bluuebunny/crossref_metadata_2025 huggingface dataset:
The Remainders in the Asymptotic Expansions of Certain Bessel Functions
Authors: Burnett D.
Published: November 2008

The usual methods of investigation of asymptotic expansions of the various types of Bessel Functions show that the remainder is less in absolute value than the first term neglected. A more refined result was obtained by Stieltjes for K0(x) and certain other functions of order zero; he found an asymptotic series for the remainder and showed that the error due to stopping at one of the smallest terms is of the order of half the first term neglected. Watson notes that it would be of some interest to obtain corresponding results for functions of any order, and this is the object of this note. The method is quite different from that of Stieltjes.
On the Cesàro summability of Fourier series and Allied series
Authors: Paley R. E. A. C.
Published: November 2008

For r>-1, let denote

If

we say that the series a0 + a1 + a2 +…+an+… is summable by Cesàro mean of order r, or more shortly summable (C, r) to sum s. If r >−1, and

we say that the series is summable by Rieszian mean of order r to the sum s. It has been shown that these two methods of summation are equivalent. Throughout this paper I shall deal with the Rieszian mean, but I shall retain the symbol (C, r). It is known† that if a series is summable (C, r), it is also summable (C, r′) to the same sum for all numbers r′ greater than r.
On the Action of the Geiger α-particle Counter
Authors: Taylor James
Published: November 2008

A very great deal has been published upon the subject of the electrical counter and its mode of action. Most explanations are incomplete in that they neglect some of the factors involved. The question of the self-restoring property of the system has also received attention, but this property has not been properly related to the general problem of intermittency in discharge tubes.
The motion of a particle in a periodic field of force
Authors: Malurkar S. L. , Hargreaves J.
Published: November 2008

The electron theory of metals revived by Sommerfeld assumes that an electron moves in a metal as though this were an equipotential medium. Considering the nuclei fixed and regularly spaced we obtain a potential periodic in space coordinates. To study the effect of such fields we may simplify the problem so as to contain only one periodic term for each coordinate in its expression for potential. This problem can be reduced further to a one-dimensional one, of which the simplest example is the motion of an electron in a field with potential cos x or sin x. Darwin has shown that a suitable combination or packet of elementary de Broglie waves is capable of moving coherently in several instances. The motion of such a packet is found to be equivalent to that of a particle in classical dynamics with the Heissenberg uncertainty relation. The wave packet is used here for the motion of the electron in a periodic field. The result obtained is equivalent to that of classical dynamics. The wave packet again moves as a particle with an uncertainty relation.
The effects produced by positive ion bombardment of solids: metallic ions
Authors: Oliphant M. L. , Rutherford E.
Published: November 2008

The following is a preliminary account of experiments made in an attempt to obtain information about the relation between the energy of a positive ion and the effect produced on a surface which it bombards. The energy relations which hold in collisions between rapidly moving positive ions and the molecules of a gas or of a solid are of extreme importance in the theory of gaseous discharges at low pressures, in particular of the initiation of the self-sustained discharge.
The Convergence of Singular Integrals
Authors: Verblunsky S.
Published: November 2008

We are concerned in this note with a problem which may be regarded as a generalisation of one which occurs in the theory of Fourier series.
The quarto-quartic transformation of four-dimensional space associated, with certain protectively generated loci
Authors: Todd J. A.
Published: November 2008

The principal object of the present paper is to give a geometrical account of a certain Cremona transformation of four-dimensional space ([4]) into itself. The transformation in question is analogous to the cubo-cubic transformation of [3] which is obtained by drawing cubic surfaces through a twisted sextic curve of genus three, . A similar transformation, in space of any number of dimensions, has been described analytically in a note by Godeaux; and most of our work in the present paper extends to the general case. The case of four dimensions is studied here as being less abstract, and more easily susceptible of geometrical treatment; in the general case it is less simple to dispense completely with algebraical methods. We shall see that the various loci which arise in the course of the work are already well known.
The theory of the extraction of electrons from metals by positive ions and metastable atoms
Authors: Massey H. S. W.
Published: November 2008

It has long been known that positive ions are capable of ejecting electrons from a metal surface, and it has recently been shown by Oliphant that metastable atoms of helium are capable of producing the same effects. In view of the importance of these phenomena in the theory of discharge processes and of their intrinsic interest, it is desirable to consider the theory of the effect in the light of the Sommerfeld theory of metals and the analytic methods of wave mechanics.
The number of contact primes of the canonical curve of genus p
Authors: Welchman W. G.
Published: November 2008

