In an earlier paper (Aitken and Silverstone, 1941) the problem of estimating from sample a parameter θ of unknown value was treated by adopting two postulates for the estimating function: (i) that it should be unbiased in the linear sense; (ii) that its sampling variance should be minimal.

Calculations are made of the resonance energy, bond order and bond length in a series of graphitic layers of varying size. Carbon–carbon bond lengths appear to vary very little in size with increasing number of carbon atoms, in agreement with experiment. But variations in resonance energy are significant, and indicate clearly that resonance, by itself, favours an approximately square, rather than oblong, shape. But in the case of such layers in equilibrium in the presence of molecular hydrogen, the most stable layer containing a given number of carbon atoms is of the long, thin polyphenyl type. Some tentative calculations suggest that polymerisation of smaller groups to larger ones should be endothermic, in agreement with the experimental fact that the formation of larger graphitic crystallites during carbonisation occurs, with emission of hydrogen, only at high temperatures.

Relativity is the study of matter in motion, and the basis of a theory of relativity can be either physical, mathematical, or logical. It is physical if some of the elementary objects and relations are concepts derived from the external world and if certain of their properties are assumed as physically obvious. If, however, the elementary objects, etc. are defined as mathematical symbols and relations, and if the subsequent theorems are mathematical deductions from these definitions, then the theory may be described as mathematical. Lastly, the basis of a theory is logical if certain terms are undefined—and clearly stated as such—and if the theory is then developed strictly deductively from an explicit set of axioms and definitions. Analogous examples taken from geometry are the Euclidean, algebraic, and projective theories. The first, as developed by Euclid, has a physical basis, while the second is mathematical, a point being defined as an ordered set of numbers (co-ordinates) and a line as the class of points satisfying a linear equation. The third is logical, the undefined elements being point and line (an undefined class of points) and the axioms being those of incidence, extension, etc. Usually a physical theory comes first, to be followed by a mathematical and then by a logical theory, this last being so constructed that it includes previous theories when its undefined elements are replaced on the one hand by the conceptual physical objects and on the other hand by the symbolic mathematical objects. The construction of such a logical theory is not merely a matter of academic interest, for it can be regarded as an analysis of the previous theories. It tests, for example, the consistency and independence of their basic assumptions and definitions. It also indicates how a theory can be modified, with as little change as possible, so as to include some feature previously excluded. This can be particularly useful in the case of a physical theory which has been constructed to correspond as closely as possible to the external world, for such a theory may need continual modification to keep in step with observational data. For this reason the axioms of a logical theory should be not only consistent and independent but also simple, i.e. indivisible.

The problem with which this paper is concerned arose in the discussion of a series of chronometric observations, but it is of more general application, and is capable of wide extension. Pairs of readings (xi yi) were taken at times ti, i = 1, 2, …, n. These readings were known to be affected by respective errors (ξi ηi) from sources different but possessing some common part. It was important to have an estimate of the consequent correlation and to assess its precision. The assumptions made in the particular experiment were that x and y were both linear in t, representable by x = a0 + a1t, y = b0 + b1t, and that the distributions of error in x and y were normal. The parameters a0 and a1, b0 and b1 were therefore obtained from two separate sets of normal equations, and the unknown correlation was then estimated from the sum of products of corresponding residuals ui, vi, one from each set. In the corresponding situation in n samples (xi,yi) from a bivariate normal distribution the mean value of is (n − 1) ρσ1σ2, where σ12, σ22 are the variances of x and y and ρσ1σ2 is their product moment. One might therefore anticipate, by analogy, that in the present case the mean value of Σuivi would be (n − 2)ρσ1σ2. So indeed it proves to be, and the sampling variance of Σuivi conforms likewise with standard results; but it is desirable, by an extension of the problem, both to see why this is so and to take notice of cases where the analogy fails to hold.

I. Introduction.—Consider a Hamiltonian system of differential equations

where H is a function of the 2n variables qi and pi involving in general also the time t. For each given Hamiltonian function H the system (1.1) possesses infinitely many absolute and relative integral invariants of every order r = 1,…, 2n, which can all be written out when (1.1) is integrated. Our interest now is not in these integral invariants, which are possessed by one Hamiltonian system, but in those which are possessed by all Hamiltonian systems. Such an integral invariant, which is independent of the Hamiltonian H, is said to be universal.

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.

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.

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.

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. 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.

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. 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.

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.

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.

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.

1. 1. An n : l is defined as a group of n particles which can be contained within a seeker length l moving around a closed line of length L on which N particles are distributed at random. An expression has been obtained for the average number of distinct n : l's per closed line.

2. 2. An expression has also been derived for the average numbers of n : l's in the corresponding problem where the line of length L containing the N particles is open and not closed.

3. 3. Analogous problems in two dimensions are considered, in which the particles are arranged at random on a plane and the place of the seeker line is taken by an orientated rectangle. Exact expressions are given for the desired averages.

4. 4. The extension of the methods used to analogous problems in three dimensions is discussed. Exact expressions have not been obtained, but approximations are given which hold when n is much greater or much smaller than x, the average number found within the seeker area.