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.

Wave functions have been obtained for the ground states of the atoms Li to Ne. These are of the simple analytical type proposed by Morse, Young, and Haurwitz. Some errors in the latter work are corrected.

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.

The gravitational field of a system of particles was investigated by de Sitter as far back as 1916. A minor alteration to the analysis was made by Eddington and Clark in 1938. The amended value of the potential g44 is the same as that derived by Einstein, Infeld and Hoffmann without making use of the energy-tensor; this agreement suggests that the revised de Sitter argument is correct. In this paper we show that this is not the case, for the de Sitter analysis completely overlooked any possible interaction terms in the stress components of the energy-tensor. We find the value of these terms, pmn, and show that the agreement mentioned above is due to the fact that the volume integral of pu vanishes.

The difficulties met in the usual treatment of quantised field theories seem to us somewhat similar to those which occurred in Bohr's semi-classical quantum mechanics of particles. In this theory the orbits were described by Fourier series in the time; there was no exact correspondence between the periodic terms of this series and quantum transitions, but only an approximate one for terms of high order. Matrix mechanics considers not the Fourier series, but the single terms which are generalised into matrix elements having not one but two indices. This generalisation is founded on Ritz's combination principle.

Hill's differential equation (1.1) derives its importance from being the prototype of the different equations of Lamé and of the equation of Mathieu, which are connected with wave and potential problems in mathematical physics. Besides this, numerous instances of its occurrence in problems of elasticity and of dynamical or statical stability are known. In the present treatment, conditions are reversed with respect to most of the older publications, since the characteristic multiplier σ of equations (1.2) is not sought as a function of the given parameters λ and γ of equation (1.1), but σ is supposed given and the corresponding values of λ and γ are regarded as unknown. Thus a linear homogeneous boundary value problem of the second order and of non-self-adjoint type ensues, the values of σ and of λ, σ being in general complex. On this latter point the present paper considerably enlarges the scope of some previous papers published by the author during the war along somewhat similar lines but for real characteristic values (Nos. 13–18 of the references at the end).

Little progress has been made in the development of a relativity theory of elasticity, although it has been realised that no disturbance can be propagated with a velocity greater than that of light. In 1917 Lorentz (1) gave a relativistic formulation of the laws of elasticity in the case of small strain and, applying the theory to the problem of a rotating, incompressible, homogeneous disc, he claimed that the radius as measured by an observer at rest on the disc undergoes a contraction. His result was accepted by Eddington (4) but was attacked by others. A great deal has been written on the subject, but it has never been pointed out that both Lorentz and Eddington were considering material in which the waves of dilatation travel with an infinite velocity. In this paper we define “incompressible” matter as that in which these waves are propagated with the velocity of light and Poisson's ratio tends to the value ½. This gives an upper limit to the modulus of compression k, which in this case is the elastic constant λ, and as a result the expansion determined by the ordinary classical theory has to be taken into account. It is found that the “relativity contraction” is exactly cancelled by the “classical expansion”. Throughout the discussion on the rotating disc the analysis is restricted to the case of small strain.

The equations of equilibrium of a continuous static distribution of matter are also investigated in the case of weak fields for which the fourth power of the density may be neglected.

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.

Thermal diffusion in aqueous solutions of raffinose, sucrose, glucose, xylose, glycerol and acetone was studied in respect of variation of concentration and of temperature. Initial rates of separation were determined as produced by an apparatus consisting of two vertical concentric glass tubes of length 130 cm. and mean annular separation o·68 mm.

With a solution of glucose of 1·7 gm. mol. per litre, the initial rate of separation for the extremes of temperature which could be experimentally applied was found to be independent of the column length for lengths greater than 100 cm.

With the exception of acetone, all the solutes concentrated at the column foot. In singlesolute solutions a maximum rate was observed at an intermediate concentration, which in the sugar series was at a lower molar concentration the higher the molar weight of the solute. With the two-solute solutions sucrose-glucose and sucrose-glycerol little or no relative separation was obtained, contrary to expectation based on the single-solute data.

Initial rates increased with rise in mean temperature and with increase in the applied temperature gradient.

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 the Proc. Roy. Soc. Edin. (Houstoun, 1941) I described a new method of measuring the velocity of light. Owing to the difficulties of the times progress has been slow, but I have now succeeded in measuring the velocity of light in water and communicate the result here.

Previous Measurements.—Every elementary textbook on Light explains how Newton's, corpuscular theory required the velocity of light in water to be μ times the velocity in air, whereas Huygens's wave theory required it to be I/μ times the velocity in air, and how Foucault's determination decided between the two theories. As a matter of fact it did, but Foucault's determination was not an accurate one and there have been only four previous determinations of the velocity of light in water, none of which can lay claim to accuracy. These are the determinations of Foucault (1850), Fizeau and Bréguet (1850), A. A. Michelson (1891), and Gutton (1911).

