bluuebunny/crossref_metadata_2025 huggingface dataset:
A Microcalorimeter
Authors: Ward A. F. H.
Published: November 2008

A microcalorimeter is described accurate to 0·0005 cal. The elevation of temperature was measured by a series of iron-constantan thermo-couples, with one set of junctions making good thermal contact with the tube-where the heat was liberated and the others in a brass ring outside, kept in a thermostat. They were connected in series with a very sensitive moving coil galvanometer. The Tian multiple walled thermostat was used—three concentric copper cylinders insulated with kapok, the inner filled with water and the outer controlled by a mercury regulator. The constancy of temperature in the inner vessel was rather better than 1/500,000° C.

Continuous heat evolutions could also be measured by passing a current through another set of thermo-couples, when the Peltier cooling compensated for the heating and the temperature was kept down to its original value, thus avoiding cooling corrections.
A Method of Generalising the Law of Mass Action for Heterogeneous Surface Reactions
Authors: Constable F. Hurn
Published: November 2008

To the conditions of reaction in homogeneous systems is added yet another condition, the adsorption of the reactants on the centres of activity of the surface. The surface is treated as if it were homogeneous, reaction taking place as if those centres on which the heat of activation is smallest were alone responsible for chemical change.

The general equation is worked out in terms of the rate of bombardment, and the mean lives of the molecules on the surface. It is incidentally shown that the same areas associated with the forward reaction must inevitablycatalyse the backward reaction.

The general solution is impracticable, so the special cases of irreversible synthesis and decomposition are considered and results are obtained.
Relation between Mean Stopping Power and Mean Range of β-rays.
Authors: Williams E. J.
Published: November 2008

Wilson's cloud method, provides a means of measuring the ranges of β-rays which is free from any uncertainties arising from the scattering of the rays, and for this reason it is an invaluable method for obtaining the true rate of loss of energy of β-rays as they traverse matter. Thetrue rate of loss of energy or ‘stopping power’ is defined as the quantity , where is the mean energy lost by β-rays of energy T in travelling a distance dx measured along their paths. The cloud method of course enables one to observe only the total ranges of β-particles, and in deducing the stopping power from the mean range it is necessary to bear in mind that owing to ‘straggling,’ which in the case of β-rays is large, the differential coefficient , where is the mean range of particles of energy T, is not exactly the same as the stopping power . is however the quantity which the cloud method enables us to observe. It is the purpose of this note to consider the exact relation between these two quantities so that the information provided by the cloud method may be rigorously interpreted.
Surface Oscillations of Water in a Rotating Cylindrical Vessel
Authors: Street R. O.
Published: November 2008

This paper is devoted chiefly to the consideration of the surface oscillations of water contained in a vessel in the shape of a circular cylinder with its axis vertical, when the motion is slightly disturbed from a uniform rotation about the axis of the vessel. The work was undertaken with the hope of finding some indication of the effect of the depth of the water in the vessel on the period of the surface waves, and for the purpose a vessel of circular cross-section was naturally chosen. It is shown that a slight change of shape does not affect the periods of the oscillations. The solution of the corresponding problem when the surface oscillations take the form of “long waves” or “tidal waves” is well known, and the present paper deals only with “short waves,” for which the horizontal velocity is not the same at all depths.
The Petersen-Morley theorem, II
Authors: Frith R.
Published: November 2008

In a paper by Mr Lyons and myself an elementary proof of the following theorem was given:

Given three lines a, b, c in space, if a′ is the line of shortest distance of b and c, b′ that of c and a, and c′ that of a and b; and if x is the line of shortest distance of a and a′, y that of b and b′, and z that of c and c′, then x, y, z have a common line of shortest distance.
Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions
Authors: Rankin R. A.
Published: November 2008

Suppose that is an integral modular form of dimensions − κ, where κ > 0, and Stufe N, which vanishes at all the rational cusps of the fundamental region, and which is absolutely convergent for The purpose of this note is to prove that

The notation employed is that of my second paper under the same general title* I refer to this paper as II.
An application of the method of coincidence counting in a β-particle spectrograph of novel design
Authors: Feather N.
Published: November 2008

First-order calculations are given concerning the possibility of designing a β-particle spectrograph in which the correlation between the photoelectrons of the stronger “natural β-ray lines” and the disintegration particles of the “continuous spectrum” might be investigated. It has been shown that, with a coincidence counting circuit of resolving time 10−6 sec., and with a source of suitable strength, this investigation should be possible in the case of any natural β-ray line the absolute intensity of which is sensibly greater than 0.01. A successful investigation of this kind would provide important data for the analysis of a complex continuous spectrum into the “partial spectra” corresponding to different modes of transformation (and might furnish direct information regarding the internal conversion coefficients appropriate to the stronger γ-radiations involved).
The adsorption of dipoles
Authors: Miller A. R.
Published: November 2008

The following correction has to be made to my recent paper on the adsorption of dipoles. Equation (10) and the sentence immediately preceding it are to be replaced by the following paragraph:

