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The problem considered is that of finding the best linear transformation to reduce a random-data vector to a vector of smaller dimension. It is assumed that the original data are Gaussian under either of two hypotheses, and that one wishes to use the transformed data to distinguish the hypotheses. The Bhattacharya distance is used to measure the information carried by the transformed data. A compromise solution is obtained for the case in which the data have both different means and different covariances under the alternative hypotheses.