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In many systems for pattern recognition or automatic decision making, decisions are based on the value of a discriminant function, a real-valued function of several observed or measured quantities. The design of such a system requires the selection of a good discriminant function, according to some particular performance criterion. In this paper, the problem of finding the best linear discriminant function for several different performance criteria is presented, and a powerful method of finding such linear discriminant functions is described. The problems to which this method may be applicable are summarized in a theorem; the problems include several involving the performance criteria of Bayes, Fisher, Kullback, and others, and many involving multidimensional probability density functions other than the usual normal functions.