bluuebunny/crossref_metadata_2025_split huggingface dataset:
The moment of the fluid forces acting on a plate which is gliding on the surface of a stream
Authors: Green A. E. 
Published: November 2008

13. Formulae have been obtained for the moment of the fluid forces which act on a plane plate which is gliding on the surface of a stream of finite or infinite depth. It has been shown that the formulae for the lift and moment forces acting on a plate which is gliding on a stream of infinite depth may be obtained by a limiting process from the formulae for the general case of gliding on a stream of finite depth. Also, by another limiting process, the lift and moment forces for the Rayleigh flow past a flat plate in an infinite stream have been obtained from the lift and moment forces for gliding on a stream of infinite depth.Numerical work has been carried out for the case when the angle of incidence of the plate to the stream is 5° and this has shown the variation of the lift and moment forces and the position of the centre of pressure with a range of values of the length of the plate, the depth of the stream and the height of the trailing edge of the plate above the bed of the stream.
A note on functions regular and bounded in the unit circle and small at a set of points near the circumference of the circle
Authors: Maitland B. J. 
Published: November 2008

The following theorem was proved by Milloux:Theorem A. Suppose that f (z) is regular and that | f(z) | < 1 in the unit circle. Suppose also that the set of points in the circle | z | ≤ r′ < 1 at whichcannot be enclosed in circles, the sum of whose radii is equal to 2eh−1. Thenwhere K is a numerical constant.
Radiating electron in a magnetic field
Authors: Eliezer C. Jayaratnam 
Published: November 2008

The paper considers the application of Dirac's classical theory of radiating electrons to examine the motion of an electron in a uniform magnetic field. The non-relativistic equations of motion are solved exactly. It is shown that for the particular case when the motion is confined to a plane, the physical motion is such that the electron describes an equiangular spiral with velocity which steadily decreases asexp {negative constant × t};and in the non-physical motion the electron spirals outwards with velocity which steadily increases asexp {positive constant × t},and ultimately the electron escapes to infinity. The relativistic equations of motion are solved approximately as a series in ascending powers of the field. It is shown that in the physical motion the velocity begins to decrease, and hence after a time the motion is given correctly by the non-relativistic solution.I take this opportunity to express my deepest gratitude to Prof. Dirac, who suggested this problem, for his patient guidance and supervision.
On the closure of
Authors: Fuchs W. H. J. 
Published: November 2008

1. A set of functions {øν (t)} (ν = 1, 2, …) is said to be closed L2 in (a, b), ifimplies f (t) = 0 p.p. in (a, b). It is well known that a set of functions is closed L2, if and only if every function of L2 (a, b) can be approximated in the mean square sense as closely as desired by finite linear combinations of the øν (t).
Random associations on a lattice
Authors: Moran P. A. P. 
Published: November 2008

P. 323, line 3 from bottom:for [8mn−7m−7n+n] read [8mn−7m−7n+4].P. 328, lines 16 and 18:for (6mn−6m−6n+l) read (6mn−6m−6n+4).line 17,in formula ending … +13m+13n−9) read …+13m+13n−8).Vol. 43 (1947), pp. 321–8.
The Parseval formulae for monotonic functions. III
Authors: Edmonds Sheila M. 
Published: November 2008

This sequence of papers is concerned with a group of theorems on Fourier transforms, and with the corresponding results for Fourier series. The transform theorems deal with the equationswhen two of the functions concerned are monotonic; and the series theorems, with the equationswhen two of the functions or sequences of coefficients are monotonic. A general survey of the problem was given in two earlier papers, which we shall call M.F. (I) and M.F. (II), and the results obtained were tabulated in M.F. (II). There is thus no need to restate the position in detail here. We shall merely recall as briefly as possible what has been proved already, so as to show the scope of the present paper.
On the theory of grand partition functions
Authors: Münster Arnold 
Published: November 2008

