In the design of multiple input-output logical systems, the authors have previously formulated and proved a computer program for obtaining optimal system representations satisfying three optimality criteria [1]. In that program, the need for obtaining prime implicants occurred. Two algorithms for this purpose are available in the literature; one due to Quine [2] and one due to Nelson [3], [4]. Nelson's, briefly, is to express the function (whose prime implicants are desired) as a product of sums of literals, multiply out, and discard repeated literals save one and proper multiples. (To obtain the function in the initial form, one can start with the complement of the function expressed as a sum of products and complement it using De Morgan's theorem.) In using these algorithms, it became apparent to us that in certain cases one of these two algorithms required less computer time than the other one. In particular, if the given system has a large number of don't cares then the Nelson algorithm seemed to be preferred [5]. Also in the case of a single function, if there are few zeroes in the truth table representation for the function, then Nelson's algorithm is preferable. It also became apparent to us that in certain situations one needs a generalized version of the Nelson algorithm.