bluuebunny/crossref_metadata_2025_split huggingface dataset:
Contractions of rotation groups and their representations
Authors: Dooley A. H. , Rice J. W. 
Published: November 2008

It is a classical result in the theory of special functions that Bessel functions are limits in an appropriate sense of Legendre polynomials. For example in (11), § 17.4, the following result is attributed to Heine:such limiting formulae are also known for certain other special functions (cf. (1), (5)). Apart from their intrinsic interest, these formulae have been used in the theory of special functions to obtain product formulae, etc. for the limit function from those of the approximating sequence.
Representation of tiled matrix rings as full matrix rings
Authors: Chatters A. W. 
Published: November 2008

It can be very difficult to determine whether or not certain rings are really full matrix rings. For example, let p be an odd prime, let H be the ring of quaternions over the integers localized at p, and setThen T is not presented as a full matrix ring, but there is a subring W of H such that T ≅ M2(W). On the other hand, if we take H to be the ring of quaternions over the integers and form T as above, then it is not known whether T ≅ M2(W) for some ring W. The significance of p being an odd prime is that H/pH is a full 2 x 2 matrix ring, whereas H/2H is commutative. Whether or not a tiled matrix ring such as T above can be re-written as a full matrix ring depends on the sizes of the matrices involved in T and H/pH. To be precise, let H be a local integral domain with unique maximal ideal M and suppose that every one-sided ideal of H is principal. Then H/M ≅ Mk(D) for some positive integer k and division ring D. Given a positive integer n. let T be the tiled matrix ring consisting of all n x n matrices with elements of H on and below the diagonal and elements of M above the diagonal. We shall show in Theorem 2.5 that there is a ring W such that T ≅ Mn(W) if and only if n divides k. An important step in the proof is to show that certain idempotents in T/J(T) can be lifted to idempotents in T, where J(T) is the Jacobson radical of T. This technique for lifting idempotents also makes it possible to show that there are (k + n − 1)!/ k!(n−1)! isomorphism types of finitely generated indecomposable projective right T-modules (Theorem 2·10).
Asymptotic expansions of generalized Bernoulli polynomials
Authors: Weinmann A. 
Published: November 2008

AbstractAsymptotic expansions of certain generalized Bernoulli polynomials are obtained, some in terms of elementary functions and others in terms of gamma functions and their derivatives. The latter results can also be written in terms of elementary functions, by using the known asymptotic expansions of the gamma functions, and the leading terms are obtained in this way. The results obtained give the asymptotic form of the coefficients occurring in all the usual central-difference formulae of the calculus of finite differences.
The volume of a lattice polyhedron
Authors: Macdonald I. G. 
Published: November 2008

Let L be the lattice of all points with integer coordinates in the real affine plane R2 (with respect to some fixed coordinate system). Let X be a finite rectilinear simplicial complex in R2 whose 0-simplexes are points of L. Suppose X is pure and the frontier Ẋ of X is a Jordan curve; then there is a well-known formula for the area of X in terms of the number of points of L which lie in X and Ẋ respectively, namelywhere L(X) (resp. L(Ẋ)) is the number of points of L which lie in X (resp. Ẋ), and μ(X) is the area of X, normalized so that a fundamental parallelogram of L has unit area.
Two-dimensional elliptic inclusions
Authors: Bhargava R. D. , Radhakrishna H. C. 
Published: November 2008

AbstractThe simple concept of minimum potential energy of the classical theory of elasticity, first applied to solve inclusion problems (1) by one of the authors (R. D. B.), who considered spherical and circular inclusions, has now been extended to solve elliptic inclusion problems. The complex-variable method of determining the elastic field, first enunciated by A. C. Stevenson in the U.K. and N. I. Muskhelishvili in the U.S.S.R., has been used to determine the elastic field in the infinite material (the matrix) around the inclusion. Strain energies are calculated. The equilibrium size of an elliptic inclusion of elastic (Lamé's) constants λ1 and μ1, differing from those of matrix, for which the constants are λ and μ, has been determined.An independent check on the calculations has been made by testing the continuity of normal and shearing stresses. The results also agree with the known results for the much simpler case when inclusion and matrix are of the same material.
The passage of electrons through small apertures
Authors: Lehmanna J. F. , Osgood T. H. 
Published: November 2008