In the polynomial expressions for f1(x) and f2(x) the coefficients of the powers of x depend on η2 and η3 which are functions of T.
The difference between consecutive prime numbers. II
Authors: Rankin R. A.
Published: November 2008

In a previous paper, under the same title, I considered the problem of how far apart two consecutive primes can be. The present paper is concerned with the opposite question. How near together can large primes lie? The published literature on this subject is scanty and, though interesting, is mainly negative in character. It appears to be very difficult to give any answer that is not trivial, or that is at all illuminating.
On the accretion of interstellar matter by stars
Authors: Hoyle F. , Lyttleton R. A.
Published: November 2008

The rate of accretion of interstellar matter by stars as proposed in a previous paper is further discussed. It is shown that this amount, while sufficient for the evolution of the majority of stars, is insufficient by a factor of the order of 10 or more to give a satisfactory description of the general evolution of massive stars and close binary systems of small mass. Consideration of the possibility of increasing the rate of accretion for such exceptional stars leads to the conclusion that this can be carried out satisfactorily only by a corresponding increase in the density of the cloud. Although we were led to this view by considering all the factors involved in accretion and showing that only a change in the density could possibly produce the required increase, it is at once clear from the accretion formula, without detailed discussion of the other quantities involved, that the density is the only factor through which effects could be introduced that do not apply to all stars quite generally. By investigating the various factors in the galaxy affecting the density, it is shown that within 100 parsecs of the galactic plane, and also in local regions, the density may rise above 10−21 g. per c.c., which gives an increase of order 100 times the normal rate for stars lying in these regions. These suggestions receive independent corroboration from investigations by Jeans relating to extra-galactic nebulae which led to average densities also of order 10−21 g. per c.c., while a further argument from geological evidence shows that the average density of material along the sun's track must be higher than 10−21 g. per c.c. It remains to be seen whether future observations will succeed in confirming these suggestions indicated by the requirements of this theory of stellar evolution.
Region formation in the ionosphere according to an attachment theory
Authors: Wilkes M. V.
Published: November 2008

The paper contains some ionization-height curves for an isothermal ionized atmosphere in which the mechanism of electron disappearance is attachment to oxygen atoms at a rate proportional to the product of the number of oxygen atoms and of electrons present. Chapman's formula for the rate of production of ions is used. The curves were obtained by means of a differential analyser.
Stratified systems of logic
Authors: Newman M. H. A.
Published: November 2008

The suffixes used in logic to indicate differences of type may be regarded either as belonging to the formalism itself, or as being part of the machinery for deciding which rows of symbols (without suffixes) are to be admitted as significant. The two different attitudes do not necessarily lead to different formalisms, but when types are regarded as only one way of regulating the calculus it is natural to consider other possible ways, in particular the direct characterization of the significant formulae. Direct criteria for stratification were given by Quine, in his ‘New Foundations for Mathematical Logic’ (7). In the corresponding typed form of this theory ordinary integers are adequate as type-suffixes, and the direct description is correspondingly simple, but in other theories, including that recently proposed by Church(4), a partially ordered set of types must be used. In the present paper criteria, equivalent to the existence of a correct typing, are given for a general class of formalisms, which includes Church's system, several systems proposed by Quine, and (with some slight modifications, given in the last paragraph) Principia Mathematica. (The discussion has been given this general form rather with a view to clarity than to comprehensiveness.)
Momentum distribution in molecular systems
Authors: Duncanson W. E. , Coulson C. A.
Published: November 2008

Theoretical calculations on the mean radial momentum distribution of the electrons are made for the molecules CH4, C2H6, C2H4 and C2H2; from these distributions the profiles of the Compton line are deduced. It is assumed that each electron acts as a single scattering centre so that the mean radial distribution for the molecule is just the sum of the various “partial distributions” for each electron separately. The electrons are supposed to be paired together in the formation of localized bonds, and the contributions from each type of bond have to be separately determined. When superposed, these give the momentum distribution for the whole molecule. Such distributions are similar in shape, but the peak value of the momentum curves moves to higher values of p as the C—C bond becomes more saturated.

A comparison of the various partial distributions for C2H4 shows that the C—H bond is an important factor in the momentum distribution of such molecules. In other hydrocarbon molecules, we may presume that the proportion of C—H bonds will be very significant in determining the breadth of the momentum distribution curve, and hence of the width of the Compton line; in fact the half-width of the Compton line decreases both with an increase in the number of the C—H bonds, and with increased saturation of the C—C bonds.

