Kaiser’s iterative algorithm for the varimax rotation fails when (a) there is a substantial cluster of test vectors near the middle of each bounding hyperplane, leading to non-bounding hyperplanes more heavily overdetermined than those at the boundaries of the configuration of test vectors, and/or (b) there are appreciably more than m (m factors) tests whose loadings on one of the factors of the initial F- matrix, usually the first, are near-zero, leading to overdetermination of the hyperplane orthogonal to this initial F- axis before rotation. These difficulties are overcome by weighting the test vectors, giving maximum weights to those likely to be near the primary axes, intermediate weights to those likely to be near hyperplanes but not near primary axes, and near-zero weights to those almost collinear with or almost orthogonal to the first initial F- axis. Applications to the Promax rotation are discussed, and it is shown that these procedures solve Thurstone’s hitherto intractable “invariant” box problem as well as other more common problems based on real data.