1. It is known, from the theory of the Riemann theta-functions, that the canonical series of a general curve of genus p has 2P−1 (2P − 1) sets which consist of p − 1. points each counted twice. Taking as projective model of the curve the canonical curve of order 2p − 2 in space of p− 1 dimensions, whose canonical series is given by the intersection of primes, we have the number of contact primes of the curve. The 28 bitangents of a plane quartic curve, the canonical curve of genus 3, have been studied in detail since the days of Plücker. The number, 120, of tritangent planes of the sextic curve of intersection of a quadric and a cubic surface, the canonical curve of genus 4, has been obtained directly by correspondence arguments by Enriques. Enriques also remarks that the general formula 2P−1 (2p −1) is a special case of the formula of de Jonquières, which was proved, by correspondence methods, by Torelli§.
On the demonstration of electric lines of force, and a new method of measuring the electric moment of tourmaline
Authors: Maurice M. E.
Published: November 2008

1. A description of a simple method of demonstrating electric lines of force around a charged body is given.

2. The instability of liquid surfaces in an electric field is used to obtain an idea of the order of magnitude of the charge developed on tourmaline on cooling through about 140° C.

3. A new method of measuring the electric moment of tourmaline is described.
The Matrix Mechanics of the Spinning Electron
Authors: Temple G.
Published: November 2008

In three earlier papers the author has developed the theory of the operational wave equation, which was originally suggested by Professor Eddington as the most extensive generalisation possible of the linear wave equation devised by Dirac. The wave function,ψ, plays a very minor rô1e in the development of this theory and, in reality, it is introduced simply to provide an operand which shall be patient of the action of the wave operators, A1, A2, A3, A4. The object of this paper is to show that the wave function may be entirely eliminated from the theory, which then takes the form of a “ matrix mechanics,” i.e. a set of relations between matrices representing the coordinates, the momenta and the spin operators.
A constant deviation X-ray vacuum monochromator
Authors: Harper G. I.
Published: November 2008

An instrument is described which provides an accurately monochromatised beam of X-rays by reflection at a crystal and permits the wave-length to be changed without moving either the X-ray tube or the measuring apparatus. This is effected by an automatic adjustment which moves a second crystal into the correct position and orientation to reflect the monochromatised beam along a fixed emergent direction. Twenty wave-lengths well distributed in the range 0·8Å.—4Å. are available, the time required to select any one being about one minute. The whole instrument is contained in a small metal box which is evacuated.

I am indebted to Professor A. M. Tyndall, F.R.S., for extending to me the facilities of his laboratory, and for his interest in this work.
On surfaces of sectional genus five
Authors: Roth L.
Published: November 2008

In a previous paper the author has examined the various types of non-singular surfaces of sectional genus four; in the present work the same method is applied to non-singular surfaces of sectional genus five. The examination of this case completes the classification of non-singular surfaces in higher space as far as those of the seventh order; for a septimic surface of sectional genus six, necessarily normal in S4, must lie on a quadric, and its characters may be determined from this fact.
Discussion of the infinite distribution of electrons in the theory of the positron
Authors: Dirac P. A. M.
Published: November 2008

The quantum theory of the electron allows states of negative kinetic energy as well as the usual states of positive kinetic energy and also allows transitions from one kind of state to the other. Now particles in states of negative kinetic energy are never observed in practice. We can get over this discrepancy between theory and observation by assuming that, in the world as we know it, nearly all the states of negative kinetic energy are occupied, with one electron in each state in accordance with Pauli's exclusion principle, and that the distribution of negative-energy electrons is unobservable to us on account of its uniformity. Any unoccupied negative-energy states would be observable to us, as holes in the distribution of negative-energy electrons, but these holes would appear as particles with positive kinetic energy and thus not as things foreign to all our experience. It seems reasonable and in agreement with all the facts known at present to identify these holes with the recently discovered positrons and thus to obtain a theory of the positron.
The orders of products and commutators in prime-power groups
Authors: Easterfield T. E.
Published: November 2008

It is known † that, if G is any p-group (group whose order is a power of the prime p) the class of which is less than p, then the order of a product of elements of G cannot exceed the orders of all the factors. This result in general ceases to hold when the class of G exceeds p − 1. In the present paper, we prove the following general results.
Considerations of the effect of space-charge in the magnetron
Authors: Moullin E. B.
Published: November 2008

When a diode thermionic tube, having cylindrical symmetry, is placed in a magnetic field parallel to its axis it is commonly called a magnetron. If there is a given potential difference between the anode and cathode of the tube, and if the magnetic field is steadily increased, a sharp and pronounced decrease of anode current occurs when the field reaches a certain value. It is easy to show that, if electrons leave without velocity from a cathode of radius b, they will just graze a concentric anode of radius a at potential V when the magnetic field H has the value given by
The calculations of processes involving mesons by matrix methods
Authors: Wilson A. H.
Published: November 2008

In a previous paper a new method, based on Kemmer's β-formalism, of calculating meson processes was given for the case in which the meson interacts with an electromagnetic field. This method is now extended to the nuclear interaction, so that the whole of the meson theory can be given either in tensor or in matrix form, the former being preferable when the wave aspect of the meson is important and the latter when the particle aspect is dominant.

