TO DEVELOP A BETTER MATHEMATICAL BASIS FOR FACTOR ANALYSIS, A MATHEMATICAL APPROACH WAS FORMULATED FOR FIXING COMMUNALITIES AND THEIR COMPLEMENTS, UNIQUENESSES, AS WELL. WHILE THE INVESTIGATOR WAS UNABLE TO PROVE THAT MINIMUM-TRACE CRITERIA FIX COMMUNALITIES UNIQUELY, A SET OF IDENTITIES AND INEQUALITIES WAS LOCATED AMONG THE COMMUNALITIES AND, EQUIVALENTLY, AMONG THE UNIQUENESSES. PROOF AND ADDITIONAL RESULTS CONCERNING THESE IDENTITIES AND INEQUALITIES APPEARED IN AN APPENDIX. THESE IDENTITIES AND INEQUALITIES WERE OF DIRECT SIGNIFICANCE SINCE THEY AFFORDED A METHOD OF OBTAINING AN APPROXIMATE SOLUTION FOR THE MINIMUM-TRACE COMMUNALITIES. NO TIME WAS AVAILABLE DURING THE PERIOD OF THE CONTRACT TO PERFORM THE NUMERICAL CALCULATIONS NECESSARY TO VERIFY THE FINDINGS. (GD)