This paper investigates optimal linear discrimination functions for populations whose members are characterized by n-dimensional feature vectors. The solution of this problem is well known if the probability distributions are Gaussian. The procedure proposed in this paper is related to the perceptron method. It yields optimal discrimination for the larger class of symmetric distributions, basically density functions that are symmetric relative to all one-dimensional projections and are the same for both populations, except for translations. The paper provides detailed proofs but no numerical examples.