bluuebunny/crossref_metadata_2025 huggingface dataset:
An Analysis of Triode Valve Rectification
Authors: Landale S. E. A.
Published: November 2008

It is shown that a quartic equation may be found which expresses the grid characteristic of a valve accurately over a considerable range, and that for modern dull filament valves the grid current curve is exponential over only a very restricted length.

Expressions are given for calculating rectified current and the amplitude of harmonics produced by a simple grid rectifier when voltages of the following wave-forms are applied: E sin pt, (A + B sin nt), and (A + B cos mt + C cos nt)sin pt. Further, a formula is given from which we may calculate the rectified current produced by a cumulative grid rectifier when a voltage E sin pt is applied.

All these formulae are applicable if the grid characteristic, is parabolic, cubic, quartic or quintic in form.

It is also shown that when an acoustically modulated radio frequency voltage is applied to a cumulative grid rectifier, even if the valve has a parabolic grid characteristic, an infinite number of harmonics of the modulation frequency are produced. These harmonics are due to the. action of the grid leak and condenser. It appears, however, that in practice the amplitude of these harmonics will be very small compared with those produced, primarily, by the curvature of the grid characteristic.
On Integrals developable about a Singular Point of a Hamiltonian System of Differential Equations
Authors: Cherry T. M.
Published: November 2008

Let

be a system of differential equations of Hamiltonian form, the characteristic function H being independent of t and expansible in a convergent series of powers of x1, … xn, y1, … yn in which the terms of lowest degree are
The relative absorbing powers of the L-levels for radiation of varying wave-length
Authors: Skinner H. W. B.
Published: November 2008

1. Changes in the relative intensities of the lines in the fluorescentL-spectrum of Cerium excited by radiation of various wave-lengths have been observed.

2. These results imply a change in the relative absorbing powers of the threeL-levels as the wave-length of the absorbed radiation diminishes from a value just below the absorption wave-length of theL-levels to a value considerably below. The absorbing power of theLI-level becomes increased relative to the absorbing powers of the otherL-levels as the wave-length diminishes. The results agree with those published by H. Robinson in a recent paper.

3. These results imply a breakdown of the law that μ/λ3is a constant (where μ is the absorption coefficient of X-rays of wave-length λ) as applied to theindividual L-levels of an element.

4. A comparison is made between the above results, and some results on the relative absorbing process of theL-levels obtained by Ellis and Skinner from β-ray spectra.
The Spectra of some Lithium-like and Sodium-like Atoms
Authors: Hartree D. R.
Published: November 2008

It is possible to obtain approximate theoretical relations between the spectra of different atoms ionised to such an extent that they possess the same electron structure. In this paper these relations are used to extrapolate values of the terms of the spectra of various lithium-like and sodium-like atoms.

From the terms so estimated lines can be calculated, and in many cases correlated with lines observed by Millikan and Bowen in the hot spark spectrum of the corresponding element. Revised term values can then be calculated from the lines so identified.

The spectra of lithium-like atoms treated in this way include those from Be II up to O VI, and those of sodium-like atoms, the spectra P V and S VI.
Notes on Optical Constants
Authors: Darwin C. G.
Published: November 2008

The tensor calculus introduced in a recent paper proves a powerful method of investigating the light scattered laterally as exemplified in the blue of the sky. The late Lord Rayleigh worked out the effect on the polarisation of having non-spherical molecules, and the present note is practically a repetition of his work but with a somewhat greater generality. The method has the advantage that it does not become unmanageable for molecules of a more complicated type of asymmetry, whereas a perusal of his work suggests that it would be a very complicated matter to give the lateral scattering from a set of arbitrary orientated “optically active” molecules. As this case has at present no practical interest, I have merely indicated the procedure so that it could easily be applied when required.
On the Motion of a Rotating Circular Cylinder filled with Viscous Fluid
Authors: Crossley A. F.
Published: November 2008

The solution for an infinite cylinder filled with viscous fluid, rotating about its axis under the action of a constant couple, has already been obtained by the present writer by the operational method of Heaviside and Bromwich, and previously by other means by T. H. Havelock.
Extension of a Geometrical Porism and other Theorems
Authors: Grace J. H.
Published: November 2008

It is a familiar fact in Solid Geometry that the problem of finding a twisted cubic curve with four given tangents is poristic. In fuller statement four prescribed tangents are apparently sufficient to determine such a curve yet

(A) In general there is no twisted cubic touching four given lines,

(B) If there is one there is an infinite number.
The elliptic products of Jacobi and the theory of linear congruences
Authors: MacMahon P. A.
Published: November 2008

In the application of Elliptic Functions to the Theory of Numbers the two formulae of Jacobi

are of great importance.
The Summation of a Fourier Integral of Finite Type
Authors: Pollard S.
Published: November 2008