In the only case where experimental results are available, CH4, the discrepancy between theory and experiment amounts to about 30%. Reasons for this discrepancy are discussed; in part the discrepancy may be attributed to the approximations used in the wave functions; but reasons are given for supposing that in part also it arises from an incorrect interpretation of the experimental results.
A theorem on finite differences with an application to the theory of Hausdorff summability
Authors: Fuchs W. H. J.
Published: November 2008

1. It is trivial that

for all integers n ≥ 0 implies sn = 0 (n = 0, 1, 2, …). This conclusion is no longer true if it is only known that (1·1) is true for an infinity of n. But we shall show that the truth of (1·1) for, roughly speaking, one-half of all positive integers, together with an order condition on the magnitude of sn, ensures sn = 0 for all n. This follows from

Theorem 1. Let a1 < a2 < … be positive integers, n(R) the number of a's not exceeding R, sn a sequence of complex numbers. If

(ii) sn = ο(nK)as n tends to infinity

then sn = P(n), where P(x) is a polynomial of degree less than K.
On a class of integro-differential equations
Authors: Pitt H. R.
Published: November 2008

We are concerned in this paper with linear integro-differential equations of the form*

and

where kr(y) are given functions with bounded variation in any finite interval, g(x) is known, and f(x) is to be determined.
On the Symmetries of the Solutions of a Certain Variational Problem
Authors: Weinstein A.
Published: November 2008

The object of this paper is to make some remarks about a minimal problem suggested by the theory of the buckling of plates. A question left open in my previous paper on this subject is thereby solved.
The intensity of X-ray spectrum lines of heavy elements
Authors: Massey H. S. W. , Burhop E. H. S.
Published: November 2008

The intensities of X-ray spectral lines resulting from transitions between the K, L and M shells have been calculated using screened relativistic wave functions. The relativistic modifications are shown to decrease the Lβ3β4 doublet ratio but do not alter the Kα1α2 and Kβ1β3 ratios, in agreement with experiment. The calculated intensities of the forbidden lines arising from quadrupole transitions are in good agreement with experiment. Lines due to transitions arising from the magnetic dipole moment of the electron are found to be very weak, but the LI → K transition might just be observable for heavy elements.
Note on optical methods of measuring the size of small water drops
Authors: wilson John G.
Published: November 2008

The very small water drops which are considered in this note are as a rule met with in the form of clouds with a high numerical density of drops. The interior of such a cloud will never be in temperature equilibrium with the walls of a containing vessel, and under these conditions measurements of the rate of fall of the top of a cloud are not measures of the Stokes free fall of a single drop. Optical methods of measurement assume, therefore, a particular importance. The angular diameter of the diffraction halo has long been used as a measure of drop size in clouds; the applicability of this method for small drops will be considered. The criterion of “Rayleigh scattering” has also been employed as showing that the diameter of the drops in question was very much smaller than the wave length of light. A third method which may be applied to drops of diameter of the order of one wave-length will be discussed.
Sufficient statistics and intrinsic accuracy
Authors: Pitman E. J. G.
Published: November 2008

It is proved that if there exists a sufficient statistic for the estimation of an unknown parameter of a population, the frequency function of the population must be of a certain type.

It is shown that some modification of previous theory of the intrinsic accuracy of statistics is necessary when the range of the population sampled is a function of the parameter to be estimated.

Finally, the theory is extended to sufficient sets of statistics, i.e. sets of statistics which together contain all the information provided by a sample about an unknown parameter.
The eigenvalues of electromagnetic angular momentum
Authors: Pryce M. H. L.
Published: November 2008

It is shown that the eigenvalues of the angular momentum of the electromagnetic field containing a number of charged particles, apart from spin angular momentum, are only the integral multiples of ħ. This is shown by using cylindrical polar variables, and taking a particular choice of the vector potential in which the radial component is zero, defined explicitly in terms of the magnetic field strengths. By expanding in terms of the Fourier functions einθ, the angular momentum is separated out into terms independent of one another, each taking on only integral values in units of ħ.

The arguments all apply equally well to a modified field theory such as that of Born and Infeld.
The envelope of the subspaces of the polyhedra of an involution on a rational curve
Authors: Baker H. F.
Published: November 2008

The following note was suggested by an interesting paper written by F. P. White, where many references are given. It refers to a theorem given by W. F. Meyer, by whom the proof is indicated as possible by generalization of an intricate analytical proof given by him for a simple case. His result is that if on the rational curve of orderr, in space [r], say the curvecr[r], there be an involution ∞k, of sets ofmpoints, expressed, suppose, by an equation

then the spaces [r− 1], formed fromrpoints of any one of the polyhedra ofmpoints, are an aggregate ∞kof primes of this space [r], which is of class (m−k,m−r), the notation (p, q) meaning the binomial coefficientp! /q!(p − q)!. By Meyer, the conditionsk<r≤mare supposed to be satisfied. But there is a theorem forr<k≤m− 1, relating toselected[r − 1], formed fromrpoints of any one of the polyhedra ofmpoints. The general theorem may be formulated thus: In a space [r], the equation of any prime may be expressed by λu+ μv+... + ρw= 0, whereu= 0,v= 0, …w= 0 are any r + 1 given independent primes, and λ, μ, …ρ are coefficients which may be described as prime coordinates of the [r].
Abstract definitions for the symmetry groups of the regular polytopes, in terms of two generators. Part II: the rotation groups
Authors: Coxeter H. S. M.
Published: November 2008

