bluuebunny/crossref_metadata_2025 huggingface dataset:
Energy Relations in Artificial Disintegration
Authors: Rutherford E. , Chadwick J.
Published: November 2008

When certain elements are bombarded by swift α particles, protons or hydrogen nuclei are liberated which are attributed to the disintegration of the nuclei of these elements. It is now generally assumed that in such an artificial disintegration of an atomic nucleus an α particle is captured by the nucleus. Experimental evidence for this capture of an α particle has been obtained by Blackett in the case of the disintegration of the nitrogen nucleus. Blackett photographed eight disintegration collisions of an α particle with a nitrogen nucleus; in each of these the track of the α particle divided into two branches, one of which was due to the residual nucleus set in motion in the collision, and the other was due to the ejected proton. No trace of a third track to correspond to the track of the α particle after the collision was observed. Blackett therefore concluded that the α particle was captured by the nitrogen nucleus.
On the Linearization of the Energy Density of the Electromagnetic Field
Authors: Born M.
Published: November 2008

The modified field equations proposed by the author can be derived from the energy density

(where is the electric displacement, the magnetic induction, the Poynting vector) with the help of the commutation laws for these vectors.
Corrigendum
Authors: Green A. E.
Published: November 2008

The Gliding of a Plate on a Stream of Finite Depth*

Page 597. The line after equation (44) should read

Proc. Cambridge Phil. Soc. 31 (1935), 589–603.
Projective characters and invariants of algebraic varieties
Authors: Roth L.
Published: November 2008

It is a familiar fact that the arithmetic genus pa and the arithmetic linear genus ω of a general surface are linear functions of its four projective characters; and we find by direct calculation that a similar property holds for the numerical invariants of a general threefold. The question thus arises, whether this result can be established a priori for any algebraic variety Vk of general type, since in that case we should have a simple means of determining its numerical invariants. It has been shown by Severi that, subject to a certain assumption, the arithmetic genus pk of Vk is a function of its projective characters, while it is known that, for k ≤ 4, pk coincides with the arithmetic genus Pa obtained by the second definition (§ 5). In the present paper we obtain, by using Severi's postulate, expressions for the arithmetic genera of a V3 and a V4 in terms of their projective characters. We obtain also the characters of their virtual canonical systems and hence derive formulae for the relative invariants Ωi. For this purpose we replace certain projective characters of Vk by others which are more easily computed and better adapted to a simple notation.
The effect of configuration interaction on the low terms of the spectra of oxygen
Authors: Hartree D. R. , Swirles Bertha
Published: November 2008

The ratios of the inter-multiplet separations for the lowest states of O, O+and O++obtained by Slater's method depart considerably from the observed values. In this method it is assumed that matrix elements of the Hamiltonian involving two different configurations are negligible, so that each state can be described by a single configuration, whereas these matrix elements are probably appreciable, and a better approximation is obtained by use of a wave function corresponding to a superposition of more than one configuration. The effect of this superposition of configurations has been called “configuration interaction”, and the general theory of it is discussed in Condon and Shortley'sTheory of atomic spectra, Chap. xv. It is shown that it occurs only between terms of the sameLandS, and of the same parity (∑leven or odd). Few quantitative applications, however, have yet been made. A calculation by Bacher for Mg shows that the effect can be considerable although the states are quite widely separated.
The effect of two isolated forces on the elastic stability of a flat rectangular plate
Authors: Leggett D. M. A.
Published: November 2008

1. Introduction and summary. The problem of the elastic stability of a simply supported rectangular plate, compressed by two equal and opposite forces acting in the plane of the plate (see Fig. 1), was first attempted by A. Sommerfeld, and later by S. Timoshenko. The former produced a solution which in a later paper he admitted to be liable to very considerable error, while the latter constructed a solution by means of the well-known strain-energy method. In many problems this method gives results in very close agreement with those obtained in a more rigorous manner, but, in the particular case considered here, it appeared likely that the error would be appreciable owing to the underlying assumption that the only stresses in the plate occurred along the common line of action of the two external forces.
The determination of intermolecular forces in gases from their viscosity
Authors: Burnett D.
Published: November 2008

A method of determining the coefficient of viscosity of a gas of spherically symmetrical molecules under ordinary conditions has been given by Chapman. His result is equivalent to

wheremis the mass of a molecule of the gas,Tis the absolute temperature,kis Boltzmann's constant 1·372. 10−16and ε is a small quantity which later investigations on a gas in which the intermolecular force is inversely proportional to thenth power of the distance have shown to vary from zero whenn= 5 to 0·016 whenn= ∞ (equivalent to molecules which are elastic spheres); it may reasonably be supposed that ε is positive and less than 0·016 in all cases which are likely to be of interest, and it will be neglected in this paper. Also

π(r) being the mutual potential energy of two molecules (that is, the repulsive force between them is − ∂π/∂r), andr0the positive zero of the expression in the denominator, or the largest such positive zero if there are several.
Über die monotone Konvergenz der Cesàro-Mittel bei Fourier- und Potenzreihen
Authors: Turán P.
Published: November 2008

