1. Introduction. In this paper we consider certain general properties of propositional calculi. Two forms of the definition of a finite model of such a calculus are discussed, these forms differing in the manner prescribed for the satisfaction of the rules of the calculus. The methods of definition are shown to be equivalent for the application of the finite model method for the demonstration of the unprovability of formulae in the calculus. It is further proved that, although the decidability of a calculus follows from the existence of a finite model counter-example for each unprovable formula of the calculus, the converse result is not true.

Let

be a real quadratic form in n variables (n ≥ 2) with integral coefficients and determinant D = |fij| ≠ 0. Cassels ((1),(2)) has recently proved that if the equation f = 0 is properly soluble in integers x1, …, xn, then there is a solution satisfying

where F = max | fij and cn depends only on n. An example given by Kneser (see (2)) shows that the exponent ½(n – 1) is best possible. A simpler proof of Cassels's result has since been given by Davenport(3), and the theorem has been improved in certain cases by Watson(4). Here I consider the inequality f(x1, …, xn) > 0, where f is an indefinite form, and obtain a result analogous to that of Cassels.

1. Introduction. The class of finite groups in which any two subgroups of the same order are isomorphic will be denoted by (C), and ‘G ∈ (C)’ will mean ‘G belongs to the class (C)’.

For a long time there has been some interest in obtaining spherical harmonics with the symmetry of the regular polyhedrons, particularly for electrostatic problema involving polyhedral conductors. Work on the icosahedral group has been done, among others, by Meyer (7), Laporte (6), Hodgkinson (3) and Poole (8), apart from the classic work of Klein (4). In the several approaches of these authors, only spherical harmonics for the totally symmetric representation were obtained, the most complete table being that of Laporte who obtained the spherical harmonics up to l = 21. New interest in the icosahedral group has arisen in connexion with the structureof some proteins (5), and we obtain here, by the recently developed methodof Altmann(1), expansions in spherical harmonics for ali the representations of this group. This has been done up to and including l = 14. (For the totally symmetric representation we have also included l = 15.) In what follows we shall use Altmann's formulae and notation.

The main purpose of this note is to show some connexions between Legendre polynomials and random walks. More specifically we shall be concerned with what may be called ‘trinomial’ random walks i.e. walks on the points 0, ± 1, ± 2, …, such that the probabilities of ‘currently’ taking steps of – 1, 0, + 1 are p–1, p0, p1.

In a previous note (Good (7)) it was shown that there is an intimate connexion between Legendre polynomials and ‘trinomial’ random walks, that is walks on a one-dimensional lattice with the same probabilities p–1p0, p1 at each point of taking a step of – 1,0,1. The purpose of the present note is to point out that if the probabilities depend on the current position of the particle, then there is scope for applications of the theory of analytic continued fractions.

1. Introduction. Problems of unrestricted random walks on lattices have been considered by many authors and methods have been discovered for the exact enumeration of the number of walks between two points, for obtaining asymptotic estimates of the distribution functions and for evaluating special parameters such as the probability of ultimate retum to the starting point. By an unrestricted walk we mean one in which the probability of the next step at any stage is not influenced by the previous choice of steps. The corresponding problems for restricted (or correlated) random walks are more difficult and fewer results have been obtained. Restricted walks are, however, of considerable physical interest in connexion with the statistical behaviour of long chain molecules, the theory of cooperative phenomena in crystals and in other applications.

Wright's model of a random process in genetica is modified by supposing that births and deaths occur individually at random so that the generations are no longer simultaneous. An exact solution is then obtained for the distribution of the number of a-genes in a haploid organism when mutation is occurring in both directions. When there is no mutation, the rate of approach to homozygosity is found in more complicated models with two sexes and diploid individuate. This rate is twice that occurring in Wright's models.

It is suggested, on heuristic grounds, that the energy-momentum tensor of a material field with non-zero spin and non-zero rest-mass should be non-symmetric. The usual relationship between energy-momentum tensor and gravitational potential then implies that the latter should also be a non-symmetric tensor. This suggestion has nothing to do with unified field theory; it is concerned with the pure gravitational field.

A theory of gravitation based on a non-symmetric potential is developed. Field equations are derived, and a study is made of Rosenfeld identities, Bianchi identities, angular momentum and the equations of motion of test particles. These latter equations represent the geodesics of a Riemannian space whose contravariant metric tensor is gij–, in agreement with a result of Lichnerowicz(9) on the bicharacteristics of the Einstein–Schrödinger field equations.

It is shown that the phenomena of double refraction—optical activity, Faraday rotation, Kerr effect, etc.—may be treated as scattering problems in quantum field theory. An expression for the scattering cross-section of an atom or molecule for photons may be easily obtained and from this expression using the idea of Stokes operators, equating their expectation values to the classical Stokes parameters, the usual quantum mechanical expressions for optical activity, etc. may be obtained. This approach does not require the ideas of refractive index or polarization of the medium, and emphasizes what are the actual observables in these experiments.

The behaviour of two disk dynamos coupled to one another is examined in relation to the earth's magnetic field. It is found that reversals of electric current and magnetic field occur, unlike the case of a single disk dynamo.

On assuming a simple engine that drives a disk dynamo, the coupling between electro-magnetism, motion and heat is discussed. A possible range of temperature fluctuation associated with a dynamo action is also estimated very roughly.

A train of long waves travelling along a canal of depth h2 encounters a vertical step at which the depth changes to h1. The wave-length is much greater than h1 or h2. The classical Long-Wave theory where vertical acceleration is neglected is clearly inapplicable, although a treatment has been given by Lamb. In the present paper the problem is treated rigorously by the linearized theory of surface waves. A singular Fredholm integral equation of the first kind is obtained for the horizontal velocity above the step and is converted into a regular equation of the second kind with a kernel which tends to zero as the wave-length tends to infinity. To achieve this we first take the formal limit λ = ∞ in the integral equation of the first kind. This corresponds to a fluid motion between rigid boundaries which can be solved explicitly by a conformal transformation. In this way we obtain the inverse operator corresponding to λ = ∞, which is then applied to the original equation where λ < ∞. An equation of the second kind results which has a kernel tending to zero as λ tends to infinity and which is soluble by iteration for large λ. Although the Long-Wave method of Lamb fails to describe the details of the motion correctly, his predicted reflexion coefficient appears as a first approximation.

The paper studies the problem of elliptical whirl of perfectly alined, finite, flooded, journal bearings, with incompressible isoviscous lubricant; it bases itself on Reynolds's equation. Explicit solutions are obtained, using the long (L/D≫ 1) and short (D/L ≫ 1) bearing approximations.

The physical situations corresponding to the solutions obtained are those of unloaded and constant-loaded bearings, with elastic, massive, frictional suspensions. The agreement with empirical findings is generally good, and the paper is published in the hope that further experimental research will be stimulated by it.