The velocities of electrons emerging through an aperture in an anode to which they were accelerated have been investigated.The velocity distribution of electrons passing through holes ranging from 0·13 mm. to 3·24 mm. in diameter in a thin copper sheet is shown in Fig. 2. In this case the accelerating field between filament and anode was 204 volts.Corresponding results for a capillary tube 0·40 mm. in diameter and 16 mm. long are shown in Fig. 3, where the curves are plotted for different accelerating potentials. Experiments were also made with shorter lengths, down to 5 mm., and with a tube 1·25 mm. in diameter. All the tubes investigated gave the same distribution of velocities of the electrons in the emergent beam, within the limits of error of the experiments.In conclusion the authors wish to express their thanks to Sir Ernest Rutherford for his interest and advice during the course of the work.
The general double integral problem in the calculus of variations
Authors: Gillespie R. P. 
Published: November 2008

In a previous paper in these Proceedings the problem of the double integralwas discussed when the function F had the formwhereIt is proposed in the present paper to extend the method to the general problem, where F may have any form provided only that it satisfies the necessary condition of being homogeneous of the first degree in A, B, C.
Note on Ramanujan's arithmetical function τ (n)
Authors: Hardy G. H. 
Published: November 2008

In his remarkable memoir ‘On certain arithmetical functions’* Ramanujan considers, among other functions of much interest, the. function τ(n) defined byThis function is important in the theory of the representation of a number as a sum of 24 squares. In factwhere r24 (n) is the number of representations;where σs(n) is the sum of the 8th powers of the divisors of n, and the sum of those of its odd divisors; and
On the polarisation of mercury lines emitted from a discharge tube in a magnetic field
Authors: Skinner H. W. B. 
Published: November 2008

1. Polarisation measurements are described on the Hg lines λ 4358, 5461, 5770, 5791 emitted from a low-pressure electron-maintained arc to which a longitudinal field of up to 3000 gauss may be applied.2. Two types of polarisation effect have been found. The first is typified by the line λ 5461. It is a polarisation of the light with the electric vector perpendicular to the lines of force. Its maximum value is 9% but it decreases to zero (a) with decreasing field strength and (b) with decreasing current in the tube. It is suggested that this effect is due to a self-reversal of λ 5461 owing to the metastability of the 2 3p2 state and that the polarisation may be caused through a slight non-uniformity of the magnetic field.The second type of effect is found for λ5770 and 5791. The polarisation is in a plane parallel to the discharge, and is found in a zero magnetic field. This effect is further discussed elsewhere. The line λ 4358 is unpolarised under all conditions tried.3. Neither of these effects implies a polarisation of the total light emitted directly from atoms in the Transverse Zeeman Effect when the atoms are excited under isotropic conditions and the experiments give definite evidence against the existence of such a polarisation effect.
The adsorption of hydrogen on tungsten
Authors: Roberts J. K. 
Published: November 2008

A method for studying the adsorption of hydrogen on a clean tungsten surface has been developed which depends on the fact that the accommodation coefficient of neon is different for a bare surface and for a surface with an adsorbed film on it.It has been shown that on a bare tungsten surface, saturation occurs at a partial pressure of hydrogen of 4.10−4 mm. of mercury and probably much lower, so that the adsorption is of the nature of chemi-sorption.Important deductions can be drawn concerning the process of activation if any for this chemi-sorption of hydrogen on a bare tungsten surface.The accommodation coefficient of neon with a tungsten surface covered with a hydrogen film of the type investigated is 0·17 at 295°K, and 0·32 at 79°K.The experiments described were carried out during my tenure of a Moseley Studentship at the Cavendish Laboratory, and I should like to thank Lord Rutherford for his interest in the work and the Council of the Royal Society for their continued support. I should also like to thank Professor Rideal for his interest in the results.
Notes on Spectrophotometry
Authors: Ditchburn R. W. 
Published: November 2008