1·1. Es bezeichne für nichtnegative ganze n und k

das n-te Cesàro-Mittel der Ordnung k der Reihe μ0 + μ1 + … Insbesondere ist also die n-te Partialsumme dieser Reihe.
Statistical theory of adsorption with long-range interaction
Authors: Wang J. S.
Published: November 2008

The statistical theory of long-range interactions between adsorbed particles on a plane lattice is worked out approximately, by treating in detail the distribution of adsorbed particles among a few sites inside and on the boundary of a circular region, and regarding the distribution outside the circle as uniform and continuous with a density Kθ per unit area, where K is the number of lattice points per unit area and θ is the fraction of surface covered by adsorbed particles. The continuous distribution begins at a distance ρ from the centre of the circle, ρ being determined by the condition that the probability of occupation of a first shell site is equal to the probability θ of occupation of the central site. Using this method, general formulae for the adsorption isotherm and the heat of adsorption are obtained. Numerical applications for dipole interactions and for quadratic and hexagonal lattices are worked out in detail and the case in which the dipole moment varies with θ is discussed.
The evaporation of concentrated films of sodium
Authors: Bosworth R. C. L.
Published: November 2008

A three-filament mechanism is described which will enable multimolecular films to be studied by the method of contact potentials and this method is applied to the study of the evaporation of sodium films of various thickness from one to twenty layers. It is shown that the vapour pressure and heat of evaporation both increase as the thickness increases from one to three layers and thereafter become practically independent of any further increase in the film thickness. The vapour pressure for a thickness of four layers is within experimental error the same as that for bulk sodium.

A difficulty due to the effect of the evaporating stream of atoms annulling the space charge around the emitter filament is discussed, and attempts are made to use this effect to measure the rate of evaporation at temperatures from 500° K. to 825°K.
The surface tension of liquid helium
Authors: Allen J. F. , Misener A. D.
Published: November 2008

The previous measurements made on the surface tension of liquid helium by Onnes(1) were carried only as far down in temperature as 1·5° K. Since we had apparatus set up suitable for such measurements and capable of reaching temperatures nearly half a degree lower, we considered it worth while to repeat the measurements. Not only was this done for the purpose of extending them to lower temperatures but also to investigate carefully the variation of surface tension in the region of the λ-point (2·186° K.), the transition temperature between Hei and Henii.
The thermal conductivity of metals
Authors: Makinson R. E. B.
Published: November 2008

The methods used to measure separately the electronic and lattice heat conductivities κeand κgin a metal are reviewed, and it is pointed out that care is necessary in interpreting the results in view of the underlying assumptions. The equations given by Wilson for κeand for the electrical conductivity σ are used to plot the theoretical values of the electronic Lorentz ratioLe= κe/σTas a function ofT, both for the monovalent metals and for a model metal with 1·8 × 10−2conduction electrons per atom, which is taken to represent bismuth sufficiently accurately for this purpose. Curves for κeand κgas functions ofTare given in both cases, and these, together with a comparison of the observed Lorentz ratio andLe, show that in the monovalent metals κgis unimportant at any temperature, but in bismuth it plays a major part at low temperatures, in agreement with experimental conclusions. Quantitatively the agreement is good for copper and, as far as the calculations go, reasonable for bismuth.

Scattering of lattice waves at the boundaries of single crystals (including insulators) at temperatures of a few degrees absolute is shown to be consistent with the experiments of de Haas and Biermasz on KCl and to be responsible for the rise in thermal resistance in this region as suggested by Peierls.

The assumption in the theory of electronic heat conduction that the lattice energy distribution function has its thermal equilibrium value is examined in an appendix. The conclusion is that it should be satisfactory, though the proof of this given by Bethe is seen to be inadequate.
The straggling of electrons due to radiation and ionization loss
Authors: Corben H. C.
Published: November 2008

The probability of an electron having an energy greater than a certain fraction of its initial energy after traversing a certain thickness of matter is calculated for different values of the initial energy. Whereas in previous calculations the energy losses by radiation and by ionization have been treated separately, and the results added, the two types of losses are here combined and their effect on each other included. The results may be compared with those of experiment for values of the initial energy up to a few times the initial energy up to a few times the critical energy of the substance traversed. The theoretical values obtained are found to give better agreement with the results of experiment than do those obtained by the less accurate method.
Superlattice formation of the type AB in an adsorbed layer
Authors: Chang T. S.
Published: November 2008

The possibility, pointed out in a previous paper, of superlattice formation in an adsorbed layer when the adsorbed atoms tend to repulse each other is developed in detail. Both Bragg and Williams's approximation and Bethe's approximation are used, but restricted to superlattices of the type AB. In a range of θ, the fraction of surface covered, a superlattice is found to be possible. Bragg and Williams's approximation shows further that the state with the lowest free energy is the one with a superlattice when the latter is possible. Rough kinetic expressions are also given. The equations derived from the principle necessary to preserve equilibrium are found to reduce to those of detailed balancing, and they also agree with the formula obtained statistically. As expected, all the corresponding results obtained from the two methods become the same when the number of nearest neighbours of a site approaches infinity, provided that the product of this number and the interaction potential of two adsorbed atoms occupying two neighbouring sites remains finite.