ABSTRACTThe grand partition function is derived by averaging over all partitions and numbers of molecules of a grand ensemble. The statistical parameters that arise are interpreted thermodynamically. A formula for the mean fluctuation of the numbers of molecules is deduced, and it is shown that the latter can become very great for singularities of the partial potentials. In that case, the system becomes statistically indeterminate, and separates out into several phases. Conversely, in the statistics of actual systems, the search for singularities, at which the partial potentials become independent of the composition, leads to the conditions under which a phase-change takes place. It is shown that, on certain hypotheses, the grand partition function is the generating function of the ordinary partition function, and that the latter can always be represented in product-form. The generating function of a factor in this product-form is called a reduced grand partition function. The significance of its parameters is discussed, and thoir use in the statistical thermodynamics of liquid mixtures is reviewed.
Zur Verallgemeinerung des Schwarzschen Lemmas auf mehrere Dimensionen
Authors: Peschl Von E. , Müller Cl. 
Published: November 2008

Das Schwarzsche Lemma der klassischen Funktionentheorie einer Veränderlichen kann als vollständige Beschreibung der inneren Abbildungen des Einheitskreises (mit Fixpunkt O) aufgefaßt werden. Ist nämlich f(z) eine in |z| ≤ 1 reguläre Funktion mit |f(z)| ≤ 1 und f(0) = 0, so muß |f′(0)| ≤ 1 sein, und es stellt diese Ungleichung bekanntlich den ersten Schritt zur Lösung des Koeffizientenproblems der im Einheitskreis gleichmäßig beschränkten Funktionen dar. Die Aussagekann andererseits so gedeutet werden, daß jede innere Abbildung des Einheitskreises mit Fixpunkt O auch innere Abbildung jedes konzentrischen Kreises |z| ≤ |ρ| < 1 ist.
Notes on gas flow through a nozzle
Authors: Binnie A. M. 
Published: November 2008

AbstractIn Fart 1 the oscillations are examined which may develop in supersonic flow through the divergent part of a nozzle. Hooker's theory is found to be confirmed by observations on a steam nozzle.In Part 2 it is shown that Taylor's approximate theory of flow between circular arcs can also analyse the throat conditions when the velocity of approach to the throat is supersonic. Limiting symmetrical flow occurs when the velocity at the centre of the throat just exceeds the local velocity of sound, no matter what the radii of the arcs and the ratio of the specific heats of the gas may be. The unique asymmetrical case is merely the reverse of Taylor's solution for the flow when the velocity of approach is subsonic.A numerical comparison has been made of the velocities across the throat as calculated by Taylor's method and by Fox and Southwell's iterative process when the velocity of approach is subsonic. The agreement is good for limiting symmetrical flow but not so satisfactory for the unique asymmetrical case.
Transonic flow past finite wedges
Authors: Mackie A. G. , Pack D. C. 
Published: November 2008

AbstractThe solution for the flow of an incompressible fluid past an infinitely long wedge with a finite sloping edge (a finite wedge) is generalized by the hodograph method. In the flow thus obtained the axis of symmetry and a sloping edge of the wedge are again part of one streamline. It becomes possible to describe the flow of an ideal gas past a finite wedge if the hypothesis is made that the first singularity on this streamline, along the sloping edge, corresponds to the shoulder of the wedge. For a given wedge, with gradually increasing velocity at infinity upstream, the singularity appears at first at subsonic velocity. Beyond a certain critical velocity at infinity the singularity is always associated with the speed of sound. The hypothesis thus implies that put forward by Maccoll(9) and supported by Busemann(l). A qualitative examination shows that the solution reproduces experimentally known features of the flow of compressible fluid past a finite wedge.
On the region defined by | xy | ≤ 1, x2 + y2 ≤ t
Authors: Ollerenshaw Kathleen 
Published: November 2008

In this paper I find the critical lattices of the region Kt defined by the inequalities | xy | ≤ 1, x2 + y2 ≤ t (t > 0). I deduce in §9 the arithmetic minima of a pair of binary quadratic formswhere f1 is indefinite, f2 is positive definite, and f1, f2 are harmonically related, that is to sayand
On eigenvalues and inverse singular values of compact linear operators in Hilbert space
Authors: Silberstein J. P. O. , Smithies F. 
Published: November 2008

1·1. In this paper we shall be concerned with the equationswhere K is a compact (completely continuous) linear operator in a Hilbert space , K is the adjoint of K, I is the identity operator, x and y are elements of ∥ x ∥ denotes the norm of x, and κ and σ are complex numbers.
Some formulae of P. Stein and others concerning trigonometrical sums
Authors: Slater N. B. 
Published: November 2008

