-cylindrical theory, that is, the boundary conditions are applied on the surface of a cylinder rather than on the body itself, the variational problems of the optimum bodies having prescribed volume or caliber are solved. The streamwise variations of cross-section area and drags of the bodies are exhibited, and some numerical results are given. The solutions are found to depend upon a single parameter involving Mach number and radius-lenght ration of the given cylinder. Variation of this parameter from zero to infinity gives the spectrum of optimum bodies with the given condition from the slender-body result of the two-dimensional. The numerical results show that for increasing values of the parameter, the optimum shapes quickly approach the two-dimensional.