The only significant result is the large value of the slope of isotherms (the rate of change of the pressure with respect to the fraction of adsorption) when there is a nearly complete superlattice.
A note on a differential equation
Authors: Hardy G. H.
Published: November 2008

1. In a note published recently in the Proceedings* Goldstein suggests the truth of the following theorem.
Some Properties of Quaternary Alloy Equilibrium Diagrams
Authors: Wilson A. J. C.
Published: November 2008

The geometrical constraints on the forms of quaternary alloy equilibrium diagrams have been derived.
Some relations between certain methods of summation of infinite series
Authors: Good I. J.
Published: November 2008

Let f(t) be defined and measurable for t > 0, and suppose that it has non-vanishing moment constants

where the integrals are Cauchy-Lebesgue integrals, or possibly if f(t) is null for t > t0 but not for t < t0 (a convention to which we adhere throughout).
Relativistic wave equations for zero rest-mass
Authors: Gårding Lars
Published: November 2008

1. In this paper I establish in a certain sense all the relativistically invariant sets of linear partial homogeneous differential equations in which every unknown function u satisfies being the relativistic Laplace operator. Analogous questions for any orthogonal group are dealt with in a previous paper, but it was thought that an independent and not too technical treatment for the Lorentz group might be of interest.
Negative probability
Authors: Bartlett M. S.
Published: November 2008

It has been shown that orthodox probability theory may consistently be extended to include probability numbers outside the conventional range, and in particular negative probabilities. Random variables are correspondingly generalized to include extraordiary random variables; these have been defined in general, however, only through their characteristic functions.

This generalized theory implies redundancy, and its use is a matter of convenience. Eddington(3) has employed it in this sense to introduce a correction to the fluctuation in number of particles within a given volume.

Negative probabilities must always be combined with positive ones to give an ordinary probability before a physical interpretation is admissible. This suggests that where negative probabilities have appeared spontaneously in quantum theory it is due to the mathematical segregation of systems or states which physically only exist in combination.
The minima of a pair of indefinite, harmonic, binary quadratic forms
Authors: Ollerenshaw Kathleen
Published: November 2008

If f is a real, indefinite, binary quadratic form of discriminant d and if κ(f) is the minimum of | f | taken over all integer values of x, y, not both zero, then it is well known that and that this is a ‘best possible’ result.
Projective collineations in a space of k-spreads
Authors: Clark R. S.
Published: November 2008

A collineation in a space of paths is defined as a point transformation which carries paths into paths. Such transformations were first studied by L. P. Eisenhart and M. S. Knebelman. Subsequently J. Douglas introduced the geometry of K-spreads and E. T. Davies has shown that the results for an affine space of paths can be extended to these more general spaces and written in a very elegant form by the aid of Lie derivation.
A note on certain stress distributions in isotropic and aeolotropic materials
Authors: Green A. E.
Published: November 2008

1. I. N. Sneddon has recently considered(11) the two-dimensional problem of the stress distribution due to a force uniformly distributed along a strip in the interior of a semi-infinite elastic medium by integrating the distribution due to an isolated force. The method of solution for the isolated force problem differs from that originally used by Melan(9), but both writers employ real variables. A solution of this problem, using complex variables, has been given by Stevenson and a considerable saving of labour is revealed. A complex variable method of solution for problems of stress distributions in an elastic medium bounded by a plane face was, however, given earlier by Sen(10), although he only made applications to problems in which the force system was applied to the plane face. Sen's method is somewhat different from that used by Stevenson and gives results more directly for many problems, including the problem considered by Sneddon.
The resultant of a large number of events of random phase
Authors: Domb C.
Published: November 2008

Rayleigh's method of deducing the probability distribution of the amplitude of the sum of n equal vibrations of random phase is generalized to the case when the amplitude of each vibration is a definite function of its phase. The same method is applied to the shot effect and it enables the distribution of random noise to be obtained. Campbell's Theorem and its generalizations can then be deduced from this.
Diamagnetism and superconductivity of a collective electron assembly
Authors: Band William
Published: November 2008

Exchange energy between electrons moving in a conductor is assumed to be such that the mean contribution per electron is proportional to the product of the total current within a defined range and the speed of the electron in the direction of the total current.

The ideal resistance of a conductor can be neutralized by such an exchange energy at sufficiently low temperatures.

The surface of the conductor, and internal surfaces of discontinuity if sufficiently marked, are regarded as sites for two-dimensional resonance states in equilibrium with the three-dimensional resonance states in the metal.

The superconducting transition occurs when, at the critical temperature, the neutralization of the ideal resistance of the surface by-passes the residual resistance of the body of the metal.