As examples of the matrix method, derivations are given of the cross-sections for the nuclear scattering of mesons and for the production of mesons from nuclei by photons. It is pointed out that the usual non-relativistic theory of the nuclear interaction is inadequate even for very small velocities.
Deviation from the Coulomb law for the proton
Authors: Ma S. T.
Published: November 2008

In a recent paper, Fröhlich, Heitler and Kahn(1) discussed the deviation from the Coulomb law for the proton on the basis of the meson theory. Instead of being considered as a point charge, the proton was assumed to be surrounded by a meson field, which interacts with an external electrostatic field. They deduced the result that the Coulomb attraction between the proton and a negative point charge goes over into a strong repulsion for distances less than one-sixth of the electronic radius. This has been found to agree with the experiment by R. C. Williams on the fine structure in the spectrum of the hydrogen atom and the interpretation of this experiment by Pasternack. Later experiments(2), however, seem to indicate that the effect is much smaller than the theory predicts.
The stress distribution due to a force in the interior of a semi-infinite elastic medium
Authors: Sneddon Ian N.
Published: November 2008

1. In this paper an analysis is given of the distribution of stress in a semi-infinite elastic medium due to the action of an external force applied to the interior of the medium. It will be assumed throughout that the force acts in a direction perpendicular to that of the boundary of the solid; the analysis is similar when the line of action of the force is parallel to the boundary and is therefore not given here. The equations of plane strain parallel to the x-y plane are employed; physically this is equivalent to assuming that there is no component of the displacement vector in a direction normal to the x-y plane or, what is the same thing, that the external force is applied along are infinite line parallel to the axis of z.
Contributions to the theory of cooperative phenomena
Authors: Temperley H. N. V.
Published: November 2008

It has been found possible to generalize Ising's (4) treatment of the linear chain with nearest-neighbour interaction, and his main result, that such interaction cannot lead to cooperative effects, has been shown to hold for very many, if not all, types of lattice in three dimensions. Reasons are given for presuming a similar result for the second nearest-neighbour model. It is found, however, that even a small interaction of the long-range type can lead to cooperative effects at high temperatures, provided that the nearest-neighbour interaction is sufficiently strong. It is therefore claimed that explicit account must be taken of both types of interaction in any acceptable theory, and a possible method of estimating their relative importance is indicated.
Second-order focusing for the mass spectrograph
Authors: Sawyer W. W.
Published: November 2008

Second order focusing can be obtained with an apparatus, which contains a circular magnetic field and differs little in its dimensions from the apparatus at present in use. The lay-out is shown to scale in Fig. 4: the essential data are collected at the end of Section V, and indications of the accuracy to be expected are given at the end of Section III.
The mechanism of the production of atomic hydrogen by hot tungsten
Authors: Roberts J. K. , Bryce G.
Published: November 2008

The fact that the rate of production of atomic hydrogen at a tungsten surface at a given temperature is proportional to the square root of the hydrogen pressure means either that the important process is the evaporation of atoms from an adsorbed film which over the whole range of experimental conditions is sparsely occupied, or that the production of atoms is in the main due to a process in which a hydrogen molecule strikes a bare tungsten atom in the surface, one atom being adsorbed and the other evaporating and the surface being almost completely covered over the whole range of experimental conditions. Either process leads to a temperature variation in the rate of atom production in agreement with experiment. A definite decision between the two processes cannot yet be made.
Superconducting cylinders
Authors: Shoenberg D.
Published: November 2008

The magnetization curves of short tin, lead and tantalum cylinders were measured by the force method, and it was shown that although for tin there was no appreciable hysteresis in the transverse position, there was a very marked hysteresis in the longitudinal position, of the same form at different temperatures but depending on the sharpness of the cylinder rims. Lead and tantalum showed additional temperature dependent features which could be attributed to impurities.

The hysteresis due to shape in the longitudinal cylinder is analogous to the hysteresis of alloy ellipsoids, in that the magnetization for a given field always lies within a certain boundary curve and varies “classically” within this boundary. This analogy suggests that the mechanism of the shape hysteresis for a pure superconductor may be similar to the sponge mechanism proposed by Mendelssohn for alloys, but it is also possible that the hysteresis is due to the formation of macroscopic superconducting rings. The considerations which fix the boundary curve are probably similar to those in the superconducting ring, and in this connection some new measurements on a lead ring are reported. The paper concludes with a discussion of the time lag between magnetization and field, and it is suggested that this is not a primary phenomenon.
Soft X-ray bands from dilute alloys
Authors: Skinner H. W. B. , Johnston J. E.
Published: November 2008

It is shown that the soft X-ray emission bands from Be and A1, present in small quantities in solid solution in Cu and A1, have a form without a sharp highenergy edge, the feature most characteristic of these bands in pure metals. The diffuse form of the bands indicates (1) that all the valence electrons of the impurity atom are given up to the lattice, (2) that the impurity atom refuses to take within it those electrons of the lattice which have energies close to the maximum energy of the conduction electrons.
A search for ionization of hydrogen by diffusion through palladium
Authors: Stansfield R. G.
Published: November 2008

