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A new technique for performing the bilinear transformation of polynomials is presented. The technique is both simple to understand as well as efficient in its computer implementation. The key to the method is the way in which the successive derivatives of a particular polynomial are computed. A simple recursion formula is used which can be done either by hand, if desired, or by computer. The order of complexity of the algorithm is found to be O(n2), while storage requirements are O(n), where n is the degree of the polynomial. The new method will handle completely general bilinear transformations. A computer implementation is presented which was found to be satisfactory for both precision and speed.