At the same critical temperature, equilibrium conditions ensure that exactly the right number of electrons enter the surface states to give the whole metal an ideal diamagnetic susceptibility.

These equilibrium conditions are disturbed by a magnetic field, and the transition temperature is shown to depend on the magnetic field in the observed manner.

The intermediate state is explained as one in which the number of electrons in the superconducting surface states is constrained by a non-uniform magnetic field to remain at an intermediate value which is inadequate to produce ideal diamagnetism.

By regarding either the outer surface or internal surfaces of discontinuity as the only regions becoming perfectly conducting, the details concerning restoration of resistance by a current are well explained, as also are the observed size effects in thin films, and the phenomena of ‘non-ideal’ superconductors among the alloys.

Finally, the observed distribution of superconductors in the periodic table receives a satisfactory qualitative explanation.
A ring in graph theory
Authors: Tutte W. T.
Published: November 2008

We call a point set in a complex K a 0-cell if it contains just one point of K, and a 1-cell if it is an open arc. A set L of 0-cells and 1-cells of K is called a linear graph on K if

(i) no two members of L intersect,

(ii) the union of all the members of L is K,

(iii) each end-point of a 1-cell of L is a 0-cell of L

and (iv) the number of 0-cells and 1-cells of L is finite and not 0.
The heat of adsorption of diatomic molecules
Authors: Miller A. R.
Published: November 2008

The heat of adsorption of particles which occupy two sites on a surface is examined by determining the statistical equilibrium. In particular, equations which are appropriate to immobile films are obtained. The heats of adsorption calculated from these equations are compared with the experimental values obtained for the heat of adsorption of hydrogen on a tungsten surface, and with the values which were obtained using an empirical method.
Note on the structure of liquid helium II
Authors: Reekie J.
Published: November 2008

A more detailed analysis has been carried out of earlier results obtained in the scattering of X-radiation by liquid He II. When certain necessary corrections have been applied to these results and a comparison made with theoretical intensity distribution functions to be expected for certain ordered arrangements of atoms in a liquid, it is found that the arrangement of atoms in liquid He II most probably follows either a diamond or a simple cubic configuration. Considerations of energy and atomic volume render the former the more likely arrangement.
On the integrals of a partial differential equation
Authors: Friedlander F. G.
Published: November 2008

The ordinary one-dimensional wave equation

has special integrals of the form

which satisfy the first-order equations

respectively, and are often called progressive waves, or progressive integrals, of (1·1). The straight lines

in an xt-plane are the characteristics of (1·1). It follows from (1·2) that progressive integrals of (1·1) are constant on some particular characteristic, and are characterized by this property.
A theoretical determination of the elastic constants of ice
Authors: Penny Angela H. A.
Published: November 2008

A simplified model of the ice crystal, equivalent to Barnes's, was used. By applying the symmetry operations of this model to the dynamical matrix, it was made to depend on six arbitrary constants. By assuming that the tetrahedra of oxygen atoms which form the lattice are regular, and hence applying a further symmetry transformation to a smaller unit of the crystal, the number of arbitrary constants was reduced to two. The elastic constants were then found in terms of these two atomic constants. Two of the experimental measurements of the elastic constants of polycrystalline (quasiamorphous) ice were used to calculate the atomic constants and hence the elastic constants for single crystals.
Three-dimensional stress distributions in hexagonal aeolotropic crystals
Authors: Elliott H. A. , Mott N. F.
Published: November 2008

The conditions for equilibrium in an elastically stressed hexagonal aeolotropic medium (transversely isotropic) are formulated, and solutions are found in terms of two ‘harmonic’ functions ø1, ø2, which are solutions of

ν1, ν2 being the roots of a certain quadratic equation.

It is also shown that in the case of axially symmetrical stress systems the solution may be expressed in terms of the third-order differential coefficients of a single stress function Φ.

The solutions for an isotropic medium may be deduced as a special case.

The problems of nuclei of strain in such a hexagonal solid are solved, and the results for zinc and magnesium contrasted with those for an isotropic solid.
Collisions through liquid films
Authors: Eirich F. R. , Tabor D.
Published: November 2008

When impact occurs between clean metal surfaces, plastic flow of the metal usually takes place at the points of real contact, so that the pressures developed are as high as the dynamic yield pressure of the metal concerned. Early experiment show that if the surfaces are covered with a thin film of a highly viscous liquid, the pressures developed and transmitted through the liquid film may be sufficiently great to produce plastic deformation of the metal, even though no metallic contact occurs through the film (1). The existence of these high pressures in the liquid layer means that extremely high rates of flow and shear may be developed in the liquid film, and that the energy dissipated in overcoming viscous flow may lead to an appreciable temperature rise in the liquid. Even in much gentler impacts, where plastic deformation of the metal surfaces does not occur, very high pressures, rates of flow and shear, etc., may be developed in the liquid film. These effects are of great interest in any study of collisions through liquids; they are of particular significance in the study of the mechanism of detonation of liquid explosives by impact.
The number of individuals in a cascade process
Authors: Good I. J.
Published: November 2008