Letwhere the frequencies vr are strictly positive and the amplitudes ar are non-zero; and letbe the number of zeros of f(t) – a for 0 ≤ t < T. In physical problems, where f(t) may represent, for example, a general coordinate in a vibrating dynamical system or a set of alternating currents in a cable, formulae are required for the frequency with which f(t) passes through a given value a; that is, for asymptotic or average values of Ga/T. The purpose here is to collect such formulae, and to sketch their background and relations in a way which may suggest extensions. The writer is particularly indebted to Prof. P. Stein for a manuscript containing new results (SI-V in §§2 and 5 below) and for his permission to publish them without his original proofs; these proofs were extensions of the method he gave (6) for the case n = 2.
Diffraction by two parallel planes of finite length
Authors: Williams W. E. 
Published: November 2008

ABSTRACTThe diffraction of a plane harmonic electromagnetic wave by two parallel, perfectly conducting planes of infinite width and finite length is considered. The wave is assumed to be polarized perpendicular to the edges of the plane, and thus the problem can also be regarded as that of diffraction of a plane sound wave. Two integral equations are obtained, and the equations are solved approximately when the length of the planes is large compared with the wave-length. The resonance of the system is also considered and an equation obtained for the resonant lengths. This equation is solved by an approximation depending on the assumption that twice the product of the wave number (k) and the resonant length (l) is always large.
On the motion of a flat plate at high speed in a viscous compressible fluid. I. Impulsive motion
Authors: Stewartson K. 
Published: November 2008

ABSTRACTThe effect of giving an infinite plate, immersed in a compressible fluid, an impulsive velocity U in its own plane is examined on the assumption that M = U/a0 ≫ 1 where a0 is the velocity of sound in the undisturbed fluid. To begin with the flow field is very complex but very soon after the motion has begun a shock wave moves at right angles to the plate gradually weakening and eventually degenerating into a sound wave. Attention is concentrated on the flow field when there is a distinct strong shock wave, it being assumed, and justified a posteriori, that the flow behind it may be divided into an inviscid region and a boundary layer region. The equation of the shock wave and a complete description of the flow behind it is found when log M, where ν0 is the coefficient of kinematic viscosity of the undisturbed air. An approximate method is also given to enable the solution to be joined up to Van Dyke's (10) which is valid when .
Contributions to the theory of stokes waves
Authors: De S. C. 
Published: November 2008

ABSTRACTThe well-known Stokes theory (9, 10) of waves of permanent form in water of finite depth has been extended to the fifth order of approximation. The solutions have been first obtained in the form of equations for the space coordinates x and y as functions of the velocity potential Φ and stream function ψ. Expressions for the complex potential W in terms of the complex variable z ( = x + iy), the form of the wave profile, and the square of the wave velocity have been obtained to the fifth order.Expressions for the three physical quantities Q, R and S, where Q is the volume flow rate per unit span, R is the energy per unit mass (i.e. g times the total head, measuring heights from the bottom and pressures from atmospheric) and S is the momentum flow rate per unit spaa, corrected for pressure forces and divided by density, have been obtained to the fifth order. The values for the dimensionless quantities r = R/Rc and s = S/Sc, where Rc and Sc refer to the values of R and S for a critical stream of volume flow Q, are tabulated for certain values of the ratios mean depth to wavelength and amplitude to wavelength. The values of r and s thus obtained have been used to calculate the ratios of mean depth to wavelength and of wave height to wavelength according to the cnoidal wave theory as recently presented by Benjamin and Light-hill(1), and the results are found to be in satisfactory agreement with that from Stokes's theory for waves longer than six times the depth.The (r, s) diagram introduced in the recent work of Benjamin and Lighthill(1) has been further considered, and the unshaded part of the diagram referred to in that paper has been mapped with a network of curves for constant values of the ratios of mean depth to wavelength and of wave height to wavelength (Fig. 2). The third barrier to the existence of steady flows, corresponding to ‘waves of greatest height’ referred to in that paper, has also been indicated in Fig. 2.
The growth of secondary circulation in frictionless flow
Authors: Hawthorne W. R. 
Published: November 2008

