Substitutional equations of the type considered by the late Alfred Young are shown to be intimately related with the theory of idempotents. Any equation LX = o possessing solutions other than X = o is shown to have the same solutions as another equation MX = o, where M is obtained from L by premultiplying the latter by a suitably chosen expression A and where the minimum equation of M is xψ(x) = o, ψ(x) being prime to x. The expression ψ(M) is then idempotent, and it is shown that the most general solution of LX = o is X = ψ(M)Y, where Y is an arbitrary expression. The number of linearly independent solutions of LX = o is X = ψ(M)Y, where Y is an arbitrary expression. The number of linearly independent solutions of LX = o is kn!, where k is the coefficient of the unit permutation in ψ(M) when that expression is expressed in terms of the permutations of the symmetric group Sn.

Corresponding results are obtained for the equation LX = R, and methods are given for solving sets of simultaneous equations of both types.

Studying the interaction of different pure fields, we have been led to some essential modifications of the ideas on which our quantum mechanics of fields is based. We shall explain these here for the example of the interaction of the Maxwell and the Dirac field.

In Part I we showed that a pure field in a given volume Ω can be described by considering the potentials and field components as matrices, not attached to single points in Ω (as the theory of Heisenberg and Pauli), but to the whole volume. Further, we assumed the total energy and momentum to be the product of Ω and the corresponding densities. In Part † we showed that this conception has to be modified; the eigenvalues of the energy and momentum as defined in Part I represent neither the states of single particles nor of a system of particles, but of something intermediate which corresponds to the simple oscillators of Heisenberg-Pauli and which we have called apeirons. The total energy and momentum of the system is a sum over the contributions of an assembly of apeirons. Mathematically the differences of the quantum mechanics of a field from that of a set of mass points (as treated in ordinary quantum mechanics) is the fact that the matrices representing a field are reducible (while those representing co-ordinates of mass points are irreducible); each irreducible submatrix corresponds to an apeiron.

R. Frisch, in a paper (Frisch, 1928) on correlation and scatter in statistical variables, made an extensive use of matrices, and in particular of the moment matrix, as he called it, of a set of variables. The matrices were square arrays, with an equal number of rows and columns. This paper of Frisch pointed the way to an even more extensive use of the algebra of matrices in problems of statistics.

What Frisch called the moment matrix may perhaps be more suitably called, nowadays, the variance matrix of a set or vector of variates, since the moments in question are all variances or covariances. In the present paper, which is illustrative of matrix methods, we explore the familiar ground of linear approximation by Least Squares, making full use of the properties of the variance matrix. We also study the linear transformations that convert crude data into smoothed or graduated values, or into residuals, or into coefficients in a linear representation by chosen functions.

The changes in his description of events brought about by an arbitrary regraduation of an observer's clock are examined, taking the axioms of general relativity as fundamental. It is shown that regraduation does not imply a change from one Riemannian space-time to another but merely a coordinate transformation within space-time. A generalisation of the “dynamical time” of kinematical relativity is a by-product of the investigation.

If is a fixed point of a Riemannian Vn of fundamental tensor gij, and if s is the geodesic distance between it and a variable point (xi), then the Vn has been called centrally harmonic with respect to the base-point if

is a function of s only, and completely harmonic if this holds for every choice of base-point . A flat Vn (gij=δij) is obviously completely harmonic, since for such a space and

1. It was remarked by me a few years ago that temporal regraduations, other than trivial changes of zero and unit, had not so far been considered in General Relativity. An interesting paper by Dr G. C. McVittie has now appeared in which regraduations are examined in certain spherically symmetric space-times. Under the assumptions made by McVittie it is shown that regraduations can exist for some but not all space-times, those for which they can exist being of a very special form which excludes many space-times generally regarded as significant or interesting. In the present paper I take the matter further and discuss the problem with more generality. It will be shown that the existence of non-trivial regraduations depends firstly upon which theory is being assumed for the derivation of the conservation equations There are two alternatives, and regraduations are found to be excluded by one, the “geodesic” theory, but not necessarily by the other, the “equivalence” theory.

This paper has for object the calculation of a ladder network, using trigonometrical functions of real multiples of (– I)¼ which, in many cases, simplify practical formulae. The work was prepared particularly for application to transmission lines, conductors in electrical machines, and isolated cylindrical conductors. The effect of the conjunction of two or more dissimilar networks is considered, leading to a method of assessing the impedance of a conductor of any shape embedded in an open slot cut in highly permeable material.

1. Introduction.—The object of this paper is twofold: firstly, to present a table of the Chebyshev polynomials Cn(x) = 2 cos (n cos−1 ½x) for n = I(I)12 and x = o(o·o2)2, values being exact or to 10 decimals; secondly, to provide a working list of coefficients and formulæ relating to these and allied functions.

Valuable accounts of applications and properties will be found in Van der Pol and Weijers, 1933, in Lanczos, 1938, and in Szego, 1939. Further applications are indicated in the following paper, by J. C. P. Miller, which also suggests methods of reducing the inconvenience caused by the present lack of tables of the allied polynomials Sn(x). It is hoped that suitable tables will be prepared later.

In this paper two computational processes are outlined in which the table of Chebyshev Polynomials Cn(x) = 2 cos (n cos−1 ½x) given in the preceding paper may be used with effect; these processes are (a) interpolation and (b) Fourier synthesis. A brief outline is also given of the idea behind the process of “Economization of Power Series” developed in Lanczos, 1938; this is related to (a). Finally the application of (b) to the calculation of Mathieu functions is considered.

The development of the idea of the chemical element is traced from its early beginnings, and the importance for this development of the Newtonian concept of invariable mass is emphasised. The emergence of the nuclear atom model is outlined, and the discovery of the complex (isotopic) nature of the majority of known chemical elements is described. Nuclear charge (Z) and mass (A) numbers are defined. Previously recognised regularities concerning mass and charge numbers of existing stable species are shown to have exact counterparts in regularities relating to the degree of instability (as measured by the energy of disintegration) of β-active species (“naturally” and “artificially” radioactive species). Naturally occurring a-active species are regarded as the analogues of the stable species for charge numbers greater than 83, and for charge numbers both greater and less than this value the limitation to the number of stable or quasi-stable isotopes of a given element (limitation of A values for a given Z) is established as essentially a question of nuclear stability as against β-emission (positive and negative electron emission). Finally, reasons are given for supposing that the number of possible chemical elements is limited (limitation to Z in the direction of Z increasing) by the susceptibility to spontaneous nuclear fission of species of sufficiently high nuclear charge.

1. Introduction.—One of the most important of Milne's discoveries is undoubtedly the significance of time-scale regraduations and, in particular, the relation between atomic t-time and gravitational τ-time. Although in Milne's work t-time is more fundamental than τ-time, this relationship is not inevitable, as was shown in an axiomatic development of cosmology given recently by the author. There the τ-scale was the more fundamental, and it was not found necessary to introduce Milne's t-scale. The object of the present paper is to discuss this primitive τ-scale still further, and to show how the t-scale may be introduced by means of an axiom. The unpublished work mentioned above is not required for this purpose because the cosmological models considered here were described in earlier papers (Walker, 1937, 1940 b). We also examine the various constants, absolute and conventional, which are connected with the different scales of time and length, and with different models.