A number of important Markoff processes, with a continuous time parameter, can be represented approximately by a discrete process, interesting in its own right, of the following type. A class of individuals gives rise seasonally (in January say) to a number of new individuals (children), the probabilities of an individual having 0, 1, 2, … children being p0, p1, p2, …. These probabilities are the same for all individuals and are independent. The individuals formed each January are regarded as a new generation, and only this generation is capable of reproducing in the next January. Let

so that F(x) is the probability generating function (p.g.f.) of the number of children of an individual. Clearly the series for F(x) is absolutely convergent when |x| < |1.
The theory of counting experiments using pulsed sources: chance coincidences and counting-rate losses
Authors: Feather N.
Published: November 2008

The use of artificially pulsed sources of particles in experiments in nuclear physics has already given rise to powerful methods of investigating short-period phenomena and of studying the effects of ‘slow’ neutrons of different energies. On the other hand, the use of such sources introduces complications in counting experiments which are not present when steady sources are employed and it is possible to assume that the time distribution of ionizing events in any counter is a strictly random one. Again, sources which are effectively steady in respect of one counting arrangement are not necessarily so in respect of all. Ordinary cyclotron sources (i.e. cyclotron sources which are not ‘artificially’ pulsed) belong to this class; they have justifiably been considered as steady in the majority of experiments hitherto performed, but the flux of primary particles reaching the target is definitely periodic in time (with the period of the R.F. Voltage applied to the dees), thus a cyclotron will appear as ‘naturally’ pulsed whenever it is used with a counting system for which the effective resolving time is sufficiently short (say < 10−7 sec.).
Classical statistical mechanics of a rectilinear assembly
Authors: Gürsey Feza
Published: November 2008

It is shown that all the classical thermodynamical properties and the equation of state of a rectilinear assembly of spherical molecules with short-range attractive forces acting only between neighbours can be derived from a single partition function which is an analytic function of the pressure and the temperature. The thermodynamical functions are simply related to the Laplace transform of the function e−E(x)/(kT), where E(x) is the interaction energy between two neighbouring molecules. Although this linear model behaves like a perfect gas at high temperatures and like a crystal near the absolute zero, there are no critical phenomena, owing to the analytic character of the partition function. It is, however, pointed out that at very low temperatures the slope of the isothermal curves can suffer extremely sharp changes which may be interpreted as changes of phase, and this point is illustrated by an example.

Finally, I should like to express my sincere gratitude to Prof. H. Jones for his guidance of this work and for his invaluable advice throughout. My thanks are also due to the Turkish Ministry of Education for the award of a scholarship.
Some metrical theorems in Diophantine approximation. I
Authors: Cassels J. W. S.
Published: November 2008

In this paper we are interested in statements about Diophantine approximation which are ‘almost always’ or ‘almost never’ true. Let

be s non-negative functions of the positive integer n, and let
A note on a singular integral equation
Authors: Heins Albert E.
Published: November 2008

In his treatise entitled Introduction to the theory of Fourier integrals, E. C. Titchmarsh derives a Fourier transform of the solution of the integral equation

The transform of the solution given by him does not satisfy the Riemann-Lebesgue lemma, and further suffers from the defect that it has certain exponential properties which prevent the actual calculation of the function φ(ξ), since the inversion integral does not exist for all ξ > 0. We shall correct these defects and show further that the solution of (1) is particularly simple and that it can be expressed as the Legendre function of an order which depends on λ.
Note on the motion of liquid near a position of separation
Authors: Dean W. R.
Published: November 2008

1. A steady motion of viscous liquid of constant density is considered in §§ 2–7; full allowance is made for the inertia of the motion, and it is assumed that the stream-function can be expanded in a double series in the coordinates x, y.
Some solutions of the equations of flow of a viscous compressible fluid
Authors: Illingworth C. R.
Published: November 2008

The exact (Navier-Stokes) equations for the flow of a viscous compressible fluid are examined in a search for simple solutions, in which the equations reduce to ordinary differential equations. Such solutions are found for the uniform shearing motion in the Couette flow between both parallel flat plates and coaxial circular cylinders in relative motion. For a compressible fluid there are no simple solutions corresponding to certain other flows (such as the Poiseuille flow between fixed parallel flat plates) described by simple solutions for an incompressible fluid. A solution is found for the flow of a gas over an infinite porous plate with uniform suction through the plate (gravity neglected), and the relation of the solution to asymptotic solutions of the boundary-layer equations is discussed. Ordinary differential equations are obtained for the problems of (i) the flow due to the rotation of a horizontal disk in a gas (dissipation neglected), and (ii) the steady circulatory flow round a porous circular cylinder with uniform suction (gravity neglected). Discussion of the gas-dynamical problems is preceded by a description of the state of a horizontally stratified gas in static and thermal equilibrium.
The lateral diffusion of a cloud of falling droplets
Authors: Davies D. R.
Published: November 2008