A Fourier integral will be described as of finite type if the range of integration of the integrals by means of which the coefficients in the Fourier integral are defined is a finite interval instead of the usual (−∞ ∞). Thus is of finite type (p, q), such that where ƒ(x) is the generating function of the series.
Continued Fractions in Several Dimensions
Authors: Paley R. E. A. C. , Ursell H. D.
Published: November 2008

Continued fractions were generalised to more than one dimension by Jacobi and others: later Perron gave an account of the existing state of the subject with a detailed discussion of periodic fractions. Quite recently the subject has been attacked afresh by Mr. Maunsell
The High-frequency Electric Discharge at Low Pressures
Authors: Taylor James , Taylor Wilfrid
Published: November 2008

Experiments have been conducted by Gutton, and later by Kirchner, and by Gill and Donaldson upon electrical discharges through gases under the influence of high-frequency oscillations of the order of 107 cycles per second. It was found that the peak voltages required to maintain bright luminous discharges were of the order of 100 volts even when the pressure was as low as that in a soft X-ray tube. The present paper deals with some further studies of these phenomena.
The electron in a gravitational field
Authors: Whittaker J. M.
Published: November 2008

The potential of an electric charge in a gravitational field has been discussed very recently by Professor Whittaker and by E. T. Copson. The latter gives the solution

for a point centre at “distance” a from a mass of gravitational radius α. The coordinates used are such that the metric is given by
A percentile table of the relation between the true and the observed correlation coefficient from a sample of 4
Authors: Allan F. E.
Published: November 2008

The exact distribution of the simple correlation coefficient was established by Dr R. A. Fisher in 1915 (Biometrika, X, p. 507). If ρ is the true correlation in the population, and r the sample value, then the frequency element is given by

where θ = cos− (−ρr).
Applications of lattice algebra
Authors: Birkhoff Garrett
Published: November 2008

By a finite lattice† is meant any finite class having the following properties:
Note on the paper “On the combination of subalgebras”
Authors: Birkhoff Garrett
Published: November 2008

Professor Oystein Ore of Yale University has kindly called my attention to the close relation between my paper and some earlier researches of Dedekind. I should like to correlate, as far as possible, my results with his.
On the easier Waring problem for powers of primes. I
Authors: Erdös Paul
Published: November 2008

The famous theorem of Schnirelmann states that a constant c exists such that every integer greater than one may be expressed as the sum of at most c primes. Recently, Heilbronn, Landau and Scherk proved that this holds with c = 71. Probably the true value of c is 3. Another well-known theorem (Hilbert's solution of Waring's problem) is that every integer is the sum of a bounded number of positive kth powers. If we omit the restriction that all the kth powers are positive, the problem is referred to as the easier Waring problem and the proof of the result is then much simpler.
The comparison of series of measures
Authors: Jeffreys Harold
Published: November 2008

We often wish to test the difference of the means of two series of measures of a quantity for a systematic difference. A suitable test, in terms of the theory of probability, can be obtained as follows. We suppose that in each series the probability of error is normally distributed with unknown standard error. Then if we have only one observation in each series, the difference of the two could equally well be interpreted as entirely due to the random error in one series, the other being right, or entirely due to systematic difference, both observations being free from random error. This suggests that we should take as one of our fundamental quantities the expectation of the difference of a single measure in one series from one in the other; the expectation of the square of the difference, however, appears still better.
The optical reflectivity of metals in the supraconducting state
Authors: Hirschlaff E.
Published: November 2008

Measurements of the reflection coefficient at normal incidence on lead and tantalum mirrors in the supraconducting and non-supraconducting state did not reveal any change within the limits of accuracy (0·2−0·5% in R). This does not agree with a suggestion of R. de L. Kronig, according to which an increase in reflectivity of about 35% could have been expected.
Notes on special systems of orthogonal Functions (III): A System of orthogonal polynomials
Authors: Hardy G. H.
Published: November 2008

Generalities. Suppose that

is real and L2(0, ∞), that

is its Mellin transform, and that

Then K(1)(x) generates a “Watson transformation”, with a “Fourier theorem”

valid for every F(x) of L2(0, ∞).
On a criterion for oscillatory solutions of a linear differential equation of the second order
Authors: Miller J. C. P.
Published: November 2008

This paper gives a criterion for determining whether real, non-zero solutions of a linear differential equation of the second order have an infinite or a finite number of zeros, or, in short, areoscillatoryornon-oscillatory,as the independent variable tends to infinity.
Note on the slow motion of fluid
Authors: Dean W. R.
Published: November 2008

