By extension of the rotational process, meaningful orthogonally related positions were found for all of the thirteen centroid factors which Thurstone extracted from his original PMA intercorrelations. Most of the original primary ability factors were more sharply delineated and corresponded more closely to the Army Air Force factors that bear similar names (demonstrating greater invariance from analysis to analysis). While such different results obtained by two investigators applying the same methods on the same data may initiate some concern, the results strengthen rather than weaken the idea that more psychological meaningfulness and greater invariance will result if centroid axes are rotated, using the concepts of a simple structure and positive manifold.