The complete (or “extended”) symmetry groups, investigated in Part I, are certain groups of orthogonal transformations, generated by reflections. Every such group has a subgroup of index two, consisting of those transformations which are of positive determinant (i.e., “movements” or “displacements”). The positive subgroup (in this sense) of [k1,k2, …,kn−1] is denoted by [k1,k2, …,kn−1]′, and is “the rotation group” (or, briefly, “the group”) of either of the regular polytopes {k1,k2, …,kn−1}, {kn−1,kn−2, …,k1}; e.g., [3, 4]′ is the octahedral group.
The theory of the single-pass cross-flow heat interchanger
Authors: Binnie A. M. , Poole E. G. C.
Published: November 2008

1. The cross-flow type of heat interchanger is shown diagrammatically in Fig. 1. One of the fluids involved is arranged to pass through a nest of small tubes, while the other streams past the tubes at right angles to them. To fix ideas we suppose that the tubes contain cooling water which enters at a temperature T1. The temperature of the water leaving a tube depends upon its position, but the water from all the tubes is mixed and is led to a single exit pipe in which the temperature is T2. In the same way the fluid to be cooled enters at a temperature t1 and leaves at a temperature t2. If the cooling water is not used again by being led through another nest of tubes also exposed to the fluid to be cooled, the arrangement is termed a single-pass heat interchanger.
Proof of a theorem of Paley
Authors: Zygmund A.
Published: November 2008

1. Let f(x) be a real function of period 2π, integrable L over (0, 2π), and let

By sn(x) and σn (x) we denote respectively the partial sums and the first arithmetic means of the series (1·1). Similarly, by and we denote the partial sums and the first arithmetic means of the series

conjugate to (1·1). By we mean the function conjugate to f(x), that is

where the integral is taken in the principal-value sense.
Concerning the velocity and temperature distributions in plane and axially symmetrical jets
Authors: Howarth L.
Published: November 2008

The calculations of the velocity and temperature distributions for the turbulent flow in axially symmetrical and plane jets are described.

For axially symmetrical jets Tollmien's results derived from calculations based on the momentum theory are stated; the calculations on the basis of the modified vorticity transfer theory and on the vorticity transfer theory with symmetrical turbulence are carried out. Unfortunately it seems to be true that none of these theories provides results in good agreement with experiment for both the velocity and temperature distributions.

For plane jets the results obtained by Tollmien and based on the momentum transfer theory are stated; the calculations on the basis of the vorticity transfer theory are carried out. The velocity distributions on both theories are identical and in fair agreement with experiment. In order to test the validity of the theories it is necessary to compare the theoretical results for the temperature with experiment. No results for the latter seem to have been published.
The disintegration of boron by slow neutrons
Authors: Bower J. C. , Bretscher E. , Gilbert C. W.
Published: November 2008

Photometrical measurements of the photographic images of cloud tracks of the disintegration of boron by slow neutrons,

have enabled the ranges of both the He and the Li particle to be determined. We find that the He particle has a range of 7·0 ± 0·3 mm. and the Li particle a range of 4·3 ± 0·2 mm. in standard air.

We deduce that the 7Li nucleus is usually formed in an excited state, with energy of excitation of either 0·5 or 0·8 M.e.V. The wide latitude in excitation energy is due to the uncertain state of our knowledge of the energy-range relation for α-particles.
Velocity and temperature distributions in the turbulent wake behind a heated body of revolution
Authors: Hall A. A. , Hislop G. S.
Published: November 2008

The distributions of velocity and temperature in the wake behind a heated body of revolution have been determined in a low-turbulence wind tunnel. Difficulty was experienced in obtaining a truly symmetrical wake, and observations have been reduced to mean values, curves of which are given.
The binding energies of the hydrogen isotopes
Authors: Wilson A. H.
Published: November 2008

The wave equation for the deuteron in its ground state is solved on the assumption that the mutual potential energy of a neutron and a proton is of the form r−1e−λr. The binding energy of the hydrogen isotope H3 is calculated approximately by the variation method.
Note on the contact between a metal and an insulator or semi-conductor
Authors: Mott N. F.
Published: November 2008

According to quantum mechanics there exists in any non-metallic crystal a band of allowed electronic energy levels which are unoccupied when the crystal is in its state of lowest energy. We call this band the conduction band; the crystal can conduct electricity if electrons are raised into the conduction level from lower levels. According to the theory of semi-conductors given by Wilson, there exist in these substances lattice imperfections at which an electron can exist in a bound stationary state below the conduction band, electrons being raised from these levels into the conduction band by the thermal agitation of the surrounding atoms.
The uncertainty of the reference frame in quantum mechanics
Authors: Corben H. C.
Published: November 2008

It is shown that, if account is taken of the uncertainty of position and momentum of the reference origin with respect to which observations are made, the self-energy and transverse self-energy of the electron become finite and the quantum theory is considerably modified for high-energy electrons and quanta.
On the stability of nuclei against β-emission
Authors: Fuchs K.
Published: November 2008

By means of purely qualitative arguments which do not depend on any particular model, the general scheme of stable nuclei and the isotopic breadth of nuclei with odd charge number are explained.

