bluuebunny/crossref_metadata_2025 huggingface dataset:
Note on the Equilibrium of Black-Body Radiation
Authors: Heitler W.
Published: November 2008

Recent discoveries have shown that two light quanta with total energy not less than 2mc2 can create a pair of electrons (inverse process to the ordinary annihilation). Hence it follows that ordinary black-body radiation no longer represents an equilibrium state. In thermodynamic equilibrium a certain number of electron pairs must be present, the number of which will depend upon the temperature. It can be calculated as follows.
A Note on the Equations of Motion in Non-Relativistic Quantum Mechanics
Authors: Powell F. C.
Published: November 2008

It is well known that the motion of a dynamical system can be pictured in two distinct ways, which Dirac names the Heisenberg picture and the Schrödinger picture. The equations of motion take different forms in the two pictures, but of course have identical physical consequences, since the motion of the system does not depend in any way on which picture we choose. It should therefore be possible to express the equations in an invariant form (independent, that is, of the picture used). It will be shown in this note that not only can this be done (equation (3)), but it can be done in such a way that it is not even necessary to introduce a picture at all (equation (3′)).
The Relation between Refractive Index and Density
Authors: Burnett D.
Published: November 2008

Varying ideas of the mechanism of the propagation of light and of the constitution of matter have led at different times to different formulae for the dependence of the refractive index n of a medium on its density ρ, notably those of Laplace, , and of Lorenz and Lorentz, . Recently Macdonald has given a theory which seems more in accord with the modern concept of atomic structure and finds a formula for a non-magnetic medium. It has therefore seemed worth while to examine a number of experimental results to see whether this new formula gives as satisfactory agreement as that of Lorenz and Lorentz; it is found that it does in most cases, and is occasionally the better.
The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part IV. Further Results relating to Terms of the Optical Spectrum
Authors: Hartree D. R.
Published: November 2008

The purpose of this note is to extend in two respects some previous work of the writer on the wave equation of a particle in a central field of force which is a Coulomb field for sufficiently large r.
On Poincaré's Theorem of “The non-existence of Uniform Integrals of Dynamical Equations”
Authors: Cherry T. M.
Published: November 2008

In Chapter v of his Les Méthodes Nouvelles de la Mécanique Céleste Poincaré proves a theorem which he calls The non-existence of Uniform Integrals. This theorem is as follows:

Let

be a system of equations in which the characteristic function F is expanded in a series

convergent for sufficiently small values of | μ |, for all real values of y1, … yn, and for values of x1, … xn within certain finite real intervals; while F0, F1; … are analytic functions of x1, … xn, y1, … yn, of which F0 is independent of y1, … yn and Fl, F2, … are periodic with respect to these variables with period 2π.
On the measurement of the critical potentials of gases
Authors: Dymond E. G.
Published: November 2008

In all methods of measuring excitation and ionisation potentials which are in common use the current-voltage characteristic curves of the apparatus are plotted. In the absence of gas these curves are smooth, but on the admission of gas at a suitable pressure sharp upward bends occur at definite voltages, which are interpreted as the critical potentials of the gas. In a variation by Franck and Hertz, the so-called inelastic impact method, these bends are downward, the critical points being represented by maxima.
Doublet and Triplet Separations in Optical Spectra as Evidence whether Orbits penetrate into the Core
Authors: Hartree D. R.
Published: November 2008

Landé has indicated a connection between the “relativity” doublets of X-ray spectra, and the doublets and triplets of optical spectra, and so has obtained a formula for the separation of optical doublets and triplets. This formula can only be expected to hold if the orbit of the series electron penetrates into the core, so it may be possible to obtain evidence on the question whether an orbit penetrates from the doublet or triplet separation of the corresponding term.

From this evidence it is concluded that, except for lithium-like atoms, p terms of all known spectra correspond to penetrating orbits; this disagrees with Bohr's assignment of quantum numbers for those terms of the spectra of neutral atoms of the Cu and Zn sub-groups. For d terms the evidence is not so definite, but it seems possible that for Cs I and Tl I the d terms correspond to penetrating orbits, in disagreement with Bohr's assignment. It is shown that the Landé formula seems to hold approximately for separations in multiplet spectra when the terms belong to a sequence of the Rydberg type.

Ideas suggested by Heisenberg in a recent paper would involve some modifications of the interpretation of the magnitudes of doublet separations, but at least the conclusions as to the p orbits penetrating still stand.
The Wave Theory of the Electron
Authors: Whittaker J. M.
Published: November 2008

In two recent papers Dirac has shown how the “duplexity” phenomena of the atom can be accounted for without recourse to the hypothesis of the spinning electron. The investigation is carried out by the methods of non-commutative algebra, the wave function ψ being a matrix of the fourth order. An alternative presentation of the theory, using the methods of wave mechanics, has been given by Darwin. The four-rowed matrix ψ is replaced by four wave functions ψ1, ψ2, ψ3, ψ4 satisfying four linear differential equations of the first order. These functions are related to one particular direction, and the work can only be given invariance of form at the expense of much additional complication, the four wave functions being replaced by sixteen.
Some experiments on the multiple scattering of alpha particles
Authors: Rose D. C.
Published: November 2008

Experiments have been performed to measure the scattering of alpha particles by surfaces on which a source of radioactive material was deposited and on surfaces in the neighbourhood of the source. It was hoped that, where a very narrow but not necessarily homogeneous beam of alpha particles was required, the effective strength of a source could be increased by including particles which are scattered out of the activated surface at glancing angles. The effect was found to be too small to be of any great practical importance.

