Let G be a graph with b+v vertices, each of the b vertices P1, …, Pb having valency k and each of the remaining v vertices Q1 …, Qv having valency r, and each edge joining a vertex Pm to a vertex Qn. Suppose also that b≥v; then r≥k since bk = vr.

It is well known that if, in a continued fraction

there is a subsequence of the an which increases very rapidly, then ξ is a transcendental number. A result of this kind follows from Liouville's Theorem on rational approximations to algebraic numbers, but the most precise result so far established is that which was deduced from Roth's Theorem by Davenport and Roth [1]. They proved (Theorem 3) that if ξ is algebraic, then

where qn is the denominator of the nth convergent to (1). Thus if

ξ is transcendental.

This paper is a sequel to a previous paper [1] on axially symmetric torsion-free stress distributions in isotropic elastic solids and applies the methods used to investigate these distributions to distributions in solids under torsion. The basis of these methods is that in a solid of revolution containing a symmetrically located crack the stresses set up in the neighbourhood of the crack by forces applied over the crack can be found by perturbing on their values in an infinite solid containing a crack of the same radius and under the same applied forces, provided the radius of the crack is small compared with a typical length of the solid of revolution. The problem of determining the stresses in the solid of revolution is shown to be governed by a Fredholm integral equation of the second kind, which holds whatever the ratio of the crack radius to the typical length, but which, when this ratio is small, is readily solved by iteration to give stresses perturbing on those in an infinite solid. A similar method can be applied to an infinite solid containing two or more cracks when the crack radii are small compared with a typical length of the crack array.

A slow steady motion of incompressible viscous liquid in the space between two fixed concentric spherical boundaries is considered. The motion arises from two point-sources of strengths ±2m at arbitrary points A, B on the outer sphere r = a. The velocity is calculated as the vector sum of the velocities in two simpler motions in each of which there is an axis of symmetry so that a stream function can be used. The force exerted by the liquid on the inner boundary r = b is similarly the resultant of two forces, each passing through the common centre of the spheres; it can be simply expressed in terms of a, b, m and the vector .

If f(s) is the analytic function defined by the Dirichlet series and if where 0 ≤ b < 1, then the series converges for Re s > 1 and f(s) is regular in the half plane Re s > b except for a simple pole with residue C ≠ 0 at a s = 1. Thus f(s) has a Laurent expansion at s = 1 and it has been shown [1] that under these conditions

where

Archbold [1] has shown how a “distance” can be denned in an affine plane over the field GF(2n) of 2n elements. In terms of this distance, he has shown how to define a group, R(2, 2n), of 2×2 “rotational” matrices which have certain properties of ordinary orthogonal matrices. In the present note we find a standard form for such matrices. Using this standard form, we show that the order of R(2, 2n) is 2n+1+2 and that it has a “proper rotational” subgroup, R+(2, 2n), of index 2. The multiples of R+(2, 2n) by elements of GF(2n) are shown to form a field, which is necessarily isomorphic to GF(22n). The groups R+(2, 2n) and R(2, 2n) are then shown to be cyclic and dihedral groups respectively.

The concept of metrisable Lie algebras was introduced in a previous paper, where some fundamental properties of metrisable Lie algebras have been given. It was shown that, associated with an admissible metric tensor of a metrisable Lie algebra, there is a unique antisymmetric tensor of the third order. A complete solution of the converse problem will be given in this paper; it is first reducedto the solution of a system of algebraic equations and then it is proved that, there always exists a unique metrisable Lie algebra corresponding to each symmetricsolution of the system, even when the solution is trivial. The Lie algebra thus obtained is a reduced metrisable Lie algebra.

In a crossed static homogeneous electric and magnetic field charged particles describing trochoidal orbits in a plane perpendicular to the direction of the magnetic field are focused, but a beam emitted by a point source in a finite solid angle spreads out indefinitely in the direction parallel to the magnetic field. The essential characteristics of trochoidal orbits can be preserved if the superimposed magnetic and electric fields are two-dimensional and orthogonal, such as derived from a vector potential, say, Ax = A(y, z), Av = Az = O, and a scalar potential Ф(y, z)= const. A(y, z). The focusing properties of such a field combination depend on the distribution of the magnetic field only. Following some general considerations, specific examples of double focusing field distributions are given, and the electron motion in one of them is treated in detail.