No ionization was found in hydrogen leaving palladium, either when the gas diffused through the metal or was evolved by warming palladium previously charged with hydrogen by electrolysis in dilute sulphuric acid.
The kinetics of adsorption with long-range interaction between adsorbed particles
Authors: Wang J. S.
Published: November 2008

The rates of condensation and evaporation of adsorbed particles when they interact with long-range forces are deduced from kinetic considerations, including also the case of adsorption of diatomic molecules with dissociation. The rate of evaporation is compared with experiment, and the variation of the dipole moment with θ required to make the theory agree with experiment is discussed.

Finally the formulae for the rates of condensation and evaporation are applied to the problem of the diffusion of gases through metals, and it is found that for the processes considered the effect of interaction does not alter the conclusion arrived at in an earlier paper concerning the diffusion equation at small p and at large p.
A note on an assumption in the theory of cooperative phenomena
Authors: Rushbrooke G. S.
Published: November 2008

When the approximate methods of Bragg and Williams or Bethe are used in theoretical work on cooperative phenomena it is customary to suppose that only the configurational energy of the assembly depends on the molecular arrangement; the contribution which a molecule makes to the partition function, apart from its share in this configurational energy, is supposed to be independent of the positions of the other molecules.
Note on the production of positron-electron pairs during the passage of β-particles through matter
Authors: Feather N. , Dunworth J. V.
Published: November 2008

The method of coincidence counting has been applied to an investigation of the possible production of positron-electron pairs by the high energy β-particles from a source of uranium X in absorbers of aluminium, brass and lead. The results are not inconsistent with the high values recently reported for atomic cross-sections for the effect, nor with the suggestion that the atomic cross-section is proportional to the first power of the atomic number rather than the second. Suggestions are made for the use of the β-particles from artificially radioactive substances in an attempt to increase the sensitivity of the method.
The “overshoot phenomenon” in superconductivity
Authors: Misener A. D.
Published: November 2008

In a previous publication (1) the author described experiments in which the “overshoot phenomenon” reported by Silsbee, Scott and Brickwedde (2) might have been expected but was not observed. The specimens used were straight wires mounted so as to be completely surrounded by the liquid helium bath. Silsbee and his co-workers used wires coiled on paper or glass cylinders. It was concluded that the “overshoot phenomenon” was due to this method of mounting the specimen and was therefore a secondary effect.
Anomalous scattering of α-particles and long-range nuclear forces
Authors: Massey H. S. W. , Mohr C. B. O.
Published: November 2008

The recent experiments of Mohr and Pringle on the anomalous scattering of α-particles in hydrogen, deuterium and helium have revealed the existence of appreciable deviations from Coulomb's law at quite large distances (of the order 10−12 cm.). One explanation of this result which immediately suggests itself is that these deviations arise from ordinary electrostatic polarization of the colliding nuclei due to their electric charges, an effect considered originally by Debye and Hardmeier by classical methods to explain anomalous scattering. It is fortunately easy to examine whether this is the correct explanation. If we assume polarization to be the main cause of long-range anomalies in nuclear interaction, it is clear that these anomalies must fall off inversely as the fifth power of the distance. From the observed scattering of α-particles in helium which have √20 units of angular momentum relative to the struck nuclei and so pass at distances where the long-range anomalies are alone important, it is possible to determine the constant in the inverse fifth power law. From this constant the polarizability of the α-particle necessary to give rise to the observed effects can be determined. Now although it is not possible to calculate the polarizability of an α-particle with any degree of accuracy, our knowledge of its structure not being sufficiently definite, we can perform the calculation for the deuteron with some accuracy. Since the a-particle is certainly a more compact structure than the deuteron there is no doubt that it has a much smaller polarizability. Comparison of polarizabilities derived from the scattering and calculated from the known structure of the deuteron can therefore decide for or against this explanation of the anomalies.
Further possibilities for the absorption method of investigating the primary β-particles from radioactive substances
Authors: Feather N.
Published: November 2008

A detailed account is given of an absorption method, using the β-radiation of radium E as a standard of comparison, by which the maximum energy and some indication of the form of the energy spectrum of the β-particles from any other radioelement may be obtained. The method is applicable alike to weak sources and to sources of finite thickness and is designed to yield the largest amount of information under such limitations. Experimental results with sources of UX2, Na24, Ag106 and P32 are treated by the method described. A modified empirical range-energy relation is given.
The Theory of Adsorption of Gases on Solids when the Potential Energy Varies Continuously over the Surface
Authors: Miller A. R. , Roberts J. K.
Published: November 2008