Experiments were also performed on the scattering of alpha particles by the walls of a long glass tube down which a beam of particles was projected. This is essentially a repetition of some experiments by Lawson and Hess in 1918, whose results indicated that the particles in being scattered by the walls of the tube lost definite amounts of energy. The present experiments indicate that they can lose any fraction of their total energy.

The experiments indicate that the most probable angle of scattering in such experiments is large, and that the average particle spends at least half and probably a greater fraction of its range in the scattering material.
A deduction of the formula for the product of two incidence symbols in the case of lines and planes
Authors: Todd J. A.
Published: November 2008

The object of this paper is to give proofs of formulae, previously obtained by Schubert (for lines) and Palatini (for planes) by degeneration methods, by a direct application of the coincidence formulae of Pieri and Severi which are referred to below. In the case of planes the essential part of the proof is the deduction of the relation (10) below, from which the required formulae follow directly. The formulae in question have as their object the expression of a product of two incidence conditions as the sum of a number of simple ones.
The Instability of a Compressible Fluid heated below
Authors: Jeffreys Harold
Published: November 2008

The vertical gradient of temperature needed to produce convection currents in a layer of incompressible liquid has already been investigated in several instances. For a compressible fluid instability cannot arise until the gradient exceeds the adiabatic one; it has been assumed usually that what matters is the excess of the actual gradient over the adiabatic. Thus the excess needed in a compressible fluid is to be found by the same formula as gives the gradient needed for instability in an incompressible fluid. It is desirable, however, to investigate the validity of this assumption.
Excitation by High Velocity Electrons
Authors: Dymond E. G.
Published: November 2008

Much attention has been focused on the excitation of atomic spectra by electrons whose velocities do not greatly exceed the critical value, and it is now possible to form a rough idea of the behaviour of the “excitation function,” or probability that an electron of given velocity shall excite on impact, when the energy of the electron is only slightly greater than that required for excitation. But the form of this excitation function for much higher velocities is still in considerable doubt.
The production of characteristic X-rays by electronic impact
Authors: Thomas L. H.
Published: November 2008

Rosseland proposed a theory of the production of characteristic X-rays. He calculated the number of atoms from which on the average an electron with definite energy will remove an electron from a particular core orbit, but neglected the velocity of the core electron in its orbit and the increased velocity of the impinging electron due to the atomic field. Taking these into account considerably alters his formula.
On the present position of the Theory of Centres of Activity in Heterogeneous Catalysis
Authors: Constable F. Hurn
Published: November 2008

The great increase in the speed of chemical change at the surfaces of specific substances has always been one of the most mysterious of all chemical phenomena, and the collection of evidence which will throw light on the nature of the process is research which lies at the foundation of chemical science, because it concerns the ultimate nature of chemical combination and dissociation.
The influence of different nuclei on the absorption spectra of organic compounds
Authors: Purvis J. E.
Published: November 2008

In various communications published in the Journal of the Chemical Society the author has discussed the effect of different nuclei on the absorption spectra of various compounds. The problem has been further investigated with a number of substances derived from aniline, o-and m-cresols, and α-and β-naphthylamines, and this communication is to give an account of the results. A condensed Cd-spark was used as the source of radiant energy, and alcoholic solutions of the substances were examined in the usual way.
Note on the modified Heine-Borel Theorem
Authors: Verblunsky S.
Published: November 2008

The Heine-Borel Theorem for one-dimensional intervals may be enunciated as follows:

(A) If a set D of intervals, all in the closed interval (a, b), be such that every point of (a, b) is an interior point of at least one interval of the set D (the end points a, b being regarded as interior to an interval when either of them is an end point of such interval), then a finite set E of intervals all belonging to D exists such that every point of (a, b) is interior to at least one interval of E.
On the transformation of certain singular surfaces
Authors: Babbage D. W.
Published: November 2008

Nöther in his classical paper Zur Theorie des eindeutigen Ent-sprechens algebraischer Gebilde has given a set of sixteen examples of the computation of his invariants pn, p(2), p(1) for algebraic surfaces in ordinary space. In the following I discuss, as a matter of some interest, the birational transformation of his surfaces into surfaces which are non-singular.
A representation of [k]'s of [m] by points of [(m − k) (k + 1)]
Authors: Room T. G.
Published: November 2008

The algebraical foundation of this representation is very simple: it depends on the theorem