This paper is an application of O. G. Sutton's theory of eddy diffusion to a case for which it was not originally developed; the theory is used to predict the dispersion of a cloud of falling droplets released from a point source at great height above the surface of the earth. Some experimental and theoretical values are given for the widths of ground area covered by liquid drops released from sources situated at heights between 1000 and 5000 ft., and agreement between theory and experiment is seen to be quite reasonable. This paper therefore provides further support for the use of O. G. Sutton's theory in problems of this type. The theory has practical applications, e.g. in the spraying of crops from aeroplanes.
A process for the step-by-step integration of differential equations in an automatic digital computing machine
Authors: Gill S.
Published: November 2008

It is advantageous in automatic computers to employ methods of integration which do not require preceding function values to be known. From a general theory given by Kutta, one such process is chosen giving fourth-order accuracy and requiring the minimum number of storage registers. It is developed into a form which gives the highest attainable accuracy and can be carried out by comparatively few instructions. The errors are studied and a simple example is given.
Notes on problems in hexagonal aeolotropic materials
Authors: Shield R. T.
Published: November 2008

Three-dimensional stress distributions in hexagonal aeolotropic materials have recently been considered by Elliott(1, 2), who obtained a general solution of the elastic equations of equilibrium in terms of two ‘harmonic’ functions, or, in the case of axially symmetric stress distributions, in terms of a single stress function. These stress functions are analogous to the stress functions employed to define stress systems in isotropic materials, and in the present note further problems in hexagonal aeolotropic media are solved, the method in each case being similar to that used for the corresponding problem in isotropic materials. Because of this similarity detailed explanations are unnecessary and only the essential steps in the working are given below.
Multiplicative norms on Banach algebras
Authors: Edwards R. E.
Published: November 2008

1. Mazur(1) has shown that any normed algebra A over the real field in which the norm is multiplicative in the sense that

is equivalent (i.e. algebraically isomorphic and isometric under one and the same mapping) to one of the following algebras: (i) the real numbers, (ii) the complex numbers, (iii) the real quaternions, each of these sets being regarded as normed algebras over the real field. Completeness of A is not assumed by Mazur. A relevant discussion is given also in Lorch (2).
On the use of convected coordinate systems in the mechanics of continuous media
Authors: Lodge A. S.
Published: November 2008

The use of a coordinate system convected with the moving medium for describing its mechanics, first proposed by Hencky (5), has since been extended by several authors, and has several advantages over the more conventional use of a coordinate system fixed in space; Brillouin(1) has shown that the relation between the strain-energy function for an ideally elastic solid and the stress tensor takes a very simple form when the latter is referred to a convected coordinate system; Oldroyd(8) has given a very general discussion of the formulation of rheological equations of state and has shown that the right invariance properties are most readily recognized when the equations are referred to a convected coordinate system; Green and Zerna (4) have similarly expressed the equations of motion and boundary conditions; and Gleyzal (2), and Green and Shield (3) have applied the formalism to certain problems in elasticity theory.
The solution of algebraic equations on the EDSAC
Authors: Brooker R. A.
Published: November 2008

This paper is an account of the methods that have been used with the EDSAC for the solution of algebraic equations. Three repetitive or iterative methods are examined: Bernoulli's method, the root-squaring method, and the Newton-Raphson method. Experience with the EDSAC has shown that, as in hand computing, quadratically convergent methods are to be preferred to those less rapidly convergent. In particular, the Newton-Raphson method has proved the most useful. Several examples are given in the appendix.
A note on the sequential estimation of means
Authors: Cox D. R.
Published: November 2008

A method is given for finding, at the end of a sequential sampling procedure, a nearly unbiased estimate of the population mean.
Generalized magneto-hydrodynamic formulae
Authors: Hines C. O. , Bondi H.
Published: November 2008

The formulae for current and force which were employed in some of the initial work on magneto-hydrodynamic theory are not valid over the whole range of phenomena which the subject itself embraces. Generalized formulae, suitable for wider application, are derived by an extension of the magneto-ionic approach, and are applied to the simple case of plane-wave propagation in the direction of a primary magnetic field. The dispersion equation which results, as well as the basic current and force formulae, are reduced by approximations in a variety of limiting circumstances. Some earlier results are thereby regained and, in particular, the region of known validity of Alfvén's propagation relation is extended.
On the flow between two rotating coaxial disks
Authors: Stewartson K.
Published: November 2008

The steady motion of a viscous fluid confined between two coaxial rotating disks is discussed both experimentally and theoretically. It is found experimentally that when the disks rotate in the same sense the main body of the fluid rotates as well, but if they rotate in opposite senses the main body of the fluid is almost at rest. An adequate theory is found to explain this.
Two theorems on Engel groups
Authors: Gruenberg K. W.
Published: November 2008