In developing the theory of adsorption taking into account the interaction between adsorbed particles it has been usual to use a physical model in which it is assumed that there is a fixed interaction energy between particles adsorbed on neighbouring sites on the surface. In this paper the differences between the behaviour of this model and that of actual surfaces are discussed by considering a one-dimensional film in which the potential energy of a single adsorbed particle varies continuously and periodically with its position on the surface and in which there is a repulsive force between adsorbed particles which varies with the distance between them according to an inverse power law. For such a physical model the variation of the heat of adsorption with the fraction of the sites occupied is considered in detail and it is shown in particular that there is much less difference between the behaviour of mobile and immobile films than is indicated by the earlier model in which a fixed interaction energy is assumed. These results are considered in connexion with the interpretation of experiments on the adsorption of hydrogen on tungsten.
The symmetrical vibrations of a thin elastic plate
Authors: Sneddon Ian N.
Published: November 2008

In recent years, the theory of integral transforms—often in the guise of ‘operator calculus'—has been employed to obtain the solutions of a wide variety of problems in mathematical physics, particularly in the theory of vibrations. The most commonly employed transforms are those of Fourier, Laplace and Mellin; in problems where the range of one of the variables is (0, ∞) or (− ∞, ∞) these transforms have often been used with success to reduce a partial differential equation in n independent variables to one in n − 1 variables and hence to simplify the boundary value problem involved. In this paper we consider some simple problems in the theory of elastic vibrations in which the Bessel, or Hankel, transform is more useful than those mentioned above and is consequently employed throughout. In the first instance we consider problems in which the variable to be eliminated from the differential equation ranges from 0 to ∞ so that the theory of Hankel transforms is directly applicable; the application of finite intervals is then extended to the Hankel transform by a method similar to that used by Doetsch in his treatment of the Fourier transform.
Approximate methods in surge-tank calculations
Authors: Binnie A. M.
Published: November 2008

The equations of motion in a surge-tank fitted to a hydraulic pipe-line are non-linear. Various methods of solving them were described by Durand: (1) simplification of the equations and the use of empirical coefficients, (2) simplification by taking the friction forces as proportional to the first power of the velocity, (3) a step-by-step method of solution. As a revolt against the serious expense of time involved in (3) (which was later developed by Jakobsen and by Cole), he was led to devise other methods including the use of models, which he dealt with subsequently in more detail. Nevertheless, there remains a need for a quick approximate method of calculating the maximum surges which may be used with confidence at least in the early stages of design. A promising line of attack was suggested by the success which has attended the approximate analysis of mechanical vibrating systems having non-linear characteristics. This method has been set out by Den Hartog, who dismisses step-by-step solutions as ‘too laborious’.
The theory of surfaces in a geometry based on the notion of area
Authors: Davies E. T.
Published: November 2008

In his tract on spaces based on the notion of area, Cartan (1) gives a theory of curves and of surfaces immersed in his 3-dimensional space. A theory of curves is possible by associating with each point of the curve the 2-dimensional orientation of the contravariant bivector (or the corresponding covariant vector density of weight − 1) of support which is normal to the curve at that point. This association is possible, since there is a unique normal 2-direction at every point of a curve in a 3-dimensional space. If the surrounding space has more than 3 dimensions, however, and the element of support is still a bivector, then a theory of curves becomes impossible since no unique normal or tangential 2-direction can be associated with the curve at every point. A theory of surfaces is still possible because a surface has a unique tangential 2-direction which can be taken to be the 2-direction of the bivector of support. It is the object of this note to consider what subspaces can be admitted by a space when the number of dimensions is n > 3.
On homotopy and deformation retracts
Authors: Hu Sze-Tsen
Published: November 2008

As a partial generalization of a theorem of Hurewicz (3), Whitehead gave an important theorem on deformation retracts ((9), p. 273) which was proved only for polytopes. A demonstration of a part of Whitehead's theorem for absolute neighbourhood retracts was published by Fox ((2), p. 42) with a remark given by Samelson(8).
The theory of the thermoelectric power of metals
Authors: Sondheimer E. H.
Published: November 2008

1. The theoretical study of the transport phenomena in metals requires the solution of a certain integral equation, which is formed by equating the rate of change of the distribution function of the conduction electrons due to the applied fields and temperature gradients to the rate of change due to the mechanism responsible for the scattering. This integral equation has so far been solved only in the simplest case where the electrons are assumed to be quasi-free, the energy being proportional to the square of the wave vector, and this assumption will be made throughout the present paper. The scattering of the electrons is due to two causes: the thermal vibration of the crystal lattice and the presence of impurities or strains. The scattering due to the impurities can always be described in terms of a free path l (or more conveniently a time of relaxation τ, were l = τν, ν being the average velocity of the conduction electrons), whereas for the scattering due to the thermal vibration this is possible only at high temperatures such that (Θ/T)2 can be neglected, where Θ is the Debye temperature (1). At very low temperatures the scattering due to the thermal vibration can be neglected compared with that due to the impurities. When a time of relaxation exists, the integral equation for the distribution function reduces to an ordinary equation, and its solution is then a comparatively simple matter. In the general case the problem is much more difficult, and no rigorous and generally valid solution has so far been obtained.
On the determination of phase shifts
Authors: Ramsey W. H.
Published: November 2008