If a coordinate system in [m] is given, then in a general [k] of the [m] a unique set of k + 1 points can be found whose coordinates form the scheme

where h = m − k − 1, and n = (m − k)(k + 1) = kh + m.
On smoothing and differentiation of tables
Authors: Jeffreys H.
Published: November 2008

In recent seismological work several questions have arisen concerning the construction of tables from data containing a certain amount of accidental error, and it seems that some of the methods may be of more general interest. The first is the problem of smoothing. A method suggested by Dr L. J. Comrie was as follows. Suppose that we have values of y for five equally spaced values of x, and that we wish to make the second differences vary as smoothly as possible. We try to find a cubic polynomial such that the sum of the squares of the deviations of y from it will be a minimum. It is found that the polynomial is less than the observed value of y at the centre of the range by of the central fourth difference; or approximately by of the fourth difference. I have since found that the method had previously been suggested by Sir G. H. Darwin. It gives a great improvement in the steadiness of the differences and thereby makes interpolation much easier.
Properties of some superconductors
Authors: Shoenberg D.
Published: November 2008

The critical-field curves of pure aluminium, gallium, thorium and zinc were measured by a magnetic induction method. An unusually marked supercooling effect was found in aluminium and to a much smaller extent also in gallium; some experiments on this effect are discussed. The first three of these metals were found to have “ideally pure” superconducting properties, and thorium, although belonging to the “hard” group of superconductors, had a value of dHc/dT of only 190 gauss/degree K. Only a very minute fraction of the volume of a titanium specimen was found to become superconducting at temperatures down to 1° K., although Meissner had found a titanium wire to have zero resistance below 1·8° K.; an explanation of this discrepancy is put forward. Molybdenum and uranium did not become superconducting down to 0·3 and 0·98° K. respectively, though, in the case of one uranium sample, a small fraction of the volume showed superconducting effects as in titanium.
A method of solving second order simultaneous linear differential equations using the mallock machine
Authors: Wilkes M. V.
Published: November 2008

The paper describes a method of integrating differential equations by means of the Mallock linear equation machine. The method, which is based on one described by Hartree, is applicable to pairs of second order simultaneous linear ordinary differential equations in two unknowns. Any required degree of accuracy in the solution can be obtained.
On the Heaviside operational calculus
Authors: Jeffreys Harold , Dalzell D. P.
Published: November 2008

Some theorems in the operational calculus, taking definite integration as the fundamental operator, are proved for discrete systems. It is suggested that it is physically more satisfactory to regard the solution for a continuous system as the limit of the solutions for a set of discrete systems, rather than as the solution of a partial differential equation. The necessary and sufficient condition for the validity in this sense of the usual operational method for continuous systems appears to be that the discrete systems considered shall not tend to infinite instability; that is, that all poles of the Bromwich integrand shall always have real parts less than some fixed positive quantity. The correction for finiteness of the number of degrees of freedom is examined for a case analogous to heat conduction and found to be unimportant.
Deuteron bombardment of silver
Authors: Krishnan R. S.
Published: November 2008

A further study has been made of the deuteron bombardment of silver. The following radioactive isotopes are formed: 2·4 min., 26 min. and 225 day along with the silver fraction; and 6·7 hr. and 1 year along with the cadmium fraction. Results of absorption measurements of the radiations emitted are given. The energy-yield curves have been determined for the 2·4 min. and the 26 min. Ag isotopes and also for the 6·7 hr. and 1 year Cd isotopes. The formation of the 26 min. body from silver has been explained on the basis of a new type of disintegration, namely the (d-p, 2n) reaction.
On the statistics of random distributions of paired events, with applications to the results obtained in the use of the interval selector with particle counters
Authors: Feather N.
Published: November 2008

If a particle counter is subject to the radiation from a pure radioactive source of sensibly constant activity, it is axiomatic to suppose that the series of events constituted of the traversals of the counter by ionizing particles is a random series in relation to distribution in time. There are various reasons why the related series of events formed by the occasions on which the counter responds to these ionizing particles is not a random one—and there may also be cases in which the primary source does not, in fact, give rise to particles traversing the counter in a random sequence. The counter responses will necessarily depart from strict randomness in time because of the finite recovery time of the counter; they may also depart from strict randomness through a not uncommon counter defect, the occurrence of ‘spurious’ discharges correlated in some way with the ‘true’ discharges brought about as a result of the ionization produced by the particles. On the other hand, the original ionizing events will not have the effect of a random sequence, even with a pure radioactive source, if secondary radiations are emitted—even in a fraction of the disintegrations—with a time delay appreciable in relation to the counter resolving time, and, clearly, they may depart considerably from true randomness in time if the source is not a pure source, and, in particular, if there is produced in the source material a ‘daughter’ radioelement of lifetime comparable with this resolving time, or with the mean time between counter discharges.
Momentum distribution in molecular systems
Authors: Duncanson W. E.
Published: November 2008