Let G be a group and let us write [x, y] = x−1y−1xy. We shall say that G is an Engel group, or that it satisfies the Engel condition, if to any pair of elements x, y in G there can be assigned an integer k (which is allowed to depend on x, y) such that
The codifying of tree-structure
Authors: Neville E. H.
Published: November 2008

1. The mathematician's tree is a collection of pairs of elements, conveniently called points. A pair of points is a link, and a collection of links is a tree if any two of the points from which the links are formed can be connected by a chain, and if there are no closed chains. A tree is determined if there is provided a complete schedule of the links composing the tree. Conversely, no description of a tree can be adequate unless a complete schedule is recoverable from the description. An arbitrary schedule of links does not usually describe a tree, and the object of this note is to give canonical schedules for the most general finite tree, and to specify these schedules as economically as possible. The fundamental result established is that the structure of a tree with n points can be encoded in a row of n − 2 symbols each of which may stand for any one of the points. Cayley's theorem, that the number of different trees with n given points is nn−2, is an immediate corollary.
On the energy scattered from the interaction of turbulence with sound or shock waves
Authors: Lighthill M. J.
Published: November 2008

The energy scattered when a sound wave passes through turbulent fluid flow is studied by means of the author's general theory of sound generated aerodynamically. The energy scattered per unit time from unit volume of turbulence is estimated (§3) as

where I is the intensity and ∧ the wave-length of the incident sound, and is the mean square velocity and L1 the macro-scale of the turbulence in the direction of the incident sound. This formula does not assume any particular kind of turbulence, but does assume that ∧/L1 is less than about 1. For turbulence which is isotropic and homogeneous, the energy scattered, and its directional distribution, are obtained for arbitrary values of ∧/L1. It is predicted that components of the turbulence with wave-number κ will scatter sound of wave-number K at an angle 2 sin−1 (k/2k). The statistics of multiple successive scatterings is considered (§4), and it is predicted that sound of wave-length less than the micro-scale λ of the turbulence will become uniform (i.e. quite random) in its directional distribution in a distance approximately .

The theory is extended (§5) to the case of an incident acoustic pulse. However, this extended theory cannot be applied directly to the case of a shock wave, for which it would predict infinite scattered energy. This is due to the perfect resonance between successive rays emitted forwards which would occur if the shock wave were propagated at the speed of sound. By taking into account (§6) the true speed of the shock wave (subsonic relative to the fluid behind it), the theory is improved to give a finite value, 0·8s tunes the kinetic energy of the turbulence traversed by a weak shock of strength s, for the total energy scattered. However, the greater part of this energy catches up with the shock wave, and probably is mostly reabsorbed by it, and only the remainder (tabulated as a function of s in Table 1) is freely scattered, behind the shock wave, as sound. The energy thus freely scattered when turbulence is convected through the stationary shock-wave pattern in a supersonic jet may form an important part of the sound field of the jet.
On multiplicative systems defined by generators and relations
Authors: Evans Trevor
Published: November 2008

The techniques developed in (9) are used here to study the properties of multiplicative systems generated by one element (monogenie systems). The results are of two kinds. First, we obtain fairly complete information about the automorphisms and endo-morphisms of free and finitely related loops. The automorphism group of the free monogenie loop is the infinite cyclic group, each automorphism being obtained by mapping the generator on one of its repeated inverses. A monogenie loop with a finite, non-empty set of relations has only a finite number of endomorphisms. These are obtained by mapping the generator on some of the components, or their repeated inverses, occurring in the relations. We use the same methods to solve the isomorphism problem for monogenie loops, i.e. we give a method for determining whether two finitely related monogenie loops are isomorphic. The decision method consists essentially of constructing all homomorphisms between two given finitely related monogenie loops.
Fixed points on rotating continua
Authors: Reifenberg E. R.
Published: November 2008

Suppose I is a bounded plane continuum whose complement is a single domain (I) and that is a (1–1) bicontinuous transformation of the plane onto itself which leaves I invariant. Cartwright and Littlewood(1) have proved the following theorem:

Theorem A. Suppose that(I) has no prime end fixed underand the frontier of I contains a fixed point P. Ifis any prime end of(I) containing P, and C is any curve in(I) converging tosuch that PeC¯, the closure of C, then for some‡ integer N the continuum I(FN) consisting of

together with the sum B(FN) of its interior complementary domains contains all fixed points in I.
Real inequalities with applications to function theory
Authors: Hayman W. K. , Stewart F. M.
Published: November 2008

Suppose that f(x) is non-negative for x ≥ 0 and let

In a recent paper (4) inequalities were proved relating fn(x) with the nth derivative f(n)(x) of f(x). However, the two main results, Theorems 3 and 4, were proved only for n = 1 and 2. In the first part of this paper we shall prove the corresponding results for all positive integral n under a slightly less restrictive hypothesis (Theorem 3). We shall then give an application showing that from the hypothesis f(x) ≤ μ(x) for all x it is sometimes possible to deduce f(n)(x) = O{μ(n)(x)} for a set of values x depending only on μ(x).
On games with almost complete information
Authors: Birch B. J.
Published: November 2008

