Certain types of 2n-dimensional Riemannian spaces admitting parallel fields of null n-planes are studied. These are known as Riemann extensions of conformal, projective or other classes of spaces of affine connection. The circumstances under which a 2n-dimensional Riemannian space admits two non-intersecting parallel fields of null n-planes are also discussed. Such spaces satisfy a condition similar to Kähler's condition in the theory of complex manifolds, and hence are called Kähler spaces. Necessary and sufficient conditions are found for a Kähler space to be a Riemann extension with respect to one of the parallel fields of null n-planes, and canonical forms are found for the metrics in the cases of Riemann extensions of conformal and projective spaces.

Kuiper's recent theory of the origin of the solar system is criticised on several grounds. Firstly, it is pointed out that the empirical relation between the ratio of the masses of two consecutive planets (satellites) on the one hand and the ratio of their distances from the sun (primary) on the other hand is not the one discussed by Kuiper. Secondly, it is shown that the densities needed for a successful application of Kuiper's theory are probably not attained in the system considered by him. Finally, some other points are discussed which enter into most theories about the origin of the solar system.

With every matrix representation of the (real) full linear group can be associated a multi-linear mapping of one affine space, Rn, into another, RN. This mapping is studied from the viewpoint of the geometry of numbers of convex bodies, and a general arithmetical property of such mappings is proved. The result generalizes my recent work on compound convex bodies.

A study is made of faithful representations of the free cyclic (non-associative) groupoid by means of bifurcating root-trees and by means of index polynomials in two indeterminates emphasizing the intimate connection between these representations. The properties of trees and index polynomials are investigated and the concept of lattice of trees is introduced.

A sequence of non-negative random variables {Xi} is called a renewal process, and if the Xi may only take values on some sequence it is termed a discrete renewal process. The greatest k such that X1 + X2 + … + Xk ≤ x(> o) is a random variable N(x) and theorems concerning N(x) are renewal theorems. This paper is concerned with the proofs of a number of renewal theorems, the main emphasis being on processes which are not discrete. It is assumed throughout that the {Xi} are independent and identically distributed.

If H(x) = Ɛ{N(x)} and K(x) is the distribution function of any non-negative random variable with mean K > o, then it is shown that for the non-discrete process

where Ɛ{Xi} need not be finite; a similar result is proved for the discrete process. This general renewal theorem leads to a number of new results concerning the non-discrete process, including a discussion of the stationary “age-distribution” of “renewals” and a discussion of the variance of N(x). Lastly, conditions are established under which

These new conditions are much weaker than those of previous theorems by Feller, Täcklind, and Cox and Smith.

The Faroe-Shetland Channel is the threshold from the north-eastern Atlantic Ocean to north-west European seas. Through it passes the main bulk of the oceanic water-mass which is the predominant influx, among several other water-masses, to these seas.

The following research into the dynamics and general hydrography of the region is based on numerous observations of temperature and salinity, from surface to bottom, taken mainly on two vertical cross-sections of the Channel between the years 1927 and 1952 inclusive, excepting the war years 1940 to 1945.

The research reveals very large scale seasonal and long-term variations in the northeastward volume-transport of oceanic water, suggests the existence on occasions of what appear to be horizontal tortional currents within the oceanic water-mass, and demonstrates (a) the intrusion of Gulf of Gibraltar (extra-Mediterranean) water into this mass over a period of years, (b) the formation of heavy oceanic water and (c) of a sub-oceanic watermass. The last-mentioned may sometimes almost entirely displace the bottom Norwegian Sea water-mass which normally underlies the oceanic mass.

One or other, or both, of two types of Arctic water may also sometimes displace bottom Norwegian Sea water as the bottom water-mass of the region, the process, like that of the above-mentioned Gulf of Gibraltar water influx, waxing and waning over a term of years and thus exemplifying the phenomenon of marine climatic change.