ABSTRACTThe appearance of a component of vorticity in the direction of flow materially alters the pattern of flow of a fluid in three dimensions. Expressions are obtained for this secondary vorticity in an inviscid compressible fluid flowing under the action of body forces. They are applied to examples such as a liquid under gravity and gas flow behind a curved shock. In compressible gas flow with varying temperature but constant stagnation pressure no secondary circulation appears. In a perfect gas atmosphere it is shown that secondary circulation may appear because of nou-adiabatic lapse rates as well as wind-speed gradients. It is also shown that in a liquid with a density gradient, gravitational effects can give rise to secondary vorticity components.
The numerical evaluation of a class of integrals
Authors: Moran P. A. P. 
Published: November 2008

Suppose thatx1, …,xnare jointly distributed in a multivariate normal distribution,f(x1…,xn), with a given correlation matrix (ρij). Then the evaluation ofis a problem which has been much discussed and has many applications (see David (1)). Whenn= 3
A method of improving Energy-level calculations for ‘series’ electrons
Authors: Douglas A. S. 
Published: November 2008

ABSTRACTA method is described for improving the approximate calculation of energy levels for atomic configurations consisting of a set of closed shells and a single ‘series’ electron. This involves the inclusion of a ‘polarization’ potential in the Hartree-Slater-Fock equation, caused by the series electron. The form of potential found in two cases investigated, namely, Si+3 and Ca+, is described and discussed.
The numerical solution of linear differential equations in Chebyshev series
Authors: Clenshaw C. W. 
Published: November 2008

ABSTRACTThis paper describes a method for computing the coefficients in the Chebyshev expansion of a solution of an ordinary linear differential equation. The method is valid when the solution required is bounded and possesses a finite number of maxima and minima in the finite range of integration. The essence of the method is that an expansion in Chebyshev polynomials is assumed for the highest derivative occurring in the equation; the coefficients are then determined by integrating this series, substituting in the original equation and equating coefficients.Comparison is made with the Fourier series method of Dennis and Foots, and with the polynomial approximation method of Lanczos. Examples are given of the application of the method to some first and second order equations, including one eigenvalue problem.
The torque on an infinite strip exposed to plane sound waves
Authors: Levine Harold 
Published: November 2008

ABSTRACTThe mean torque on an infinite fixed strip due to plane harmonic sound waves is calculated, employing simple acoustical theory as a basis and neglecting viscosity. In the range of wavelength λ small compared to the strip width δ, the torque per unit length iswhere α denotes the angle between the incident wave normal and that of the strip and ρο the equilibrium density of the medium. This oscillatory character of the torque in its dependence on δ/λ is not to be found at long wavelengths where, as is well known, the torque acts only in the sense of aligning the strip broadside to the incident wave normal.
Random processes in genetics
Authors: Moran P. A. P. 
Published: November 2008

ABSTRACTWright's model of a random process in genetica is modified by supposing that births and deaths occur individually at random so that the generations are no longer simultaneous. An exact solution is then obtained for the distribution of the number of a-genes in a haploid organism when mutation is occurring in both directions. When there is no mutation, the rate of approach to homozygosity is found in more complicated models with two sexes and diploid individuate. This rate is twice that occurring in Wright's models.
Elliptical whirl of flooded journal bearings
Authors: Halton J. H. 
Published: November 2008

ABSTRACTThe paper studies the problem of elliptical whirl of perfectly alined, finite, flooded, journal bearings, with incompressible isoviscous lubricant; it bases itself on Reynolds's equation. Explicit solutions are obtained, using the long (L/D≫ 1) and short (D/L ≫ 1) bearing approximations.The physical situations corresponding to the solutions obtained are those of unloaded and constant-loaded bearings, with elastic, massive, frictional suspensions. The agreement with empirical findings is generally good, and the paper is published in the hope that further experimental research will be stimulated by it.
Quadratic equations in several variables
Authors: Birch B. J. , Davenport H. 
Published: November 2008