The breadth of the isobaric region can be obtained if the numerical values of certain energies are known. Though these can be estimated only very roughly, the values for the breadth of the isobaric region obtained in this way are in good agreement with the experimental values. The increase in the breadth of the isobaric region from light nuclei to heavier nuclei can be explained, but no plausible explanation has been found for the fact that the breadth decreases again for the heaviest nuclei.
The application of statistical methods to immobile adsorbed films
Authors: Roberts J. K. , Miller A. R.
Published: November 2008

Bethe's method is applied to determine statistically the rate of condensation of diatomic molecules which are adsorbed with dissociation to form an immobile film, the two atoms of one adsorbed molecule occupying two neighbouring sites. A lower limit to the number of gaps, or isolated unoccupied sites, in the final film is also obtained. The results calculated statistically agree with those obtained earlier by using models.
Electronic states at the surfaces of crystals
Authors: Goodwin E. T.
Published: November 2008

In the present paper application of the work of paper I is made to a specific problem, namely the excitation of an adsorbed atom by the surface electrons of the underlying metal. It is shown that the corresponding mean life-time in a typical excited level is 3·7 × 10−11 sec., which is comparable with the value 1·7 × 10−12 sec. for deactivation by the conduction electrons when the Brillouin zone is only partly filled. Thus, though this is an essentially surface phenomenon, the effect of the surface electrons is begligible. This suggests that their effect can, in general, only become appreciable in semi-conductors or insulators.
Momentum Distribution in Molecular Systems
Authors: Coulson C. A.
Published: November 2008

A general discussion is given of the momentum wave functions of simple molecules in which only single bonds occur; the momentum functions are obtained by transformation of the space wave functions, for which two distinct approximations are possible. In each case it appears that the presence of the bond decreases the mean component of the velocity in the direction of the bond, increases the mean component of velocity perpendicular to the bond, and increases the mean momentum averaged over all directions. The analysis is illustrated by consideration and H2; in the latter case the strong dependence upon the mutual orientation of the two electronic momenta is emphasized.
Momentum Distribution in Molecular Systems
Authors: Coulson C. A.
Published: November 2008

The discussion of momentum distribution is extended to cover electrons in double bonds and conjugated bonds of fractional order. The mean radial distribution function is obtained, and it shows that the mean momentum of the electrons is less than if they were rigidly attached to their own particular nuclei. Polar diagrams show that, if electronic mobility is high in any direction in a molecule, the momentum is generally small in that direction.
On the theory of the liquid state
Authors: Fürth R.
Published: November 2008

The theory of holes, developed in part I of this paper, leads to the existence of a metastable state of a liquid. It is identified with the superheated state, the main properties of which can be accounted for in this way. A theory of the supersaturation of gases in liquids is given on the same line.
Two-centre integrals occurring in the theory of molecular structure
Authors: Coulson C. A.
Published: November 2008

There are many types of problem in the study of molecular structure in which integrals involving two or more centres of force arise. We may mention the calculation of molecular energy levels, the Stark effect in molecules, the van der Waals forces between molecules, and, more recently, the internal arrangement of neutrons and protons in nuclei. During the last few years the writer has been accumulating a table of these integrals, and the explicit forms for many of them are given in this paper.
The energy levels of ‘holes’ in liquids
Authors: Auluck F. C. , Kothari D. S.
Published: November 2008

The Schrödinger equation for a ‘hole’ in the hole-theory of liquids is constructed and solved by the B.W.K. method. The energy levels are found to be discrete, and the eigenvalues are obtained in terms of the density and surface tension of the liquid. The relation between the energy of the ground state and the temperature of melting is considered.
Optical characteristics of a two-cylinder electrostatic lens
Authors: Goddard L. S.
Published: November 2008

In this paper formulae are developed for the first and second focal lengths, and the positions of the first and second principal planes of a type of electrostatic lens which has been the subject of study (mostly experimental) in several previous papers. The lens, which is commonly used in electron optical devices, lends itself to a theoretical study, although this does not appear to have been attempted before. It consists of two equal semi-infinite cylinders placed end to end so that their axes coincide and the ends are separated by a small gap. If the cylinders are at potentials V1 and V2 and we write σ = V2/V1, the system behaves as an electron lens when σ > 0 and as an electron mirror when σ < 0. In the latter case some experimental results have been given by Nicoll(1) who also studied the focusing action in the case σ > 0 and, in particular, the formation of intermediate images when σ ≪ 1 and when σ ≫ 1. But for the precise formulation of the relationship between σ and the number of cross-overs a theoretical study, based on the paraxial equation, would be necessary. The problem will be indicated below. An experimental determination of the lens characteristics for values of σ from about 2 to 15 and for several gap widths has been made by Spangenberg(2), whose results will be compared with those obtained in the present paper. The two-cylinder lens has also been studied by Klemperer and Wright(3) using an experimental and a numerical (trigonometrical) method, and some crude analytical results have been given by Gray(4).
Waves on the surface of a compressible liquid
Authors: Bondi H.
Published: November 2008