The two regular representations of quaternions give rise to a classical set of sixteen 4 × 4 matrices that have fairly recently reappeared in a paper by S. R. Milner. He uses them as the basis of a calculus of “Ɛ-numbers”, which he develops for the purpose of making physical applications. The covariantive nature of his calculus is, however, not always fully apparent, and raises some points of interest of which an examination is made in the present paper in terms of 3-dimensional projective geometry. The theory that emerges is the classical one of the collineations of projective 3-space that transform a quadric into itself, but the formulation is different from that of existing theories based on the same set of matrices and having the same or a similar geometrical background. For example, the present theory is quite different from that of 4-component spinors. The constants of multiplication γijk of quaternion algebra make their appearance in a generalized form and in a geometrical setting. In the final section an indication is given of possible generalizations to Riemannian geometry, and of the connection of the present work with the theory of Kähler manifolds of two complex dimensions.

Given a power series with real non-negative coefficients and having radius of convergence p, a summability method P is defined as follows:

The main concern of this note is to establish conditions sufficient for one such method to include another.

A Lie group is said to be metrisable if it admits a Riemannian metric which is invariant under all translations of the group. It is shown that the study of such groups reduces to the study of what are called metrisable Lie algebras, and some necessary conditions for a Lie algebra to be metrisable are given. Various decomposition and existence theorems are also given, and it is shown that every metrisable algebra is the product of an abelian algebra and a number of non-decomposable reduced algebras. The number of independent metrics admitted by a metrisable algebra is examined, and it is shown that the metric is unique when and only when the complex extension of the algebra is simple.

A theorem, due to Dr Grace, about a configuration of lines in [3], has been shown by him to be a rewording of a theorem about a configuration of points and spheres in [4]; in this six spheres pass through each point. The present paper discusses the analogous configuration in which seven spheres pass through each point.

The National Institute of Oceanography owes its formation to a growing need for more detailed and systematic knowledge of the physics as well as the biology of the oceans, and the feeling that the United Kingdom, in co-operation with the Commonwealth countries, should play a part in marine science more in keeping with its tradition of interest in the oceans and their navigation. The Institute is controlled by the National Oceanographic Council, which was granted a Royal Charter in 1949 and operates with the aid of Government grants.

The emphasis of the programme is on long-term research under the main headings:

Interchange of energy between the atmosphere and oceans

Response of the sea surface to wind and pressure changes

General circulation of the oceans

Free and forced oscillations in the oceans

Distribution of marine organisms

Organic production in the oceans

Bionomics of whales.

There are many other activities and as much as possible is done to work out the practical implications of the research, and to help other marine scientists. The headquarters of the Institute are at Wormley, near Godalming, Surrey, and its research ship, the R.R.S. “Discovery II”, is based at Plymouth.

In a recent paper J. L. Synge (1952) has shown that the theories of Einstein and Whitehead predict practically the same phenomena in the solar field. He finds that the rotation of perihelion, the deflexion of light rays and the red-shift in the sun's spectrum are the same in both theories. In this paper the motion of the centre of mass of a double star is investigated. It is found that Whitehead's theory predicts a secular acceleration in the direction of the major axis of the orbit towards periastron of the larger mass. The possibility of a binary star having such an acceleration has already been considered by Levi-Civita (1937), and he has given an example in which it may become detectable in less than a century. On the other hand, it has been shown by Eddington and Clark (1938) that there is no secular acceleration according to Einstein's theory.

The logarithmetic L of a non-associative algebra or class of algebras S has been previously defined as the arithmetic of the indices of powers of the general element when indices are added (non-associatively) and multiplied by certain conventions similar to those of ordinary algebra. With respect to addition, L is a homomorphic image of the “most general” logarithmetic B, the free additive groupoid with one generator 1, and in the case of algebras of one operation is essentially the same as the free algebra in one variable on S. The definition is now extended so that L is defined when S is any subset of an algebra or class of subsets of algebras, with the result that every homomorph of B is a logarithmetic ; but a distinction has then to be drawn between closed logarithmetics in which as before both addition and multiplication are defined, and other logarithmetics in which there is only addition. L is its own logarithmetic (taken with respect to addition) only if L is closed. For subsets of palintropic algebras, L is necessarily closed.

