Suppose that in a given non-negative definite symmetric matrix R = [rij] of order n the diagonal entries are replaced by arbitrary quantities x1, x2,…, xn so that the matrix assumes the form

Matrices of this type are met with in the statistical technique known as Factorial Analysis (Thurstone, 1935; Thomson, 1939); there the nondiagonal elements rij(i≠j) are the correlation coefficients of certain tests and are given by observation. The diagonal entries of the “correlational matrix,” which is always non-negative definite, are originally all equal to unity, but, on the hypothesis which underlies the process of Factorial Analysis, it is permissible to diminish the diagonal entries arbitrarily provided the modified matrix is still non-negative definite, it being assumed that the amount which has been deducted from the diagonal cells is due to the variance of the specific factors, the investigation of which is not the primary aim of the theory.

In the past two methods have been applied to the study of differential fertility. One may be called the absolute method, recently applied to Scottish data by the writer. The other may be called the comparative method, subsequently applied to the same statistics by Mr Barclay and Dr Kermack. Since the conclusions obtained by the two methods are not always consonant, it is important to evaluate the scope and reliability of each.

The pioneer work of Stockard and Papanicolaou (1917, 1919) forms the basis of most subsequent observations on the œstrous cycle of the guinea-pig, and their results are, in the main, accepted by later workers—Ishii (1920), Selle (1922), Nicol (1932), etc.

It is well known, and fundamentally important in the justification of the current mode of deriving Student's t-distribution, that the mean m of n values xi all obeying the same normal law of probability, and the associated estimate of variance, namely

are statistically independent.

Any solution of the Lamé equation

in which sn u has the modulus k, may be termed a Lamé function, but this paper is devoted mainly to functions of real periods 2K or 4K. We suppose that k2 ≤ 1; the number n is real, but not necessarily an integer, and it is sufficient to take n ≥ − ½. These periodic solutions exist for certain characteristic values of h; a table of such values is included in this paper.

1. When a battery of intelligence tests is administered to a set of persons it is a common practice among psychologists to “explain” the scores obtained in terms of a number of “factors.” Thus if we suppose that there are altogether t tests and that xi, denotes the score of any one person in the ith test, then it is assumed that

where f, g, …, h represent the person's measures in one or more general or group factors, and Si, is the person's specific ability in the ith test. It is further assumed that for a given and supposed infinite population of persons all the factors, specific and otherwise, are distributed normally and independently and that they are standardised, i.e. that their standard deviations are unity.

Hitherto the Lamé functions of u, of real periods 2K and 4K, and the Lamé polynomials in particular, have been expressed as series of integral powers of sn u (multiplied, in appropriate cases, by cn u, dn u, or cn u dn u). The necessity to use these, or the corresponding Weierstrassian forms, is a serious handicap to research, because the terms of the expression are not mutually orthogonal. In particular, results that can be deduced from the orthogonal properties of the Fourier series in which periodic Mathieu functions are developed, find no analogues in the Lamé functions as expressed in the customary forms. For instance, the integral equations that generate the Mathieu functions have proved to be valuable sources of information, but those for the Lamé functions cannot be much more than scientific curiosities until the development of Lamé functions as series of orthogonal functions enables them to be put to use.

This paper contains an attempt to formulate more rigorously the postulate of reciprocity (Born, 1939) and to solve the problem of finding reciprocal wave functions. We have not yet completely succeeded in this task, though considerable progress has been made; but it turns out that the original discussion has to be modified in several important respects.

In this paper we consider a new class of Riemannian spaces which arise in the theory of the solution of the tensor generalisation of Laplace's equation ∇2V = o. To obtain this generalisation Beltrami's second differential parameter is defined in terms of the metric

of the associated n-dimensional Riemannian space by the usual formulæ

where denotes the Christoffel symbol . The generalised Laplace's equation is then Δ2V = o. For simplicity the quadratic differential form (1.1) is taken to be positive definite, which involves no essential loss of generality.