The possibility of the propagation of disturbances along a free surface with a velocity exceeding the velocity of sound in the liquid is examined. Simple necessary conditions for the occurrence of such ultrasonic propagation are found in terms of the boundary condition. It is shown that the conditions are not satisfied in the case of gravity waves, and a detailed examination is made of the pressure and surface waves due to a localized disturbance. Capillary waves are shown to satisfy the conditions for ultrasonic propagation, the existance of a ‘faster-than-sound’ wave is proved, and its main properties are examined.
On the theory of spinning particles
Authors: Shanmugadhasan S.
Published: November 2008

A classical theory of a spinning particle with charge and dipole moment in an electromagnetic field is obtained by working symmetrically with respect to retarded and advanced fields, and with respect to the ingoing and outgoing fields. The equations are in a simpler form than those of Bhabha and Corben or those of Bhabha, and involve fewer constants. On the assumption that the spin angular momentum tensor θμν satisfies the equation θ2 ≡ θμν θμν = constant, the value of the dipole moment Zμν is chosen to be Cθμν, where C is a constant. The theory is generalized to the case of several particles with charge and dipole moment. By using a suitable Hamiltonian equation, the classical equations of motion, obtained on the assumption that θ is a constant, are put into Hamiltonian form by means of the ‘Wentzel field’ and the λ-limiting process. The passage to the quantum theory is effected by the usual rules of quantization. The theory is extended to the case of particles with charge and dipole moment in the generalized wave field by defining the Wentzel potential in terms of the generalized relativistic δ-function.
On the algebra of the meson matrices
Authors: Harish-Chandra
Published: November 2008

It is shown that the algebra of the Duffin-Kemmer matrices can be handled quite neatly by introducing an extra dimension and by using the Γ-formalism of an earlier paper (3). By this means it is possible to derive identities similar to the De Broglie-Pauli identities for the Dirac matrices. To complete the analogy a Γ-formalism is also set up for the Dirac matrices.
A note on Hausdorff dimension functions
Authors: Dvoretzky A.
Published: November 2008

1. We recall the meaning attributed by F. Hausdorff to the statement: ‘The (linear) set S is of dimension [h(x)].’
A campanological problem in group theory
Authors: Rankin R. A.
Published: November 2008

1. The problem investigated in this note was suggested by a study of the mathematical basis of change ringing. Its campanological application is discussed in § 4.
The yield of 139Ba in the fission of natural uranium by fast and slow neutrons
Authors: Broda E. , Kowarski L. , West D.
Published: November 2008

The yields of 139Ba per fission in natural uranium by unslowed neutrons from a mixed radium-beryllium source and by C-group neutrons were compared. In each case the fission rates measured with an ionization chamber were correlated with the β-ray activity of 139Ba, chemically extracted from a mass of uranium oxide exposed to an identical neutron spectrum. The ratio of the two fission yields is found to be 0·84 ± 0·03. The possibility of using 139Ba as a local or integrating indicator of fission in extended uraniferous media, and of distinguishing between the contributions by fast neutrons and by slow neutrons is discussed.
Combinants of a pencil of quadric surfaces (IV)
Authors: Todd J. A.
Published: November 2008

In this paper we determine the complete system of combinantal line-complexes associated with a pencil of quadric surfaces. The method employed is that already used in (I) and (II) to obtain the complete systems of combinantal covariants and contravariants.
On one-dimensional regular assemblies
Authors: Rushbrooke G. S. , Ursell H. D.
Published: November 2008

The grand partition function of any statistical assembly may be defined by the equation

where E denotes any value of the energy of the assembly, k is Boltzmann's constant, T the thermodynamic absolute temperature and λi a parameter which is later to be connected with the chemical potential, μi, of the ith species in the assembly by the simple formula
The propagation of steady disturbances in a supersonic stream bounded on one side by a parallel subsonic stream
Authors: Howarth L.
Published: November 2008

The problem of the propagation of steady disturbance in a semi-infinite supersonic stream moving parallel to and bounded on one side by a subsonic stream is discussed and the complete solution obtained for any assumed form of incident disturbance. It is shown that a local initial disturbance generally produces a reflected wave extending to infinity both upstream and downstream of the initial disturbance. A (positive) pressure pulse as an incident wave produces pressure upstream and rarefaction downstream and the magnitude of the effects is calculated in a particular case. Conditions along the axis of symmetry are calculated for the subsonic region in this case.
Quantum mechanics of the anharmonic oscillator
Authors: McWeeny R. , Coulson C. A.
Published: November 2008

A method is presented for the accurate numerical treatment of molecular vibration problems in which the potential energy function is of the form

The treatment is carried through in detail for the case V(q) = Aq2 + Bq4, which, in most instances affords an adequate description of the potential. There is no restriction on the relative magnitudes of quadratic and quartic terms so that the method is equally applicable to the purely anharmonic oscillations occurring in types of ring bending (1), where V(q) = aq4.