An investigation is made of the motion of a one-dimensional finite gas cloud which is initially at rest and is allowed to expand into a vacuum in both directions. The density of the gas at rest is assumed to rise steadily and continuously from zero at the boundaries to a maximum in the interior of the cloud.

If the subsequent motion is continuous, it is completely specified by analytical solutions in seven different regions of the x-t plane joined together along characteristics. The motion of one of the boundaries is discussed, and conditions found for it to have (i) an initial stationary period or (ii) a final constant velocity of advance into the vacuum. The gas streams in both directions from a dividing point at zero velocity. This point ultimately tends to the mid-point of the initial distribution.

The possible breakdown of the continuity of the motion is discussed, and a condition on the initial density distribution found for shock-free flow to be maintained.

In this note we study the asymptotic behaviour of a product of matrices where Pj is a matrix of transition probabilities in a non-homogeneous finite Markov chain. We give conditions that (i) the rows of P(n) tend to identity and that (ii) P(n) tends to a limit matrix with identical rows.

A number of analogues to the simple fluid compressibility equation are deduced by considering fluctuations in a binary mixture; and their simplest expressions are found to be in terms of the binary mixture direct correlation functions. The accuracy of these results is tested with the aid of the appropriate extension of the approximate Born-Green theory, which facilitates the demonstration of consistency with the first three terms of the virial expansion. The problem of the evaluation of the corresponding radial distribution functions by means of X-ray or optical scattering is taken as far as the determination, in principle, of a concentration correlation function from observations of critical opalescence.

The classical theorems of Vitali and Blaschke are shown to be simple consequences of an inequality of an interpolatory character due to J. M. Whittaker.

Theorems generalising one of Montel relating to functions bounded in a half-plane and tending to zero at a sequence of points are established by similar methods.

The temperature gradient in the lower atmosphere can be directly determined by measuring the optical refractive index of the air. This method is suitable for use on the Greenland ice sheet where errors introduced by water vapour are small, and where the strong solar radiation reflected by the snow surface makes it difficult to measure temperature differences over height differences of about I metre.

The refraction was measured by observing the apparent vertical angle of each of a set of targets at distances up to 4 km. from a theodolite. The refraction was found to vary linearly with the distance of the target. The true vertical angle to the targets was determined when a second theodolite was available and reciprocal sights could be taken with it from the site of target to the fixed theodolite. The true vertical angle varied with time due to slow descent of the theodolite as the firn slumped; a correction for this was made. The standard error of the temperature gradient measurements was about 1.5 × 10−2 C.° per metre. It is considered that the method could be developed and improved so that over a range of only 100 metres temperature gradients could be measured to an accuracy of about 0·1° C. per metre.

The writer's theory of unimolecular dissociation rates, based on the treatment of the molecule as a harmonically vibrating system, is put in a form which covers quantum as well as classical mechanics. The classical rate formulæ are as before, and are also the high-temperature limits of the new quantum formulæ. The high-pressure first-order rate k∞ is found first from the Gaussian distribution of co-ordinates and momenta of harmonic systems, and is justified for the quantum-mechanical case by Bartlett and Moyal's phase-space distributions. This leads to a re-formulation of k∞ as a molecular dissociation probability averaged over a continuum of states, and to a general rate for any pressure of the gas.

The high-pressure rate k∞ is of the form ve-F/kT, where v and F depend, in the quantum case, on the temperature T; but v is always between the highest and lowest fundamental vibration frequencies of the molecule. Concerning the decline of the general rate k with pressure at fixed temperature, k/k∞ is to a certain approximation the same function of as was tabulated earlier for the classical case, apart from a constant factor changing the pressure scale in the quantum case.