Several theorems concerning properties of the communaltiy of a test in the Thurstone multiple factor theory are established. The following theorems are applicable to a battery of n tests which are describable in terms of r common factors, with orthogonal reference vectors.

1. The communality of a test j is equal to the square of the multiple correlation of test j with the r reference vectors.

2. The communality of a test j is equal to the square of the multiple correlation of test j with the r reference vectors and the n—1 remaining tests.

Corollary: The square of the multiple correlation of a test j with the n—1 remaining tests is equal to or less than the communality of test j. It cannot exceed the communality.

3. The square of the multiple correlation of a test j with the n—1 remaining tests equals the communality of test j if the group of tests contains r statistically independent ests teach with a communality of unity.

4. With correlation coefficients corrected for attenuation, when the number of tests increases indefinitely while the rank of the correlational matrix remains unchanged, the communality of a test j equals the square of the multiple correlation of test j with the n—1 remaining tests.

5. With raw correlation coefficients, it is shown in a special case that the square of the multiple correlation of a test j with the n—1 remaining tests approaches the communality of test j as a limit when the number of tests increases indefinitely while the rank of correlational matrix remains the same. This has not yet been proved for the general case.