The source of light used as a background is an important factor in determining the convenience and accuracy of ultra-violet absorption work, etc. If a source of light of constant intensity is available, a direct comparison method can be used and it is only necessary to calibrate the plates. If the source of light is not constant in intensity, it is necessary to divide the light into two beams and use one to check the variations of intensity while the other goes through the absorbing substance or (during calibration) the reducing sector or wedge. This latter method requires much more complicated apparatus and if the variations in the source are at all large it becomes inaccurate. In addition to being constant in intensity a good background for ultra-violet absorption spectra should possess the following qualities:(1) Most of the energy should be emitted in the form of a continuous spectrum.(2) It is desirable to be able to use one photograph of the whole region to be investigated. For this purpose it is necessary that the variations of intensity in different parts of the spectrum should be small enough for it to be possible to arrange the exposure so that all parts of the spectrum are within the correct exposure range, i.e. it must not be necessary to over-expose any part in order to get a strong enough intensity at another wave-length.The hydrogen continuous spectrum possesses both these qualities and is an excellent background for the region on the short wave-length side of 3200 A.U. It may be used for longer wavelengths, but the hydrogen secondary lines are apt to prove trouble-some unless a fairly large dispersion is used.
The Upper Limits of Energy in the β-ray Spectra of Actinium B and Actinium C″
Authors: Sargent B. W. 
Published: November 2008

Absorption curves in paper have been obtained for the β-rays of actinium (B + C) and actinium C′. Some inferences regarding the shape of their energy distribution curves have been drawn, and the ranges of the β-rays of actinium B and actinium C′ fix their maximum β-ray velocities at Hρ 3430 and 6140. The potency of the method is emphasized for elements which cannot be easily investigated directly in a magnetic spectrograph.The γ-ray ionizations of actinium B and actinium C′ have been compared under the same conditions with the β-ray ionization of actinium C′.In conclusion, the writer wishes to thank Professor Sir Ernest Rutherford for his kind interest in the work. Thanks are also due to Dr J. Chadwick and Dr C. D. Ellis for many valuable suggestions and discussions.
Note concerning the β particles of very small Energy emitted during Radioactive Transformation
Authors: Feather N. 
Published: November 2008

A preliminary investigation is described in which a type of gold leaf source suitable for use with an expansion chamber is developed. Such a source is necessary for the detailed examination of β particles of energy less than about 50,000 electron volts by the Wilson method. Some results obtained with sources of radium D and thorium B + C of this type are presented in outline.The writer would like here to acknowledge the help and encouragement he derived from many discussions with Professor Sir Ernest Rutherford and with Dr J. Chadwick during the course of the investigation, and further to express his thanks to Mr G. R. Crowe for assistance in the preparation of some of the sources.
Concerning some Solutions of the Boundary Layer Equations in Hydrodynamics
Authors: Goldstein S. 
Published: November 2008

The boundary layer equations for a steady two-dimensional motion are solved for any given initial velocity distribution (distribution along a normal to the boundary wall, downstream of which the motion is to be calculated). This initial velocity distribution is assumed expressible as a polynomial in the distance from the wall. Three cases are considered: first, when in the initial distribution the velocity vanishes at the wall but its gradient along the normal does not; second, when the velocity in the initial distribution does not vanish at the wall; and third, when both the velocity and its normal gradient vanish at the wall (as at a point where the forward flow separates from the boundary). The solution is found as a power series in some fractional power of the distance along the wall, whose coefficients are functions of the distance from the wall to be found from ordinary differential equations. Some progress is made in the numerical calculation of these coefficients, especially in the first case. The main object was to find means for a step-by-step calculation of the velocity field in a boundary layer, and it is thought that such a procedure may possibly be successful even if laborious.The same mathematical method is used to calculate the flow behind a flat plate along a stream. The results are shown in Figures 1 and 2, drawn from Tables III and IV.
The stability of viscous fluid flow between rotating cylinders
Authors: Goldstein S. 
Published: November 2008