In this paper an approximate formula is developed for the phase shifts required in the calculation of the scattering of particles according to wave mechanics. It is shown to be more accurate than the current approximations and it is not difficult to handle. The same method can also be used to compute any quantity which is sensitive to the form of the wave function near the centre of force (e.g. a matrix element).
Combinants of a pencil of quadric surfaces (III)
Authors: Todd J. A.
Published: November 2008

The two previous papers of this series dealt with the determination of the complete system of combinantal covariants and contravariants of a pencil of quadric surfaces. The present paper is concerned with combinantal forms involving line-complexes, and with certain syzygies which exist between these forms. It is essentially a preparation for the determination of the complete system of combinantal line-complexes of the pencil, which will be carried out explicitly in the next paper of the series.
Remarks on the Dirac δ-operator
Authors: Hamburger Hans Ludwig
Published: November 2008

1. In this note we shall prove the

Theorem 1. Letbe a linear space of (real or complex) functions f(s) defined in the interval 0 ≤ s ≤ 1 subject to the following two conditions:

(i) every function of the infinite sequence 1, s, s2, …, sn, … is an element of;

(ii) two elements, f(s) and g(s), ofare to be considered as distinct if, and only if, they differ on a set of positive measure in the interval 0 ≤ s ≤ 1.
The condition of a certain matrix
Authors: Todd John
Published: November 2008

1. The following sentence occurs in a recent paper ((1), p. 54): we do not understand precisely what it means, but we shall show that the idea behind it can be put on a reasonably quantitative basis: ‘In the vocabulary of solvers of simultaneous equations, the operator (1, −2, 1) tends to lead to ill-conditioning when the number of equations is large.’
Some metrical theorems in Diophantine approximation. III
Authors: Cassels J. W. S.
Published: November 2008

Let

be w sequences of differentiable functions defined in the ranges

where aj, bj may be − ∞, + ∞ respectively. For each j and each m ≠ n let

be monotonic and let there be a constant K independent of m, n, j, θj such that
The oscillatory behaviour of moving averages
Authors: Moran P. A. P.
Published: November 2008

The theory of graduation discusses methods of obtaining a smoothed series of values of a function from a given empirical set of values. This is usually done by replacing each observation by a weighted average of it and neighbouring observations. Thus if {xi} (i = 0, ± 1, …) is a sequence of values, we can replace them by the series

where the A's are constants which are chosen in some suitable manner. If we regard the xi as the sum of a functional part fi and an error term εi we may attempt to choose the A's in such a way that the functional part is reproduced as well as possible, e.g. that

for some desired type of function. If this is so and the error terms εi are independently distributed with zero mean and finite standard deviation σ, we find that

where the ηi are a series of random variables which are no longer independent of each other but which have zero mean and a standard deviation σ1 such that
A note on the relativistic invariance of quantized field theories
Authors: de Wet J. S.
Published: November 2008

In an earlier paper (1), which will be referred to as A, the present author has demonstrated the relativistic invariance, for general transformations of coordinates, of the Einstein-Bose and Fermi-Dirac quantizations of linear field equations derived from higher order Lagrangians. The proof consisted of the identification of the commutation relations with the generalized Poisson brackets introduced by Weiss (2) and proving the invariance of the latter.
Two maximal subgroups of a collineation group in five dimensions
Authors: Hartley E. M.
Published: November 2008

Several recent papers have been concerned with a group G, of order 28.36.5.7, which can be represented as a collineation group in five dimensions. This collineation group is generated by harmonic inversions (projections) which leave fixed a point (the vertex) and a prime (said to be conjugate to the vertex). There are 126 projections in G and the set of 126 vertices form a configuration which is described in a paper (Hamill (3)) in which the operations of G are expressed as products of at most six projections. The group leaves invariant six algebraically independent primals; these have been determined (Todd (6)), and the equations of the simplest are given. The simplest invariant is a sextic primal, and some properties of this have been recorded in an earlier paper (Hartley (4)).
On the asymptotic probability distribution for certain Markoff processes
Authors: Ledermann Walter
Published: November 2008

The type of Markoff process which is considered in this paper corresponds to a system capable of n states, the time being regarded as a continuously varying parameter. At any instant t the probability distribution is represented by the vector
On the line-comitants of a space cubic
Authors: Weitzenböck R. W.
Published: November 2008

In a short paper recently published in these Proceedings, E. J. F. Primrose determines amongst other things a projective invariant N of a space cubic K and a linear complex L, N being of the third degree in the coefficients cik of the complex L. The purpose of the following remarks is to establish the connexion between this invariant N and the well-known projective comitants of the cubic K.
The general motion of a sphere in a viscous liquid
Authors: Drazin M. P.
Published: November 2008