1. It is well known that a game with perfect information has an equilibrium-point of pure strategies; this was first proved for two-person games by Zermelo (9), and later extended to n-person games by Kuhn(3). More recently, Dalkey(1) and Otter and Dunne (8) have published the stronger result (Theorem 6 below): If in the complete inflation of a game Γ every player has complete information about every other player, then Γ has an equilibrium-point of pure strategies.
Sur les algèbres dérivées d'une algèbre de Lie
Authors: Dixmier Par J.
Published: November 2008

Introduction. Dans cet article, K désigne un corps commutatif quelconque. Toutes les algèbres de Lie considérées sont des algèbres de Lie de dimension finie sur K. Si g est une algèbre de Lie, on désigne par (D0g, D1g, D2g, …) la suite des algèbres dérivées de g (D0g = g; Dig = [Di−1g, Di−1g]). Si h est un idéal de g, et si x ↦ g, on désigne par adh, x l'application y → [x, y] de h dans h.
On the use of asymptotic approximations of Green's type when the coefficient has zeros
Authors: Jeffreys Harold
Published: November 2008

Allowance is made for the error terms in the cases where χ has a simple zero at x = 0, or two simple zeros at a; = ± 1 (with χ either positive or negative for − 1 < x < 1). The errors of the approximations given by neglecting the error terms are found to be small in all cases, sometimes much smaller than there was any previous reason to expect.
The final period of decay of non-homogeneous turbulence
Authors: Phillips O. M.
Published: November 2008

In this paper, a study is made of the properties of the motion due to a general localized disturbance in an infinite fluid initially at rest. Two invariants of the motion are found, which are identified with the net linear momentum, or effective dipole strength, and the net angular momentum imparted to the fluid, and an equation is derived expressing the rate of change of the effective quadrupole strength of the motion. Solutions are obtained for the final period of decay of such a motion, and it is shown that, when the net linear momentum of the fluid is non-zero, the motion corresponds to a type of viscous vortex ring. The length scale increases with time t as [ν(t – t0)]½, and the total energy decays as [v(t–t0)]-1, where ν is the kinematic viscosity and t0 is a virtual time origin. If the net momentum is zero, the total energy decays as , and expressions are given for the velocity field in this case also.

It is shown that, if it is attained, the final period of any turbulent motion can be found from the vortex ring type of solution described above. The final period of homogeneous turbulence is clearly one particular example of such a motion. The final period of an axially symmetrical turbulent wake is treated in this way and is shown to consist generally of the superposition of two types of motion: (i) a mean streaming motion whose energy per unit length of the wake decreases as [ν(t – t0)]-1, and (ii) a superimposed fluctuating motion whose energy per unit length decays as [v(t–t0)]-1. Expressions are given for the distribution of mean-square vorticity and the energy spectrum tensor in the wake.
On Taylor's theorem
Authors: Grimshaw M. E.
Published: November 2008

In the expression of Taylor's theorem

there are three well-known forms of the remainder R(n, a, b). The first is the Schlömilch form
General principles of antithetic variates
Authors: Hammersley J. M. , Mauldon J. G.
Published: November 2008

1. In (2) the idea of antithetic variates in Monte Carlo work is introduced and applied to the estimation of integrals. The present paper studies the general theoretical structure of antithetic variates. We have made only limited progress and present various unsolved problems as a challenge to the reader.
Renormalization in meson theories
Authors: Taylor J. C.
Published: November 2008

By a generalization of methods recently applied in electrodynamics, it is proved that the ‘bare-particle’ coupling constant in renormalizable meson theories becomes imaginary at infinite cut-off. Analytical assumptions made are discussed.
Bending of a circular plate with an eccentric circular patch symmetrically loaded with respect to its centre
Authors: Bassali W. A. , Dawoud R. H.
Published: November 2008

The complex variable method is applied to obtain solutions for the deflexion of a supported circular plate with uniform line loading along an eccentric circle under a general boundary condition including the clamped boundary , a boundary with zero peripheral couple , a boundary with equal boundary cross-couples , a hinged boundary and a boundary for which , η being Poisson's ratio. These solutions are used to obtain the deflexion at any point of a circular plate having an eccentric circular patch symmetrically loaded with respect to its centre. Expressions for the slope and cross-couples over the boundary and the deflexions at the centres of the plate and the loaded patch are obtained.
On the Sturm-Liouville equation with two-point boundary conditions
Authors: Douglas A. S.
Published: November 2008

In numerical solution of a Sturm-Liouville system, it is necessary to determine an eigenvalue by a method of successive approximation. A relation is derived between the estimated accuracy of an approximate eigenvalue and the accuracy at every point of its corresponding eigenfunction. A method is also described whereby the correction to a trial eigenvalue, required for convergence to its true value, can be automatically determined. This method has been successfully used in solving radial wave-function equations, both with and without ‘exchange’, arising from the Hartree-Slater-Fock analysis of Schrödinger's equation.