An approximate formula is derived for the energy levels in the general case. The case V = aq4 affords an illustration of the accurate treatment; the first five eigen-values and eigen-functions are computed and from them the associated transitionprobabilities. Numerical results are presented in a form convenient for further application. On comparison the approximate formula is found to yield results in error by approximately 1 %, the error decreasing for the higher levels for which the formula tends to a B.W.K. expression. In conclusion a ground-state eigen-function of simple analytical form is found to approximate remarkably well to the case of purely quartic vibrations.
Certain integral representations for Whittaker functions
Authors: Shanker Hari
Published: November 2008

1. Shastri (1) has shown that if and then

But (Goldstein (2))

hence substituting from (1·2) in (1·1) and changing the order of integration, which we suppose to be permissible, we find that if

then

assuming of course that the integrals converge.
The thermodynamics of magnetization
Authors: Livens G. H.
Published: November 2008

Recent studies by Debye (1), Bloch (2), Stoner (3) and Guggenheim (4) expressing somewhat divergent views suggest that there is still some confusion and uncertainty in the method of application of the thermodynamic argument to magnetic phenomena. The first effective contribution to the subject was made by Larmor (5) over fifty years ago, and developing this further he finally proposed in 1929 a sufficient basis for a general theory of the whole range of phenomena. At about the same time (1927), and apparently in ignorance of Larmor's earlier work, Debye(1) also gave a brief outline of the thermodynamic aspects of the subject, starting, however, from much more tentative magnetic ideas and formulae which are now known to be without adequate physical justification. In 1933, Bloch (2) elaborated the argument still further, using, however, basic ideas more akin to those employed by Larmor, but derived by a very specialized argument. Then in 1935, using the same fundamental equation as Debye, Stoner revived the discussion with a systematic survey of the theoretical results bearing on the thermodynamic aspects of the subject, attempting later (6) to justify the basic magnetic ideas by a discussion which he now admits is not entirely satisfactory.
Some statistical properties of random noise
Authors: MacDonald D. K. C.
Published: November 2008

A considerable volume of knowledge is now available on random fluctuations (noise) as regards the behaviour in amplitude. Familiar names in this field are those of Uhlenbeck and Ornstein(7), Fürth(1) and Rice (4), although very many others have made valuable contributions. A particular class of problem, of considerable practical importance, exists when the frequency spectrum is limited to a relatively narrow range. The resulting noise has then the character of a more or less regular oscillation modulated randomly in amplitude and phase. In this case, if we write the fluctuation in the form

(where R(t) and θ(t) are variables changing slowly in comparison with sinω0t), it is clear that the magnitude of the envelope R(t) and the phase θ(t) are now the significant quantities. Rice (4), among others, has made a study of the statistical properties of R, deriving in particular the correlation function R(t) R(t + τ) in terms of the characteristics of the (power) spectrum ω(f). Fürth and the writer (2) have extended this work and carried out a collateral experimental investigation.
A note on podolsky electrodynamics
Authors: Matthews P. T.
Published: November 2008

Three formalisms have been considered for the ‘extraordinary’ field introduced by Podolsky in his generalized electrodynamics.

The c-representation (tildons) for negative energy particles (§ 3), leads to an unstable vacuum in electron hole theory and must be discarded (§ 7).

The d-representation suggested by Podolsky is a hole theory for bosons. This leads at once to inconsistencies and it is impossible to introduce a Fock representation (§ 6).

The least unsatisfactory is the b-representation (tilde photons) used by PKMG. Tilde photons can be created at high energies but decay immediately by pair creation. The introduction of tilde photons does not interfere with any agreement already established between experiment and ordinary quantum electrodynamics. Evidence might be obtained for the value of the constant a from consideration of the 2S level shift of the hydrogen atom and from the theory of showers where the deviation from the usual theory is greatest (§ 7).

However, the formalism involves the introduction of an indefinite metric for the tilde photon field and consequently ‘negative probabilities’ for states in which an odd number of tilde photons are present. Approximate results can be obtained by assuming that ‘negative probabilities’ in the theory are positive probabilities in the real world. This leads to a small but fundamental error and it does not seem possible that a theory in this form can have any deep seated validity (§ 5).