The stability of the motion of viscous incompressible fluid, of density ρ and kinematic viscosity ν, between two infinitely long coaxial circular cylinders, of radiiaanda+d, whered/ais small, is investigated mathematically by the method of small oscillations. The inner cylinder is rotating with angular velocity ω and the outer one with angular velocity αω, and there is a constant pressure gradient parallel to the axis. The fluid therefore has a component velocityWparallel to the axis, in addition to the velocity round the axis. A disturbance is assumed which is symmetrical about the axis and periodic along it. The critical disturbance, which neither increases nor decreases with the time, is periodic with respect to the time (except whenW= 0, when the critical disturbance is a steady motion). As Reynolds number of the flow we take ||d/ν, whereis the average value ofWacross the annulus, and we denote bylthe wave-length of the disturbance along the axis, by σ/2π the time period of the critical flow, bycthe wavelength of the critical flow, byωcthe critical value of ω, and we putapproximately, if α is not nearly equal to 1.
A note on the absorbability in cadmium of neutrons slowed down in water and boric acid solutions
Authors: Westcott C. H. 
Published: November 2008

Measurements have been made on the proportion of the effect induced in silver which is due to neutrons strongly absorbed by cadmium, using cylinders of water and boric acid of different sizes to slow down the neutrons. In addition to furnishing data for use in the foregoing paper, the results show that practically all the neutrons of groups A and B must have energies greater than any of those of group C.The author desires to thank Lord Rutherford for his interest in the work and for the use of a Ra + Be source, and Dr Oliphant for discussion and advice. He is also indebted to the Department of Scientific and Industrial Research for a Senior Research Award.
On the magneto-resistance effect in single crystals of cadmium
Authors: Milner C. J. 
Published: November 2008

A method of growing single-crystal resistance specimens free, to a large extent, from strains and impurities is described. The magneto-resistance effect in cadmium single crystals has been studied in some detail at the temperature of liquid nitrogen, using sufficiently high magnetic fields to observe the linear effect found by Kapitza.Although it proved impossible, in general, to determine the orientation of individual crystals, the experiments suggest that the “critical field” of the linear effect is dependent on the orientation of the crystal with respect to the current and magnetic field, and not on the perfection of the crystal lattice.
On two classical lattice point problems
Authors: Ingham A. E. 
Published: November 2008

1. Let r(n) be the number of representations of n as a sum of two squares, d(n) the number of divisors of n, andwhere γ is Euler's constant. Thus P(x) is the error term in the problem of the lattice points of a circle, and Δ(x) the error term in Dirichlet's divisor problem, or the problem of the lattice points of a rectangular hyperbola.
Deuteron bombardment of the heavy elements.
Authors: Krishnan R. S. , Nahum E. A. 
Published: November 2008

The results of the bombardment of mercury, lead and thallium by 9 M.e.V. deuterons are reported. The following radioactive isotopes have been detected: 5·5 min., 48 min., 36 hr., 60 day mercury isotopes; 4·4 min., 10·5 hr., 44 hr., and 13 day thallium isotopes; 10·25 min., 2·75 hr., and 54 hr. lead isotopes; 18 hr. and 6·35 day bismuth isotopes. The 10·25 min. lead isotope is positron emitting, an interesting result in an element of high atomic number. Absorption measurement have been made of the radiations emitted by many of these isotopes and assignments have been made in most cases.In conclusion we wish to thank Dr N. Feather for valuable discussions, and also for making for us a Ra E source. We are indebted to Dr Lewis for advice in setting up the thyratron scale of eight counter. This paper would be incomplete without a sincere acknowledgement of our indebtedness to the hard work of past members of this laboratory who have been mainly responsible for setting up the cyclotron.One of us (R. S. K.) is grateful to the Royal Commissioners for the Exhibition of 1851 for the grant of an overseas scholarship which made this work possible.
A method of calculating the integral breadths of Debye-Scherrer lines: generalization to non-cubic crystals
Authors: Stokes A. R. , Wilson A. J. C. 
Published: November 2008

