A rational theory of discrimination learning is developed for the special case in which the subject must discriminate between two stimuli which differ with respect to one variable such as size or brightness. It is shown that the previous equations developed by Gulliksen and Thurstone are special cases of the present one. It is predicted that the ultimate level of accuracy of the discrimination is inversely related to the difference, as determined psychophysically, between the two stimuli. Other implications of the theory for experimental work are presented.

A correlation matrix may be expanded as the weighted sum of a series of ‘unit hierarchies’. The properties of the ‘unit hierarchy’ are not only of theoretical interest for themselves, but lead to simpler modes of practical calculation. The analysis is analogous to a spectral set of projective operations in quantum-theory: and the analogy itself suggests many further problems and solutions.

The theory of error elimination developed by N. Rashevsky is extended and generalized. Theoretical conclusions obtained are compared with the experimental data available.

Certain matrix algebra, pertinent to multiple factor theory, is presented.

The problem of solving simultaneous linear equations is one that frequently confronts the scientist. Whenever it is desired to state the parameters as expressions of the constant terms of these equations, the problem resolves itself into the discovery of the inverse of the matrix of coefficients. The method here described gives a routine for the solution of this problem.