Hughes and Mann(1) have shown that, under certain conditions, the distribution curve of the energies of electrons scattered inelastically by an atom or molecule has the same shape as that of the X-ray Compton line. Hughes and his collaborators (1–5) have applied the method to an examination of a number of atoms and molecules in the gaseous state under which condition it can be assumed that each atom or molecule acts as an independent scattering centre. Coulson and the author (6–11) have applied quantum methods to derive a theoretical shape of the Compton line for certain of the molecules for which wave functions are available and have discussed the discrepancy between the experimental and the theoretical results(10, 11).
Notes on special systems of orthogonal functions (IV): the orthogonal functions of whittaker's cardinal series
Authors: Hardy G. H.
Published: November 2008

Suppose that n runs through all integral values, that (øn) is a system of normal orthogonal functions for the interval (−∞, ∞), and that ψn is the Fourier transform of øn. Then, by Parseval's theorem for Fourier transforms,

and (ψn) is also a normal orthogonal system.
On the limiting values for infinite pitch of a parameter occurring in airscrew theory
Authors: Goldstein S.
Published: November 2008

In an improved method of strip-theory calculation of the performance of an airscrew a factor K occurs in the formula

which, for an airscrew with N blades, connects the total interference velocity w1 and the circulation Γ at radius r. In this formula ø is the inclination to the plane of rotation of the airscrew of the velocity of the air relative to the airscrew blade; for infinite pitch ø=½π For large but finite pitch the calculation of K is difficult; in computing tables of K it is therefore useful to have the values for infinite pitch, which can be calculated by a much simpler theory than that needed for the general case†.
An inequality relating means
Authors: Bronowski J.
Published: November 2008

1. Let a, b be positive constants; and let y1, y2, …, yn be real exponents, not all equal, having arithmetic mean y defined by

(here, and in what follows, the summation ∑ extends over the values i = 1, 2, …, n). Then it is clear that

since the right-hand sides are the geometric means of the positive numbers whose arithmetic means stand on the left-hand sides. I know of no results, however, which relate the ratios and and I have had occasion recently to require such results. This note gives an inequality between these ratios, subject to certain restrictions on a and b.
Thermionic Emission through double layers
Authors: Georgeson W.
Published: November 2008

The effect of surface layers in modifying the thermionic emission has been explained by Nordheim on the assumption that the atoms in the surface contamination create an electrical double layer on the surface. If the layer is electro-positive with respect to the metal the work of final extraction of a given electron is reduced. This reduces the value of the work function χ, but at the same time requires the escaping electrons to pass through a region in which the potential energy is greater than their total energy. It follows that the emission coefficient D (W) is decreased at the same time. The details of the connection between χ and D (W) have been examined further by Fowler.
The distribution of range of the α particles from radium C′ and thorium C′
Authors: Feather N. , Nimmo R. R.
Published: November 2008

From measurement of tracks formed in an expansion chamber range distribution curves are obtained relative to 2134 α particles from thorium C′ and 729 α particles from radium C′. These curves exhibit marked differences in the proportion of short range particles which each contains. The origin of these differences is discussed. Estimates are made of the linear straggling coefficients relative to the absorption of the two groups of particles in air (in the experimental arrangement other media entered as well). These results very definitely confirm the conclusions reached by Briggs from measurements of the straggling of velocity of the particles. The sources of error which enter when the expansion chamber is used for the measurement of straggling coefficients are briefly reviewed.
The Errors of Approximation in Jeffreys's Phases
Authors: Arnot F. L.
Published: November 2008

The errors occurring in a large number of Jeffreys's phases for neutral chlorine have been determined. A discussion of the source of these errors is given, from which it appears that errors in the phases of zero order and of order unity are due to a cumulative error over the range of integration. A method of correcting for this error is obtained. This method is of easy practical application, and it reduces the error almost to zero for phases of these orders.

The error occurring in higher order phases is found to be mainly due to errors in the two π/4 terms, which take into account the exponential tails of the waves in the totally reflecting region. For phases of high order these errors in the two π/4 terms cancel each other, and consequently the total error in phases of high order is practically zero.

A correction graph for the zero-order phase is given from which the error in this phase for atoms of atomic number up to 36 can be read off. The graph is such that an extrapolation to higher atomic number would not involve much error.
Note on radiationless transitions involving three-body collisions
Authors: Smith R. A.
Published: November 2008

When an electron makes a transition from a continuous state to a bound state, for example in the case of neutralization of a positive ion or formation of a negative ion, its excess energy must be disposed of in some way. It is usually given off as radiation. In the case of neutralization of positive ions the radiation forms the well-known continuous spectrum. No such spectrum due to the direct formation of negative ions has, however, been observed. This process has been fully discussed in a recent paper by Massey and Smith. It is shown that in this case the spectrum would be difficult to observe.
An attempt to produce artificial radioactivity by an electron beam, with some notes on the behaviour of newly made Geiger-Muller counters
Authors: Lewis W. B. , Burcham W. E.
Published: November 2008

