By means of a general theorem, the space of the variables of classification is separated into population regions such that the probability of a correct classification is maximized. The theorem holds for any number of populations and variables but requires a knowledge of poppulation parameters and probabilities. A second theorem yields a large sample criterion for determining an optimum set of estimates for the unknown parameters. The two theorems combine to yield a large sample solution to the problem of how best to discriminate between two or more populations.