1. In his classical investigation of the motion of a sphere moving (without spin) in a fluid at rest at infinity, Stokes showed (using his method of ‘stream functions') that the sphere experiences a thrust − 6πμau, where a is the radius of the sphere, and μ, u have their usual meanings.
The relativistic mass of a rotating cylinder
Authors: Gilloch Josephine M. , McCrea W. H.
Published: November 2008

An investigation by G. L. Clark is found to provide exceptionally instructive illustrations of the calculation of the variously defined ‘densities’ and ‘masses’ occurring in general relativity theory and of their physical significance. In particular, the effects of motion and stress upon gravitational mass are elucidated. An explicit example shows how kinetic energy is included in the gravitational mass.
On the interaction of colloidal particles IV. general mathematical theory for two identical particles
Authors: Levine S.
Published: November 2008

A general theory of the interaction of two charged identical colloidal particles of arbitrary shape is developed. An expression for the Helmholtz free energy of the electric double layers is obtained by the methods of statistical mechanics. The condition that there is equilibrium between the ions adsorbed on the surfaces of the colloidal particles and those dissolved in the dispersion medium is accounted for by requiring that the free energy of the whole system be a minimum with respect to variation of the ionic density on the surfaces.

The theory presented here is a further development of the work of Verwey and Overbeek. The conclusions of this paper are that in dilute sols, to which the present investigations are restricted, the results of these authors require extension in two directions. First, there is a correction to the mutual energy of two particles, due to the Coulomb interaction of the ions in the bulk of the solution. Secondly, no special assumption concerning the relation between the surface potential (or charge) and interparticle separation need be introduced. The equations set up to determine the free energy of interaction at the same time yield the ‘adsorption isotherm’ for the ion type which is common to the solution and the surface of the particle.
Humbert's plane sextics of genus 5
Authors: Edge W. L.
Published: November 2008

1. In 1894 Humbert encountered a twisted curve C7, of order 7 and genus 5, the locus of points of contact of tangents from a fixed point N0 to those twisted cubics which pass through five fixed points N1, N2, N3, N4, N5. The cubics of this family which touch an arbitrary plane do so at points on a conic, and it was by investigating this complex of conics that Humbert was led to study C7.
On the interaction between shock waves and boundary layers
Authors: Stewartson K.
Published: November 2008

The effect on the boundary-layer equations of a weak shock wave of strength ∈ has been investigated, and it is shown that ifRis the Reynolds number of the boundary layer, separation occurs when ∈ =o(R−i). The boundary-layer assumptions are then investigated and shown to be consistent. It is inferred that separation will occur if a shock wave meets a boundary and the above condition is satisfied.
On the law of decay of homogeneous isotropic turbulence and the theories of the equilibrium and similarity spectra
Authors: Goldstein S.
Published: November 2008

The requirements of Kolmogoroff's theory of the equilibrium spectrum are satisfied only at very high Reynolds numbers, higher than any at which experiments have yet been done. In particular, when the theory holds, the rate of decay of the mean-square vorticity ω must be negligible compared with either its rate of increase due to the stretching of the vortex filaments or the rate of dissipation due to viscosity.

An extended version of Kolmogoroff's hypothesis may be proposed, in which the statistical properties of the turbulence in a range of wave-numbers (range of eddy sizes) depend not only on the rate of dissipation ∈ per unit volume and the viscosity ν, but also on the time rate of change d∈/dt of ∈. The result is to introduce a dependence on the Reynolds number R of the turbulence into quantities and constants which, on Kolmogoroff's original hypothesis, were independent of R. The Reynolds number R is defined from the decay law; u2t, with an origin of time suitably chosen, is a function of t, finite when t = 0, and R is defined as (u2t)t=0/ν. Lin's decay law follows logically from the extended hypothesis, according to which the rates of change of ω−1, due to the causes mentioned above, are constant during the decay; Lin's decay law would also follow if this less general part only of the extended hypothesis be assumed. The same decay law is also obtained if the similarity spectrum of Heisenberg is taken to apply not to the whole of the energy-bearing eddies, but only to the energy-dissipating eddies. But it is suggested that further generalization of the theory of the similarity spectrum, and of the decay law, is necessary; that the similarity spectrum is probably only asymptotically correct for a range of large wave-numbers, the range depending on the initial conditions and decreasing as the decay proceeds; that the general decay law is u2t = μRd(t), where d(t) is an integral function of t, such that d(0) = 1, and with an asymptotic value for large t to give correctly the law of decay in the final period; and that d(t), and the number of constants needed to specify it approximately, depend on the initial conditions. An experiment is suggested to test the dependence of the law of decay on the initial conditions. It is also suggested that the recently observed constancy of u2t in the initial period in the turbulence behind a single grid is only approximate, and this approximate constancy still needs explanation. Remarks are also included on the range of application of the equilibrium spectrum, for which some formulae are given when there is a definite cut-off in the spectrum.
On matrices whose eigenvalues are in arithmetic progression
Authors: Landsberg P. T.
Published: November 2008

