Papers involving factorial methods appearing in current literature often misuse factorial methods. These meaningless results are reached because of neglect of certain conditions basic to factor theory. The conditions are:

1. (1) The number of basic factors must be smaller than the number of tests.

2. (2) The diagonals of the correlation matrix must be regarded as unknown.

3. (3) The axes must be rotated into a simple configuration.

4. (4) Each factor must be overdetermined by appearance in several tests.

5. (5) Tests should have simple factorial composition.

A table to facilitate the computation of Fisher's t function for the comparison of two means is presented. The derivations leading to the construction of the table, and the necessary instructions for its use, are given.

Since the factor problem is reduced mathematically to the expression of the obtained correlational matrix in terms of a matrix of lower rank, criteria for the determination of this lower rank are of first importance. These criteria are investigated by means of certain mathematical deductions, and brought into relation with Spearman's and Thurstone's factor theories.

When several criteria are available, it is ordinarily necessary (1) to select one of them as the criterion, (2) to use several and thus arrive at several different sets of weights, or (3) to combine them into a single measure. A formula is derived for the determination of a unique set of weights. The use of these weights will produce the highest possible average coefficient of correlation between the various criteria and two (or more) weighted independent variables. If desired, the criteria may be assigned any predetermined weights. The weights then derived for the independent variables are such that the weighted average of the correlation coefficients between the various criteria and the independent variable composite will be a maximum. In the use of these formulas, no assumptions are necessary regarding the interrelationships existing among the criteria and it is not necessary to compute the intercorrelations among the criteria. A numerical example is included.

Offner's demonstration of the excitation time equivalence, for all currents, of the theories of Hill and Rashevsky is here extended to a more general case which includes both. It is found that any system of this general type may be replaced by an equivalent one of the Rashevsky type, thus effecting considerable simplification in the mathematical detail.

From Thurstone's theoretical learning curve, a solution for the difficulty of the problem and the learning constant of the subject has been developed. The curve is an equilateral hyperbola. Therefore the semi-major axis represents the learning situation in one constant. The vertex of the curve is a point where all of the animals are equated, since they are all making errors at the rate of one error per trial.

By the use of the Seashore tests of Pitch Discrimination, Intensity Discrimination, Time Discrimination, and the test of Tonal Memory, it is shown that the easiness of an item, as determined by absolute scaling methods, is proportional to the logarithm of the magnitude of the stimulus. It is proposed that this is a case of Fechner's psychophysical law and that the unit of absolute scaling as applied to test items may become a satisfactory unit of all S-scales in the more traditional psychophysical problem.

A nomograph for the computation of point biserial r, biserial r, and fourfold correlation coefficients is presented, with instructions for use.