A new proof of the Cayley-Hamilton theorem avoiding the use of determinants makes it possible to apply this theorem to matrices over a commutative semi-ring. The relationship of this theorem to a theorem by Lunts concerning switching matrices is investigated.

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.

In this paper we examine the general paraboloidal co-ordinate system, in which the normal surfaces are elliptic or hyperbolic paraboloids, including as special cases the “parabolic plate” and the “plate with a parabolic hole”. We then show that normal solutions of Laplace's equation in these co-ordinates are given as products of three Mathieu functions, and apply this to the solution of boundary-value problems for Laplace's equation in these co-ordinates. In a subsequent paper the corresponding treatment of the wave equation will be given.

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 seven-dimensional Euclidean space considered as the space of purely imaginary Cayley numbers is called a Cayley space. The six-dimensional sphere in a Cayley space admits an almost complex structure which is not integrable. Moreover the algebraic properties of the imaginary Cayley numbers induce an almost complex structure on any oriented differentiable hypersurface in the Cayley space. The Riemannian metric induced on the hypersurface from the metric of the Cayley space is Hermitian with respect to the almost complex structure.

It is proved that the induced Hermitian structure of an oriented hypersurface in the Cayley space is almost Kaehlerian if and only if it is Kaehlerian, that a necessary and sufficient condition for a hypersurface in a Cayley space to be an almost Tachibana space is that the hypersurface be totally umbilical, and that a totally umbilical hypersurface in a Cayley space admits a complex structure when and only when it is totally geodesic.

For a hypersurface in the Cayley space with the induced Hermitian structure which is an *O-space it is proved that all the principal curvatures of the hypersurface are constant, and from this is deduced a classification of such *O-spaces.

This paper formulates a general solution, within the scope of classical elastostatic theory, for the problem of layered systems subjected to asymmetric surface shears. As an illustrative example the solution for the problem of an elastic layer supported on an elastic half-space is presented for the particular loading consisting of a surface shearing force uniformly distributed over a circular area. Numerical results are included indicating some displacement and stress components of interest.

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.

An attempt is made to develop the statistical mechanics of the liquid state based not on the usual concept of a “radial distribution function” but on that of a “next neighbour distribution function” which is closely linked up with Bernal's ideas on the characteristic features of liquid structure. Making certain simplifying assumptions it is indeed possible to construct a partition function for an atomic liquid in this way and from this to derive the thermodynamic properties of the system according to the principles of classical statistical mechanics. It is shown that the free energy, the equation of state, the specific heat and entropy as obtained from the theory are consistent with the expected behaviour of such liquids. It is further shown that the computed next neighbour distribution function for close packing is in good agreement with the one derived empirically from a model by Bernal and Mason.

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 solution of the dual series relations

where {λn) is the sequence of positive zeros of the Bessel function Jν(αλ), arranged in order of increasing magnitude, þ and ν are real numbers (−1 <þ < 1, ν >0), the functions, f1(ρ), f2(ρ) being prescribed, is obtained by giving an integral representation of {αn} in terms of a single function g(t). The problem is reduced to that of solving a Fredholm integral equation of the second kind for the auxiliary function g(t).

This paper is concerned with the theory of the probability distribution of the total number of electrons in the avalanche produced by the release of a single electron in the gas of a proportional counter. The disagreement between existing theory and experimental results is discussed and a new theory is proposed, based on the fact that fluctuations in the number of electrons at a given point in the avalanche are accompanied by fluctuations in the average electron energy. This aspect of the problem is incorporated directly into a simple one-dimensional model of the multiplication process, and the resulting distribution function has a mathematical foim in agreement with that observed experimentally.

The fluctuation in the number of electrons predicted by this theory is not constant, but is determined by a parameter which, for large mean values, is essentially the mean fraction of electrons in the avalanche having energies above the ionization energy of the counter gas. Limits on the variation of this parameter are obtained by calculations of the mean values using a particular two-dimensional model, in which electrons are divided into two classes according as they have energies above, or below, the ionization energy. The experimentally observed fluctuation lies within the predicted range and close to the lower limit; it is concluded that there is little scope for improvement in the resolution to be obtained from the conventional type of proportional counter.

Let R be a ring and let ΓR be the Jacobson-Perlis radical. It is shown that the radical of the ring of row-finite matrices over R is the ring of row-finite matrices over ΓR if and only if ΓR is right-vanishing. This is done by extending the results of an earlier paper. One interesting consequence of the theorem is noted.

The synthesis of 3-bromofluorenone is described and attention drawn to erroneous statements by previous workers in the field. The reduction of the ketone to 3-bromofluorene has been investigated and shown to be readily effected by heating with hydrazine hydrate in diethylene glycol. This appears to be the method of choice for reducing fluorenones to fluorenes.

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.

The problem discussed is that of determining the sequence {an} such that

where (λn) is the sequence of positive zeros of the function λJν(λ) + HJν(λ), arranged in order of increasing magnitude, þ, ν and H are real constants (−I <þ < I, ν > −½) and f1(ρ), f2(ρ) are prescribed. By expressing the sequence {cn} in terms of a sequence of integrals involving a function g(t) the problem is reduced to the solution of a non-singular Fredholm integral of the second kind for g(t).

The methods developed in I, II of this series of papers are applied to a solution of a variety of dual series relations involving trigonometric series. In general the problem is reduced to one of solving (usually by numerical methods) a Fredholm integral equation of the second kind for an auxiliary function g(t), but for certain values of the parameters it is possible to obtain analytical solutions of the integral equations and these cases are considered in detail.

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.

The methods employed in papers I–IV of this series are modified to provide the solution of certain dual equations involving trigonometric series. It is necessary to introduce a modified form of the conventional operators of fractional integration and to discuss their relation with generalized Schlömilch series expansions of an arbitrary function. These general methods are illustrated by detailed reference to a particular special case.

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.