The factors found by the Hotelling method of principal components present the same necessity for rotation as those found by the Thurstone method of multiple factors.

The Thurstone method of multiple factors is here found superior to the Hotelling method of principal components, and is suggested as the method of choice for all problems requiring the determination of elemental components from a group of related variables, since it was found on every point of comparison that, when applied to a correlation matrix of unknown communalities, the Thurstone method was more accurate than the Hotelling method, and also required much less time.

A further study of the perceptual factor, previously isolated in a factor analysis of a battery of fifty-six tests, is made in a manner designed also to determine whether the same seven primaries would be found in a different population of subjects and with another battery of tests. The tests are described, and the results of the analysis are given in detail. Much attention is given to the matter of the orthogonality of primary factors and to their psychological meaningfulness.

A new rotational method, particularly appropriate when a positive manifold can be assumed, is presented in theory and in computational detail. It is applied to a five-dimensional factorial matrix in which a simple configuration is known to exist. The new technique is much more efficient in yielding the simple configuration than are the methods formerly used. Landahl's transformations can be combined with this method to advantage.

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.

When the first of a uniform chain of neurones is excited with a stimulus intensity S1, it is found that, according to the nature of the chain, either the excitation is always transmitted with decreasing intensity, or else the intensity for succeeding neurones of the chain approaches a fixed finite value, characteristic of the chain and independent of S1, whenever S1 exceeds a certain “threshold” value.

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.

An initial transformation for facilitating the analysis of a factor problem is discussed. Such a transformation is used in the rotational procedure developed by L. L. Thurstone in an accompanying article.

Necessary and sufficient conditions are given for a set of numbers to be the mutual distances of a set of real points in Euclidean space, and matrices are found whose ranks determine the dimension of the smallest Euclidean space containing such points. Methods are indicated for determining the configuration of these points, and for approximating to them by points in a space of lower dimensionality.

The theory of discrimination learning developed in Part I* of this paper is extended to more complex learning situations such as those used for differentiating relative from absolute habits. The greater difficulty of learning an absolute discrimination than of learning a relative discrimination is predicted. The theory is also extended to cover cases of transfer. It predicts that under the ordinary training conditions transfer to new stimulus situations will be on a relative basis, and it predicts the type of transfer to be expected under other training conditions. A number of additional established facts of learning and transfer can be deduced from the theory. Methods of testing the adequacy of the theory by new and more crucial experiments are suggested.

A method is developed for extending any type of factor solution to new tests. The theoretical basis for this approximating scheme is thoroughly investigated, and then a simplification in the technique is introduced for practical purposes. An example is presented which illustrates the procedure of extending a factor solution to three new tests simultaneously.

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

When no criterion variable is available, the combination of tests or other variables by the use of multiple correlation is not possible. Three methods of combining variables are described mathematically, and discussed with reference to the linear combination of tests. Iterative computational schemes are outlined and illustrated.

A battery of thirteen tests of motor abilities are analyzed by means of the Thurstone factor methods. Interpretations of the nature of the six factors extracted are offered, with reservations. The high degree of specificity of motor abilities results in very narrow group factors.

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

On the basis of previous mathematico-biophysical studies, a theory of visual perception is outlined. The theory is applied to the aesthetic values of polygonal patterns and found in fair agreement with experimental results.

The abac is presented with instructions for use.

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.

A formula for the stimulus is derived. The essential postulates and assumptions are based on Gestalt theory.