Letbe a quadratic form in n variables (n ≥ 3) with integral coefficients. One of us gave recently (2) a simple proof of a result of Cassels (l), in the following form: if the equation f = 0 is properly soluble in integers, then it has a solution satisfyingwhere γn–1 denotes Hermite's constant.
On prime-power groups with two generators
Authors: Blackburn N. 
Published: November 2008

Let G denote a group of order a power of the prime p, and let G′ be the derived group of G. The lower central series of G will be writtenFor any subgroup H of G we denote by P(H) the subgroup of H generated by all elements xp as x runs through H, and by Φ(H) the Frattini subgroup of H. We write (H:Φ(H)) = pd(H); thus d(H) is the minimal number of generators of H.
Series expansions and general transforms
Authors: Miller J. B. 
Published: November 2008

A number of writers have noticed Fourier sine or cosine transform relations between certain pairs of series. For example, Duffin(4) has shown that in a variety of circumstancesare a pair of sine transforms, and Guinand(7) thatare cosine transforms, using L2 theory. Other examples are given in (2) and (5); and in (6) power series and general transforms are considered.
The interaction of light with neutral systems
Authors: Power E. A. , Shail R. 
Published: November 2008

A recent investigation of the interaction between the Maxwell radiation field and neutral atoms or molecules has led us to consider the relation between the straightforward interaction Hamiltonianbetween particle (k) and the radiation field, and the more physical, and explicitly gauge invariant, couplings of the form q. E and µ. H, etc., where q and µ are the electric and magnetic dipole moments of a system of particles. That these latter form of interaction energies follow from well-formulated approximations for atoms in external fields has been known for many years (Goeppert-Mayer(1); see also Rosenfeld(2)). When the Maxwell field is an internal and dynamical field further discussion is necessary, especially concerning the Coulomb interactions. Their advantage has recently been shown in dealing with dipole interactions in the long range Van der Waals force between two neutral systems where the complete coupling, in dipole approximation, is given by — q.ET for each neutral system, where ET is the transverse electric field. The short-range London limit is contained in this coupling despite the absence of longitudinal electric fields (Power and Zienau (3)). In this note we extend these arguments to higher electric multipoles and magnetic multipoles. We also demonstrate their use in the quantum theory of scattering of light, and in particular the derivations of Stephen (4) of the phase angle between emergent beams in optical double refraction phenomena are shown to follow immediately.
A thin circular plate normally and uniformly loaded over a concentric elliptic patch
Authors: Bassali W. A. , Nassif M. 
Published: November 2008

ABSTRACTThis paper is concerned with the small transverse displacement of a thin circular plate elastically restrained along the circular boundary and subject to uniform normal pressure distributed over the area of an ellipse concentric with the plate. The method of complex potentials is used and the general assumptions relating to the small deflexion theory of thin plates are adopted. Explicit formulae for the bending and twisting moments and shearing forces are given at any point on the boundary and at the centre of the plate.
Bending of a thin circular plate under hydrostatic pressure over a concentric ellipse
Authors: Bassali W. A. 
Published: November 2008

ABSTRACTAn exact solution in finite terms is derived within the limitations of the classical thin-plate theory, for the problem of a thin circular plate acted upon normally by hydrostatic pressure distributed over the area of a concentric ellipse, and subject to boundary conditions covering the usual rigidly clamped and simply supported boundaries.
Flexural problems of circular ring plates and sectorial plates. II
Authors: Bassali W. A. , Gorgui M. A. 
Published: November 2008

ABSTRACTIn a previous paper solutions have been derived for the deflexion of a circular annular plate under various combinations of boundary conditions and subject to certain line loadings spread along the circumference of a concentric circle. These solutions, which were used in the analysis of circular ring plates and sectorial plates acted upon by a concentrated load or a concentrated couple at any point, are utilized in this paper to obtain solutions appropriate to lateral loadings of the typedistributed over the entire circular ring plate. Solutions corresponding to the case φ(r) =rmare then explicitly found, and the principle of superposition can be applied to get the solution when the load on the annulus can be expanded into a Laurent-Fourierseries in the polar coordinatesr, θ. The method of images is used to deduce solutions for an isotropic plate bounded by two arcs of concentric circles and two radii, and loaded by the normal pressurep=p0rmwhen the plate is simply supported along the straight edges and is subject to various conditions along the circular edges.