Beginning with fundamental results obtained by Mason for the effect of self-cooling on the evaporation of drops, and by Fuchs for the diffusional retardation of evaporation for small droplets of any radius, explicit expressions for the effect of the transport of heat on the rate of quasi-stationary growth or evaporation, are discussed.

The simplest algebraic formulation of the results lends itself to interpretation as expressing a resistance to evaporation, the total resistance being the sum of four resistances in series. Two of these resistances, one to diffusion and one to the conduction of heat, are offered by the gaseous phase in bulk; and there are two corresponding resistances at the interface. Corrections are formulated for the effect of the heating of the droplet by radiation. These corrections may be expressed as a (finite) resistance in parallel with the other two resistances to the transfer of heat. Simplified equations are obtained for the evaporation of a liquid whose latent heat of vaporization is very large.

Some remarks are made on the formation of a monodisperse aerosol by the growth of smaller droplets. Integrated expressions are obtained for particular cases of the evaporation of a droplet over a finite period of time.

Empirical regularities are sought amongst the experimental data relative to the spontaneous fission rates and the nucleon-pair binding energies of the heavy even-even nuclei. Certain regularities are found which have not hitherto been noted; in particular it appears that the isotopic number D is a significant parameter in relation to these various quantities. The speculation is made that possibly the saddle-point deformation of the protons in the nucleus is greater than that of the neutrons. Predictions are made concerning the spontaneous fission rates of the even-even isotopes of thorium.

Methods which are widely used in the mathematical analysis of random noise are used here to obtain expressions for the coincidence-counting rates which could be obtained in studies of the intensity correlations between plane-polarized, parallel beams of nearly monochromatic light. It is shown that, with presently available circuit techniques, delayed coincidence measurements could provide information about the breadths of lines radiated by atomic beam light sources, and in prompt coincidence experiments it should be possible to observe interference between beams of incoherent light from sources whose line widths are comparable with those of the Hg198 electrodeless discharge.

The apparent thermal conductivity of a polyatomic or isomerizing gas (as measured in a given apparatus) may decrease at low pressures for two distinct reasons. There may be accommodation effects at boundary surfaces, and there may be relaxation effects arising because molecules with excess energy do not yield it up fast enough to maintain local “chemical” equilibrium. If the apparatus is such that the temperature measured at points in the gas and not in the walls, relaxation effects may be observed directly, and accommodation effects are (in theabsence of relaxation effects) absent.

A detailed analysis is made of the apparent thermal conductivity measured in such an apparatus with cylindrical symmetry. Expressions are obtained in closed form. Numerical calculations show that, for a gas of relatively long relaxation time in an apparatus of reasonable dimensions, the apparent thermal conductivity would decrease appreciably at readily-attained pressures.

A study is made of the generalization of the entropic law (xy)(zw)—(xz)(yw) to an identity connecting two operations. It is shown that such an identity is equivalent “in the large” to the condition that the set of endomorphisms with respect to one operation is closed with respect to the other. Furthermore, for such entropic operations, each may be regarded as a generalized endomorphism of the other and various generalizations of elementary properties of endomorphisms are obtained. Examples of entropic pairs of operations are quite common in mathematics and a number of these are discussed. An important aspect of the paper is the matrix notationintroduced to facilitate what would otherwise be extremely cumbersome computations with entropic operations.

After the detection of correlations in two coherent light beams by Hanbury Brown and Twiss, objections were raised by Brannen and Ferguson on the basis of the experiments of Adam, Janossy and Varga and their own experiments in which no correlations were detected. It is pointed out here that the different groups were looking for two entirely different effects, one being quadratic, the other one linear in the number of photons involved; the quadratic effect (discovered by Hanbury Brown and Twiss) is in agreement with quantum theory while the linear effect is not. It was shown by Purcell and by Hanbury Brown and Twiss that the choice of parameters in the experiments which gave negative results was inadequate to show the quadratic effect. It is shown in this paper that their experiments were also inadequate to decide between the existence or nonexistence of the linear effect.

Nemilov and Pisarevskii (1957) have reported a 390 keV γ-ray in coincidence with neutrons from a Po-Li source, and interpret this as evidence for a level in B10 at 390 keV. A search for confirmatory evidence of this γ–ray has been unsuccessful, and it is estimated that such n–γ coincidences would have been detected in the experiment here described if they involved 8 per cent or more of the neutrons from the Po-Li source.