The same analysis applies to the extension of the Podolsky theory to the Proca field by Green (5). In order that the energy should be positive definite it is necessary to introduce an indefinite metric for one of the two superposed meson fields. Fundamentally the difficulties of interpretation are the same but in practice they will be more obvious because the coupling constants are greater.
The thermodynamics of magnetization
Authors: Guggenheim E. A.
Published: November 2008

A paper under the above title (1) by G. H. Livens was published recently in this journal. The main object of this paper appears to be to show that the treatment of the same subject (2) by myself is unsound. I shall not attempt to refute Professor Livens's opinions. His views concerning magnetic energy have already been criticized by others (3) more expert than myself. I merely wish to draw attention to two completely untrue statements of fact in Professor Livens's paper, namely the following:

(a) On p. 538 “Guggenheim writes therefore
Ionic hydration
Authors: Moelwyn-Hughes E. A.
Published: November 2008

In applying the general theory of intermolecular forces to the complicated problem of ionic solvation, the minimum requirements are an exact knowledge of (1) the force laws governing the interaction of two isolated solvent molecules as a function of their separation and the various angles of mutual inclination, (2) the force laws governing the interaction of an ion with one solvent molecule, also as a function of their distance apart and of their mutual inclinations, (3) the average disposition of solvent molecules in the pure liquid and (4) the disposition of the solvent molecules in the force field of the ion, particularly in its immediate vicinity. None of these theoretical requirements is available for systems of practical interest. The established laws of intermolecular force are those relating to spherically symmetrical fields, applied to gases (J. E. Lennard-Jones; see (1)), in which the molecular arrangements are random, and to crystals (M. Born (2)), in which the arrangements are orderly. In seeking to extend these laws to liquids and solutions, we are hampered at the start by our ignorance of directional forces in general and of the degree of disorder prevailing in condensed fluid systems. What is attempted here is a simplification of the problem of applying intermolecular force theory to ionic hydration by the introduction of assumptions which can claim some experimental warrant. How far the introduction of these assumptions may swamp other factors must remain an open question. At any rate, the attempt removes many anomalies, and allows of the absolute calculation of ionic dimensions in aqueous solution from intermolecular force constants determined without reference to the condensed states of matter.
The formation and growth of shock waves in the one-dimensional motion of a gas
Authors: Pillow A. F.
Published: November 2008

In many compressible fluid flow problems the classical solution breaks down completely owing to the formation of regions of infinite acceleration in the flow field. The actual behaviour of the fluid in such cases does not seem to have been investigated mathematically and this is largely due to the difficulties which enter with non-uniform shocks. It is with these difficulties that this paper is principally concerned.

The way in which discontinuities may arise mathematically in a flow field is first discussed. The equations governing the one-dimensional motion of a gas due to an accelerating piston are then set up. It is shown that when allowance is made for varying entropy conditions due to the presence of non-uniform shocks these differential equations reduce (outside the shock wave) to three first order quasilinear ones.

The initial solution breaks down when a point of infinite acceleration occurs in the flow field. From this point onwards a shock wave grows in the fluid and behind it three different sets of characteristics are required to describe the flow. By working in the plane of two quantities that are constant along two different sets of characteristics, we can use the shock-jump conditions to determine the equation of the shock-line in this plane and to reduce the equations of the characteristics to three differential equations, which would be linear if the relation between the entropy and other flow variables were known.

In the case of a constantly accelerating piston a first approximation is found by neglecting reflexions and entropy variations behind the shock. Using this as a basis we then find a second approximation for the entropy in the neighbourhood of the initial portion of the shock-line and show that the problem reduces to the solution of a second order linear partial differential equation. The introduction of a Riemann function and the satisfaction of the boundary conditions at the shock lead to an integral equation whose solution enables us to determine the position of the shock as a function of the time. The solution is in the form of a power series and is valid provided the shock wave does not become too strong.

Finally, it is shown that if the piston is given a constant terminal velocity a reflected wave from the shock is reflected again from the piston and eventually overtakes the shock and reduces its velocity to a final steady value which is in agreement with the value arising from an impulsive start.
Elastic stability of infinite strips
Authors: Hopkins H. G.
Published: November 2008

In this paper some elastic stability problems are solved for infinitely long, rectangular, isotropic plates, simply supported at the long edges. The problem is solved when the same arbitrarily prescribed distribution of normal stress is applied to each long edge, with no stresses and normal displacement at infinity. The extensional deformation of the middle surface during buckling is neglected; this leads to an underestimation (at most of the order of 10%) of the critical forces, but is unlikely to influence the general conclusions. A strain-energy method of analysis is used. The critical stress distributions are shown to be given by the eigenvalues of an ordinary differential equation, or, alternatively, of an integral equation. The general problem is then specialized to that of sets of concentrated normal forces, and to that of a normal force distributed uniformly over a finite length of each long edge; some numerical results are given for these cases. In the first of these problems there is negligible error in treating pairs of concentrated forces individually when they are separated by distances greater than twice the plate width. In the second problem there is little change in the critical stress when the length of edge stressed exceeds four times the plate width; also, if the length of edge stressed is less than one-quarter of the plate width, the distributed force required to produce instability does not differ appreciably from the concentrated force required to produce instability.