An unsuccessful attempt has been made to detect artificial radioactivity in aluminium after bombardment with electrons at 300 kV. During these experiments a transitory activity produced by cleaning the walls was observed in a Geiger counter. It is suggested that this might be due directly or indirectly to a progressive oxidation of the inner surface of the walls.
On an inequality in the elementary theory of numbers
Authors: Heilbronn H. A.
Published: November 2008

Let a1, a2, …, an be a set of n positive integers. Then it is easily seen that the set of positive integers not divisible by any aν has a density, i.e. that if Nn(z) is the number of such integers not exceeding z, then z−1Nn(z) tends to a limit when z → ∞; and that

where

and where [u1, …, uμ] denotes the least positive common multiple of the positive integers u1, …, uμ.
The velocity of sound in liquid nitrogen
Authors: Hirschlaff Ernst
Published: November 2008

This note describes measurements of the velocity of sound in liquid nitrogen by a method which has previously been used in the case of liquid oxygen(1). The method depends on the diffraction of light by suprasonic waves.
An extension of a theorem of Kulakoff
Authors: Easterfield T. E.
Published: November 2008

It has been shown by Kulakoff that if G is a group, not cyclic, of order pl, p being an odd prime, the number of subgroups of G of order pk, for 0 < k < l, is congruent to 1 + p (mod p2); and by Hall that if G is any group of finite order whose Sylow subgroups of G of order pk, p being odd, are not cyclic, then, for 0 < k < l, the number of subgroups of G of order pk is congruent to 1 + p (mod p2). No results were given for the case p = 2. In the present paper it is shown that analogous results hold for the case p = 2, but that the role of the cyclic groups is played by groups of four exceptional types: the cyclic groups themselves, and three non-Abelian types. These groups are defined as follows:

(1) The dihedral group, of order 2k, generated by A and B, where

(2) The quaternion group, of order 2k, generated by A and B, where

(3) The "mixed" group, of order 2k, generated by A and B, where
The electrical resistance of a metal at low temperatures and Matthiessen's rule
Authors: Dube G. P.
Published: November 2008

By solving the fundamental integral equation an expression is obtained for the electrical resistance which takes into account, to the first approximation, the mutual influence of the impurities and the lattice vibrations. It is found that deviations from Matthiessen's rule are to be expected and that these deviations are surprisingly large. The formulae derived only indicate the trend of the resistance curve, but this trend is not confirmed by experiment. Whereas the theory indicates that the mutual influence of the impurities and the lattice vibrations should decrease the electrical resistance, the experimental results of Grüneisen on copper show that, when deviations from Matthiesen's rule occur, the resistance is increased.
The sputtering of oxide-covered magnesium surfaces
Authors: Ditchburn R. W. , Roulston K. I.
Published: November 2008

The rate of sputtering of magnesium in oxygen was observed with an apparatus which permitted the field immediately in front of the target to be reversed. Change in the sign of the field did not affect the rate of sputtering which remained abnormally low. It is therefore concluded that, while the low rate is in some way due to the presence of an oxide film, it is not due to charged particles being returned to the cathode by the electric field, as had been suggested by other workers.
The disintegration of the separated isotopes of boron under proton and deuteron bombardment
Authors: Smith C. L. , Murrell E. B. M.
Published: November 2008

The disintegration of the separated isotopes of boron, under proton and deuteron bombardment, has been investigated, and it is shown that the groups of disintegration particles previously discovered had been attributed to the transmutation of the correct isotopes. In addition, experimental evidence has been obtained for a previously unknown excitation level of 8Be at about 7·5 × 106 e.V. above the ground state.
On the derivation of discontinuous functions
Authors: Dickinson D. R.
Published: November 2008

Let f(x) be a real function of the real variable x, let P be any point lying on the graph of f(x) and let l be a ray from P making an angle θ (− π < θ ≤ π) with the positive direction of the x-axis. We say that θ is a derivate direction of f(x) at the point P if the ray l meets the graph of f(x) in a set of points having a limit point at P.
A problem in conduction of heat
Authors: Carslaw H. S. , Jaeger J. C.
Published: November 2008

The problem of the conduction of heat in a solid sphere with a concentric core of a different material, the surface kept at a constant temperature, and the initial temperature of the whole zero, has already been solved in these Proceedings.
Elastic collisions of mesons with electrons and protons
Authors: Massey H. S. W. , Corben H. C.
Published: November 2008

The cross-sections for the elastic scattering of mesons by protons and electrons are calculated by analogy with Møller's theory of collisions between electrons, and the results are compared with those of Bhabha for the collisions with protons and electrons of Fermi-Dirac particles with the same mass as the meson. For collisions with electrons, the difference is inappreciable for energies of the meson less than 100 times its rest energy, but for collisions with protons the difference is important at much lower energies. The ratio of the corresponding cross-sections for scattering by a Coulomb field is found to tend to infinity as the square of the energy of the incident particle, giving a finite cross-section for the scattering of a meson of infinite energy by such a field. It is pointed out that, if an exact solution of this problem is possible for the meson case, the question of the validity of meson theory for high energies would be considerably clarified.
Note on the sensitive time of a Wilson expansion chamber
Authors: Williams E. J.
Published: November 2008