The following matrix problems are well known in quantum mechanics:

(a) The one-dimensional harmonic oscillator. Given

determine the eigenvalues hjj of H, and the matrix elements of X, P if H is diagonal. It is found (Wigner (4)) that
Parametric surfaces
Authors: Reifenberg E. R.
Published: November 2008

1. In this paper I consider various properties of the area of a surface including the relationship between Hausdorff measure and the Lebesgue area. The results here obtained will enable me, in a subsequent paper, to investigate the tangential properties of parametric surfaces. For simplicity of exposition I confine my attention to surfaces homoeomorphic to a disk, but the general case follows easily from cyclic element theory.
On direct decompositions. II
Authors: Goldie A. W.
Published: November 2008

The theory developed in our first paper is applied here to the case of algebras A such that every pair of congruences on A commute. The general refinement theorem which we obtain is already known when the ascending and descending chain conditions hold for the lattice of congruences. This result is due to O. Ore. We here weaken this finiteness assumption to a point comparable with that of group theory when both chain conditions are assumed for subgroups of the centre. Our path to the theorem is obtained by adapting to a different situation methods developed by Kurosch, Baer, and Jónsson and Tarski. From Kurosch (4) we take the concept of the centre of a pair of direct decompositions, which we extend and formulate as a pair of mappings π [α, β; γ, δ] over the set of congruences. In this connexion, Theorems 12·5 and 12·6 appear to be new, even for groups. From Baer (1) we take the concept of canonical refinements and generalize it to our case. The final approach (§ 14) to the refinement theorem is essentially following Jónsson-Tarski (3). Possibly our most interesting conclusion is that the elegant methods using endomorphisms, introduced by Fitting into group theory, generalize naturally to our case.
A method for the determination of converging factors, applied to the asymptotic expansions for the parabolic cylinder functions
Authors: Miller J. C. P.
Published: November 2008

1. The method of converging factors, for hastening the convergence of slowly convergentseries and improving the accuracy of asymptotic expansions, was introduced by J. R. Airey and is well known to computers (see Airey(1) and Rosser(2)). The principle is as follows. It is required to compute a quantity which is expressed as an infinite series

The series may be either convergent or asymptotic and divergent.
On partitions of bipartite numbers
Authors: Auluck F. C.
Published: November 2008

1. In statistical mechanics, we usually deal with assemblies containing a fixed number of particles in which energy is the only conserved quantity. Recently, Fermi (1) has shown that the angular distribution of the pions produced in high-energy nuclear collisions can be explained if one takes into account the conservation of angular momentum in addition to the conservation of energy. We are, therefore, led to discuss the thermodynamical properties of assemblies characterized by the conservation of two or more parameters. The simplest assumption of this type that we can make is that there are two parameters, say E and P, which are conserved, and that each particle of the assembly can occupy the levels (r, s) (r, s are non-negative integers), where the contribution of the level (r, s) to E is rε0 and to P is sη0. In order to find the entropy, and hence other thermodynamical properties of the system, we have to enumerate the distinct number of ways, p(m, n), in which an assembly of particles corresponding to given values of E = mε0 and P = nη0 can be realized. In this paper we find asymptotic expressions for p(m, n) in the following cases: (a) m is a fixed number, (b) m and n are of the same order. It is assumed here that the number of particles is greater than m and n. We deal with the case (a) in §2 and the case (b) in §4. §3 deals with the asymptotic expansions of the generating function for p(m, n) which are used in §4.
Fredholm solutions of partial integral equations
Authors: Salam Abdus , Smithies F.
Published: November 2008

The following ‘partial’ integral equation arises in the quantum theory of fields:

where f(x1x2) = f(x2x1), K(x1x2, y) = K(x2x1, y), and a symmetric solution of the equation is desired. The solution of (1) will be obtained as a special case of the solution of the following equation:
The diamagnetism of quasi-bound conduction electrons
Authors: Wilson A. H.
Published: November 2008

The diamagnetism of the conduction electrons gives rise to some of the most difficult problems in the theory of metals, the complete solution of which has not yet been found. Formally, the problem is equivalent to determining the density matrix

and the exact expression for ψ(r′, r, γ) for perfectly free electrons in a constant magnetic field H has recently been found by Sondheimer and Wilson(2). The extension of the theory to deal with quasi-bound electrons for all values of H seems to be out of the question, but an approximate partition function was given by Peierls (1), excluding terms of higher order than H2. In The theory of metals ((3), referred to as T.M.) I gave a more powerful and simpler method of dealing with the problem, based upon the properties of ψ(r′, r, γ), but since the solution was obtained as a power series in μ0Hγ, where it could at best determine only the normal diamagnetism.