Stokes and Wilson(1,2) have shown that the integral breadths of the Debye-Scherrer lines produced by small or imperfect crystals of the cubic system are given byor, what is the same thing, that the apparent particle sizes are given byIn these equations λ is the X-ray wave-length, θ is the Bragg angle, Vt is the volume common to the crystal and its ‘ghost’ shifted a distance t in the hkl direction(1), and Jt is the mean value of the product FF of the structure amplitudes of two cells separated by a distance t in the hkl direction (2). The purpose of this note is to outline an argument which establishes the same result for crystals of any symmetry.
On the origin of the solar system
Authors: Hoyle F. 
Published: November 2008

3. The essential requirement that the nova must satisfy in the above theory is that the total mass in the form of diffuse gaseous material must be of the order of 1/10 the solar mass, which requirement seems to be consistent with the observations of novae. Thus, although the theory applies explicitly to novae that break into two pieces of stellar mass having a gaseous filament drawn out between them (the pieces may have a mass ratio as great as 5/1), it seems clear that the discussion could be adjusted to include the case of novae following other models.The process described above can be applied to the formation of non-solar planets. The mass of available planetary material depends upon such factors as the mass of the nova and its separation from the companion before outburst. It is to be expected that variations in these factors can lead to planetary masses that vary over a fairly wide range. It follows that since an appreciable fraction of the stars of mass comparable or greater than the solar mass are known to be members of binary systems, the number of planetary systems must be at least of the same order as the number of novae that have occurred in this class of star.
Adsorption Isotherms. Critical Conditions
Authors: Fowler R. H. 
Published: November 2008

1. In a recent note I showed that Langmuir's adsorption isothermwhere θ is the fraction of the surface covered by adsorbed gas, p the gas pressure in equilibrium with it, and A (T) a specified function of the temperature, can be derived as a theorem in statistical mechanics without any appeal to the mechanism of deposition and re-evaporation. Necessary and sufficient assumptions for the truth of (1) are that the atoms (or molecules) of the gas are adsorbed as wholes on to definite points of attachment on the surface of the adsorber, that each point of attachment can accommodate one and only one adsorbed atom, and that the energies of the states of any adsorbed atom are independent of the presence or absence of other adsorbed atoms on neighbouring points of attachment. Under these assumptions the explicit form of (1) iswhere m is the mass of the adsorbed atom or molecule, bg(T) the partition function for its internal states in the gas phase, and vs(T) the partition function for its set of adsorbed states. These sets of states are to be so specified that the energy zero is assigned tot the lowest state of each set in constructing bg(T) and vs(T), and then X is the energy required to transfer a molecule from the lowest adsorbed state tot the lowest gas state. Quite another adsorption isotherm was shown to hold when adsorption of a molecule takes place as atoms and requires two or more points of attachment.
The resolution of monoidal Cremona transformations of three-dimensional space
Authors: Babbage D. W. 
Published: November 2008

A Cremona transformation Tn, n′ between two three-dimensional spaces is said to be monoidal if the surfaces of order n in one space which form the homaloidal system corresponding to the planes of the second space have a fixed (n − 1)-ple point O. If the surfaces of order n′ forming the homaloidal system in the second space have a fixed (n′ − 1)-ple point O′, the transformation is said to be bimonoidal. A particularly simple bimonoidal transformation is that which transforms lines through O into lines through O′, and planes through O into planes through O′. Such a transformation we shall call an M-transformation. Its equations can, by suitable choice of coordinates, be expressed in the formwhere φn−1(x, y, z, w) = 0, φn(x, y, z, w) = 0 are monoids with vertex (0, 0, 0, 1).
Variation of ionization with range of α-particles, protons, deuterons and 3H particles
Authors: Bower J. C. 
Published: November 2008

Photometrical determinations of the variation of density along photographs of expansion champer tracks have enabled the positions of maximum ionizing efficiency for single α-particles, protons, deuterons, and 3H particles to be determinated.The maximum rate of ionization for an α-particle in air occure when it has a residual range of 3·7 mm. in standard air.The distance of the position of maximum ionizing efficiency from the end of the particle range is shown to be approximately in the ratio of the particle mass in the case of the particles 1H, 2H, and 3H.These experiments were carried out in the Cavendish High Tension Laboratory.