By “sensitive” time of a cloud chamber is meant the length of time for which the supersaturation in the chamber remains sufficient to cause condensation along the tracks of ionizing particles. If the cooling expansion is made rapidly this sensitive time depends on the rate at which the temperature of the gas in the chamber rises after the expansion. This problem of the warming of a gas in a vessel, after an initial cooling, has been previously considered by Rayleigh (1). To follow the whole history of the warming until the gas reaches the temperature of the vessel requires an elaborate mathematical treatment which can only be carried out for specially shaped vessels. Rayleigh considers two special cases, viz. a gas contained between two parallel infinite planes, and a gas contained in a spherical vessel. The results obtained by Rayleigh for these cases are, however, not very helpful in dealing with the cloud-chamber problem under consideration. This is so, not so much because specially shaped vessels are assumed, but because the treatment does not lend itself to an evaluation of the rate of warming in the early stages of the process. It is this initial warming that we are concerned with here, since a cloud chamber ceases to be sensitive when the average temperature of the gas has risen by only 5% or less of the cooling produced by the expansion. Rayleigh's result for the average rise in temperature of the gas at any instant is represented by an infinite series which converges very slowly when the rise in temperature is small.
The dynamics of a simple system. I. Observables
Authors: Powell F. C.
Published: November 2008

Any particular form of mechanics (e.g. classical or quantum) makes use of a particular ‘formalism’ or set of rules governing the use of the symbols representing dynamical variables and the symbols (such as +, =) representing relations between dynamical variables. In the present paper an attempt is made to examine the physical content of this formalism, particularly that of quantum mechanics. This is done by building up a formalism as the direct expression of a number of physical postulates having direct operational meaning. It is shown that, in a simple case, with any two observables A and B can be associated a unique sum observable A + B and a unique (real) symmetric product observable A.B (= B.A).

It is next shown that, if, like a classical Hamiltonian system, the system has the property that the time rate of change of an observable depends linearly on some observable H when the environment in which the system moves is varied, then a (real) skew product observable A × B (= − B × A) can be defined.

Finally, if the equations of motion are ‘of second order', it is possible to express the second rate of change of any observable as an algebraic function of that observable and H. This leads to algebraic identities from which it follows that ‘complex multiplication’, defined by AB = A.B + iA × B, is associative (but not commutative). Observables are thus shown to possess the properties usually ascribed to them in quantum mechanics. These properties make possible a representation by Hermitian matrices.
The Fourier transform solution of an elastic wave equation
Authors: Sneddon Ian N.
Published: November 2008

1. In a recent paper(1) the partial differential equation governing the symmetrical vibrations of a thin elastic plate was reduced to an ordinary differential equation by the use of the Hankel transform method. By the discussion of the solutions of the latter equation and by the use of the Hankel inversion theorem an account was given of the free and forced vibrations of the plate under symmetrical conditions.
Notes on the theory of series (XXIV): a curious power-series
Authors: Hardy G. H. , Littlewood J. E.
Published: November 2008

1. This note originates from a question put to us by Mr W. R. Dean concerning series of the type

where θ is irrational. Series of the type

are familiar: it is well known, for example, that

may have any radius of convergence from 0 to 1 inclusive, according to the arithmetic nature of θ. It is natural to ask how this radius is connected with that of

and, more generally, how those of (1·1) and (1·2) are connected.
A note on the Petzval field curvature in electron-optical systems
Authors: Goddard L. S.
Published: November 2008

1. The Petzval field curvature produced in a compound glass optical system of axial symmetry is given by a well-known formula. If the system consists of a number of media, of refractive indices n0, n1, n2, …, having spherical faces whose radii of curvature are r0, r1, r2, …, the formula is [see for example, (1) or (2)]

and this is in common use amongst those concerned with the design of optical equipment. The analogous integral expression in the electron optical case has not, however, received the attention it deserves, in spite of the developments in electron optics during the past decade. In 1935 Glaser(3) presented the third-order error theory of an axially symmetric electron optical system.
Collective electron assemblies in a metal with overlapping energy bands
Authors: Band William
Published: November 2008

In Part I of this series it was shown that in a metal with overlapping energy bands, one of which is nearly full and the other nearly empty, there exists a critical temperature below which spontaneous magnetization will be present, Tc being given approximately by the equations

where kθ is the exchange energy term postulated by Stoner, b is the ratio of excess vacancies in the d-band to the maximum possible number of parallel spins, and ζd0 is the equivalent energy depth of the vacant levels in the d-band.
Notes on Fourier series (IV): Summability (R2)
Authors: Hardy G. H. , Rogosinski W. W.
Published: November 2008

