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The theory of the design of optimal cascaded transformer arrangements can be extended to the design of continuous transmission-line tapers. Convenient relationships have been obtained from which the characteristic impedance contour for an optimal transmission-line taper can be found. The performance of the Dolph-Tchebycheff transmission-line taper treated here is optimum in the sense that it has minim reflection coefficient magnitude in the pass band for a specified length of taper, and, likewise, for a specified maximum magnitude reflection coefficient in the pass band, the Dolph-Tchebycheff taper has minimum length. A sample design has been carried out for the purposes of illustration, and its performance has been compared with that of other tapers. In addition, a table of values of a transcendental function used in the design of these tapers is given.