The systems of “partitive numbers” introduced in this paper differ from ordinary number systems in being subject to non-associative addition. They are intended primarily to serve as the indices of powers in algebraic systems having non-associative multiplication, or as the coefficients of multiples in systems with non-associative addition, but are defined more generally than is probably necessary for these purposes. They are essentially the same as root-trees (Setzbäume) with non-branching knots other than terminal knots ignored, with operations of addition and multiplication defined.

Partitive numbers are of two kinds, partitioned cardinals and partitioned serials, defined respectively as the partition-types of repeatedly partitioned classes and series. For each kind, multiplication is binary (i.e. any ordered pair has a unique product) and associative. Addition is in general a free operation (i.e. the summands are not limited to two, and indeed, assuming the multiplicative axiom, may form an infinite class or series); but it is non-associative, which means that for example a + b + c (involving one operation of addition) is distinguished from (a + b) + c and a + (b + c) (involving two operations). A one-sided distributive law is obeyed:

Partitioned cardinals are commutative in addition.

In this paper the curvature tensor Rijkl in a Riemannian Vn is used to define a quadratic complex of lines in an (n – I)-dimensional projective space Sn–1. Work in this direction has been done for a V4 by Struik (1927–28), Lamson (1930), and Churchill (1932). Of these, Struik and Lamson both use 3-dimensional projective geometry, the former for the purpose of defining sets of “principal directions” in V4 by means of the Riemann tensor, and the latter for the purpose of discussing some of the differential and algebraic consequences of the field equations of general relativity. Churchill considers the geometry of the Riemann tensor from the point of view of 4-dimensional Euclidean geometry. In this paper an indication is given of the nature of the general n-dimensional theory, which, by way of elementary illustration, is then applied in moderate detail to a V3. A few general formulae are also obtained for a V4.

The structure of commutative associative linear algebras is well known and is usually derived from more general results concerning non-commutative algebras (Cartan, Frobenius). The novelty of the present treatment is that while it avoids the complexities of the non-commutative case, it exhibits the essential relationship between the theory of commuting matrices and that of commutative algebras.

While theorems 1 and 2 of this paper are implicit in the writings of Voss (1889), Taber (1890), and Plemelj (1901), it has been considered worth while to recapitulate these results in the explicit form required for the discussion of commutative algebras. In doing so, some new facts emerge.

An elementary theory of thermal diffusion applicable to gaseous and liquid systems has been developed. This is based on the difference of diffusional characteristics of a molecule considered as diffusing through two different temperature regions, when the pressure is constant throughout.

For gaseous systems, the resultant expression is shown to be in general accordance with experimental variation of temperature, mass, and diameter factors, and is further developed to include isotopic separation, change of sign of separation with concentration, and general force law considerations.

A similar approach to thermal diffusion in solution, combined with the convection effect of a “cascade” system, gives an expression which is in general agreement with the results of experimental variation of mean temperature and temperature gradient for aqueous solutions of sucrose, glucose and glycerol. The simple expression does not account rigidly for the sign of separation or the effect of altered concentrations. These discrepancies are discussed in relation to the general formula; it is concluded that in addition to the diffusion diameters, the relative thermal expansions of solute and solvent are of importance in this connection.

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.

6. The expected score of an individual on a test consisting of a large number of items is assumed to be given by a formula involving the ability of the individual and also two quantities constant for the test. An expression is then derived for the covariance between two tests measuring different abilities. It appears that if a factorial analysis is performed on a set of tests of unequal difficulty, using the matrix of variances and covariances, a spurious factor will tend to be introduced depending mainly on the differences in difficulty. The effect of this is removed by transforming the variances and covariances to a new set of coefficients. A numerical example of the process is given.

In conclusion I should like to thank the Carnegie Trust for the Universities of Scotland for a grant to cover the cost of the setting and printing of mathematical formulæ in a paper previously published in the Society's Proceedings (LXI, A, 1943, 273–287).

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.

A general formula is obtained for the interference velocity when an aerofoil with elliptically distributed circulation is in a closed or open wind tunnel of any cross-section. The mapping of the section on the interior of a circle is given in terms of the Jacobian elliptic functions appropriate to the ellipse and rectangle. The result is worked out for an aerofoil which spans the focal distance in a tunnel whose section is an ellipse.

I. Throughout this paper k1, …, k3 will denote s ≥ I fixed distinct positive integers. Some years ago Pillai (1936) found an asymptotic formula, with error term O(x/log x), for the number of positive integers n ≤ x such that n + k1, …, n + k3 are all square-free. I recently considered (Mirsky, 1947) the corresponding problem for r-free integers (i.e. integers not divisible by the rth power of any prime), and was able, in particular, to reduce the error term in Pillai's formula.

Our present object is to discuss various generalizations and extensions of Pillai's problem. In all investigations below we shall be concerned with a set A of integers. This is any given, finite or infinite, set of integers greater than 1 and subject to certain additional restrictions which will be stated later. The elements of A will be called a-numbers, and the letter a will be reserved for them. A number which is not divisible by any a-number will be called A-free, and our main concern will be with the study of A-free numbers. Their additive properties have recently been investigated elsewhere (Mirsky, 1948), and some estimates obtained in that investigation will be quoted in the present paper.