1.1. There are two familiar methods of summation of divergent series usually called the methods (R, 1) and (R, 2). If sn = u0 + u1 + … + un and, as it will be convenient to suppose throughout, u0 = 0, then

when h → + 0§: the convergence of the series for small positive h is presupposed.
The flexure and torsion of aeolotropic beams
Authors: Green A. E.
Published: November 2008

Many writers have studied special cases of the Saint-Venant torsion and flexure problem for an isotropic beam of constant cross-section. An excellent summary of this work has been given by Stevenson (4) who also greatly extended the field of application of the theory by showing that the general Saint-Venant problem can be reduced to the solution of boundary problems involving six simple ‘canonical’ flexure functions. Miss Rosa Morris (2, 3) has applied the Saint-Venant theory, as extended by Stevenson, to the solution of the flexure and torsion problem for beams possessing two fairly general types of cross-section, her results including many previous solutions as special cases.
Note on waves on the surface of running water
Authors: Dean W. R.
Published: November 2008

The method for the solution of problems of surface waves on water with which this note is concerned can be sufficiently illustrated by a particular example given by Lamb (1). In Lamb's example it is supposed that a steady stream in which the x, y components of the velocity are U, 0, where U is constant, is slightly disturbed by the application of a constant pressure ρPU2 to a band of finite width 2a of the surface of the stream; ρ denotes the density of the water and P a non-dimensional constant. In the solution Lamb, using an artifice due originally to Rayleigh and widely used since, assumes that ‘the deviation of any particle of the fluid from the state of uniform flow is resisted by a force proportional to the relative velocity’, and states that this ‘law of friction does not profess to be altogether a natural one, but it serves to represent in a rough way the effect of small dissipative forces; and it has the great mathematical convenience that it does not interfere with the irrotational character of the motion’.
A note on the theory of dislocation in metals
Authors: Dean W. R. , Wilson A. H.
Published: November 2008

1. Since many writers, among them G. I. Taylor (1) and W. L. Bragg (2), have based theories of the strength of metals upon the assumption that there is some type of dislocation in the metal, it has seemed desirable to give an account of a dislocation in which displacement and stress are given, according to the classical theory of elasticity, by simple closed formulae.
Some remarks on Schottky's theorem
Authors: Hayman W. K.
Published: November 2008

1. We denote by the class of all functions regular in |z| < 1, and never taking the values 0 or 1 in |z| < 1. Then Schottky's(1) theorem states:

Theorem A. Let f(z) = a0 + a1z + … belong to . Then

and hence

where

and the constantsK(a0), K(a0,ρ) depend only on the quantities indicated.
Combinants of a pencil of quadric surfaces. II
Authors: Todd J. A.
Published: November 2008

1. In the first paper of this series I have explained a method by which the complete irreducible system of combinants of a pencil of quadric surfaces may be obtained, and have determined explicitly the combinantal invariants and covariants. The present paper deals with combinantal contravariants. The notation used is that of I, a knowledge of which will be assumed.
Notes on the molecular orbital treatment of linearly coupled systems
Authors: Coulson C. A. , Rushbrooke G. S.
Published: November 2008

It is well known (1) that for an unsaturated hydrocarbon of the type CnHn+2 (where n is the number of carbon atoms in the linear chain), the theory of molecular orbitals applied to calculate the binding energy of the mobile π-electrons leads to the secular determinant, of continuant type,

where β is the resonance integral between adjacent carbon centres and x stands for the quantity generally written E0 − E, E being the energy of a mobile electron, which we seek to calculate. The equation for the allowed energies E is, of course, simply Pn = 0.
The relation between different types of Abel summability
Authors: Kuttner B.
Published: November 2008

1. Let {μn} be a sequence of positive numbers increasing to infinity. If the series

converges for s > 0, and if f(s) → l as s → 0, then ∑an is said to be summable (A, μn) to l; this generalization of Abel summability, first introduced by Hardy, has long been familiar. Suppose that, for x ≥ μ0, ø(x)is a positive increasing function of x, which tends to infinity as x → ∞. We write throughout λn = ø(μn), and

if this exists.
On a class of probability distributions
Authors: Chandrasekhar S.
Published: November 2008

1. The statement of the problem. A problem which arises in certain physical (5) and astronomical (1, 3) contexts relates to the probability distribution of the infinite sum

where the rj's are the position vectors from a fixed origin of an infinite uniform Poisson distribution of points with a given constant mean space density N. While certain special cases of the sum (1) have been considered in the literature (cf. (2)) the study of the general case discloses properties of a class of probability distributions which appear to have some interest. For example, we shall show that the probability distribution of F exists only for that the moments of F (= |F|) exist only for orders 0 ≤ p < 3/n, and finally that an explicit formula can be given for all the finite moments.
Shearing displacement of a rectangular plate
Authors: Mann E. H.
Published: November 2008

A solution is given of a problem already solved by Inglis, a rectangular plate stressed by the uniform longitudinal relative displacement of two opposite edges, the other pair of sides remaining free. The method used here yields a more accurate solution with greater rapidity than did the previous paper. The general form of the solution is given for a plate of any dimensions, and a plate of sides in the ratio 2:1 is considered in more detail. An approximate solution is obtained by taking four terms of an infinite series, and the consequent errors discussed and compared with those of Inglis.
The Compton scattering by particles possessing charge and dipole moment
Authors: Shanmugadhasan S.
Published: November 2008

The simplest exact classical theory, of a particle possessing charge and dipole moment proportional to the spin leads to a wave equation with an explicit spin interaction with the field. This wave equation is used in this paper to calculate the scattering probability coefficient in the Compton effect by two methods. The first method uses the unquantized radiation field; the second uses Dirac's scheme of field quantization. The scattering probability coefficient is given in the general case in terms of the charge e, the constant of proportionality in the dipole moment C, and the quantities k, l specifying the spin. Some special results are considered for particles of spin ½ and 1. Formulae are given for the scattered intensity as a function of the angle of scattering, and the total scattering cross-section as a function of the energy of the unpolarized incident radiation. Graphs are given for these formulae when the magnetic part of the dipole moment takes the empirical values in the case of the electron, proton and neutron. In the case of the meson, normal and abnormal values of the dipole moment are used.
Rectilinear flow of non-Bingham plastic solids and non-Newtonian viscous liquids. I
Authors: Oldroyd J. G.
Published: November 2008

In a previous paper on steady rectilinear plastic flow of a Bingham solid (1), the point of view was adopted that the study of all possible velocity distributions was equivalent to a study of the geometry of velocity contours in a section normal to the flow. This led to an analysis of the differential geometry of velocity contours in rectilinear plastic flow under zero pressure gradient, and the problem of finding all possible velocity distributions was eventually reduced to the problem of finding the general integral of a single linear second-order partial differential equation with two independent variables. But the form of this differential equation (equation (25) of (1)) was such that only a very limited number of solutions could be found by standard methods. The particular solutions which were obtained corresponded to those flow patterns in which the velocity contours could be identified with a family of parametric curves in a curvilinear coordinate system obtainable by conformal transformation of a cartesian frame of reference. In effect, this means that the results so far achieved by using the new approach to the problem could have been obtained by more direct methods.
Numerical integration by systematic sampling
Authors: Moran P. A. P.
Published: November 2008

Recent investigations by F. Yates (1) in agricultural statistics suggest a mathematical problem which may be formulated as follows. A function f(x) is known to be of bounded variation and Lebesgue integrable on the range −∞ < x < ∞, and its integral over this range is to be determined. In default of any knowledge of the position of the non-negligible values of the function the best that can be done is to calculate the infinite sum

for some suitable δ and an arbitrary origin t, where s ranges over all possible positive and negative integers including zero. S is evidently of period δ in t and ranges over all its values as t varies from 0 to δ. Previous writers (Aitken (2), p. 45, and Kendall (3)) have examined the resulting errors for fixed t. (They considered only symmetrical functions, and supposed one of the lattice points to be located at the centre.) Here we do not restrict ourselves to symmetrical functions and consider the likely departure of S(t) from J (the required integral) when t is a random variable uniformly distributed in (0, δ). It will be shown that S(t) is distributed about J as mean value, with a variance which will be evaluated as a function of δ, the scale of subdivision.
Some asymmetric inequalities
Authors: Blaney Hugh
Published: November 2008

Let α, β, γ, δ be real numbers with Δ = |αδ −βγ| > 0, and let ξ, η denote the linear forms
Summation of slowly convergent series
Authors: Cherry T. M.
Published: November 2008

Let ∑un be a convergent infinite series which is not summable in finite form. In principle its sum can be found, to within any preassigned error ε, by adding numerically a sufficient number of terms; but if the series is slowly convergent, the ‘sufficient number’ of terms may be prohibitively large. A plan to deal with this case is to separate the series into a ‘main part’ u0+u1+ … +un−1 and a ‘remainder’ Rn = un+un+1+…; the main part is evaluated by direct summation, while the remainder is transformed analytically into a series which is more rapidly ‘convergent’, in the practical sense, and so evaluated. For example, the Euler-Maclaurin sum-formula gives such a transformation. It commonly happens that the new form of the remainder Rn is a divergent series, but that it represents Rn asymptotically as n ˜ ∞. It is for this reason that the transformation is applied to Rn instead of to the whole series; for practical use we have to choose n sufficiently large for the error inherent in the use of the asymptotic series to be below the preassigned bound ε.
Space-time on the rotating disk
Authors: Kurşunogğlu Behram
Published: November 2008

1.Introduction.In a recent paper Clark (1) has dealt with the problem of the rotating disk, the material of which is such that the waves of dilatation in this particular material travel with the velocity of light. The material of the disk is supposed to be under an isotropic stresspwhen in a strained state, and the relation between stresspand the dilatation Δ is found to be connected by an expression

wherea= density in the unstrained state, and Δ is given by

Ui(i= 1, 2, 3) are the components of the strain.