In this paper we consider the slow two-dimensional motion of viscous liquid past a sharp edge projecting into the stream, the motion being one of uniform shear apart from the disturbance caused by the projection. A special form is assumed for the boundary so that a method lately developed by N. Muschelišvili can be used in solving the biharmonic equation; a simple expression in finite terms is found for the stream function ψ1. Fig. 1 shows the section ABC of the fixed boundary of the liquid, the equation of the curve ABC is given in § 2, and ψ1 in § 3.
The stability of crystal lattices. III
Authors: Born M. , Fürth R.
Published: November 2008

The energy density of a cubic lattice, homogeneously deformed by a force acting in the direction of one axis, is calculated, and the equilibrium conditions and the stability conditions for any arbitrary small additional deformations are derived. A special assumption is made as to the law of force between the atoms, and the numerical calculations are performed for the face-centred lattice. In this way the strain as a function of the deformation is calculated and, from the stability conditions, the tensile strength is determined. The results are not in agreement with the experimental facts, and the possible reasons for this disagreement are discussed.
The thermodynamics of crystal lattices
Authors: Born Max
Published: November 2008

In 1939 I published a paper in the American Journal of Chemical Physics (Born(1)) in which I tried to develop the thermodynamics of a crystal lattice in the domain of classical (Boltzmann) statistics. Definite formulae and numerical tables for the temperature dependence of the elastic constants up to the melting point were obtained; nevertheless the work was unsatisfactory not only because the approximations used were rough and their accuracy not known, but because a fundamental difficulty with respect to lattice stability turned up. This led to a series of investigations by my collaborators and myself, published in these Proceedings under the title ‘On the stability of crystal lattices’ (quoted here as S I to SIX), by which the difficulty mentioned has been removed. It is now possible to return to the original problem, to which the present series of papers is devoted. In this introduction I wish to recapitulate the whole situation and to explain the plan of the following papers which will be published by my collaborators.
On a conjecture of Mordell concerning binary cubic forms
Authors: Davenport H.
Published: November 2008

Let f(x, y) be a binary cubic form with real coefficients and determinant D ≠ 0. In a recent paper, Mordell has proved that there exist integral values of x, y, not both zero, for which

These inequalities are best possible, since they cannot be satisfied with the sign of strict inequality when f(x, y) is equivalent to

for the case D < 0, or to

for the case D > 0.
A type of stress distribution on the surface of a semi-infinite elastic solid
Authors: Dean W. R. , Parsons H. W. , Sneddon I. N.
Published: November 2008

1. In this paper we consider the conditions at the surface of a semi-infinite elastic body due to the action of an external force applied, in a direction at right angles to the surface, to the interior of the body.
The thermodynamics of crystal lattices
Authors: Gow Margaret M.
Published: November 2008

Using the results of the preceding paper of this series (by M. Bradburn), the equation of state and the elastic constants for a monatomic face-centred cubic lattice are calculated. Central forces with a potential of the form −ar−m+br−n are assumed to act between the particles. Numerical results are obtained for five sets of values (m, n) and represented in tables and diagrams. The general features of these are discussed and compared with previous computations.

I take this opportunity of thanking Prof. Born who suggested this problem to me, for his interest in my work and his advice on many occasions. I am also indebted to Dr R. Fürth for his numerous helpful suggestions.
Note on the multiplication of series by Cauchy's rule
Authors: Hardy G. H.
Published: November 2008

1. I proved in 1908(1) that if A = Σam and B = Σbn are convergent, and

for large m and n, then

is convergent (necessarily to AB); and this theorem has been extended in a number of directions both by other writers and by myself. Thus we may replace (1), when am and bn are real, by

we may use conditions unsymmetrical in am and bn; we may put the same problem for the product of any number of series; and we may consider modes of ‘Dirichlet multiplication’ based on sequences λm and μn, reducing to Cauchy's when λm = m and μn = n. The appropriate references will be found in(2).
Corrigenda to the paper On the stability of crystal lattices:
Authors: Born M.
Published: November 2008

There are several mistakes and misprints in the above paper.

(1) In introducing Voigt's notation for the strain components the well-known factor 2 in the normal components has been omitted, and (3·11) should read

The following formulae in which this notation is used are correct as they stand.
Solution of a Moment Problem for Bounded Functions
Authors: Verblunsky S.
Published: November 2008

1. Given a sequence of variables a1, a2, …, we define a sequence of polynomials B1, B2, … by

For every positive integer n, Bn is a polynomial in a1, …, an, with rational coefficients. We write Bn ≡ Bn (a1, …, an) to indicate the variables on which Bn depends. From the polynomials Bn, we form a new sequence of polynomials Dn in accordance with the following definitions.
Movements of electrically charged cloud particles
Authors: Gott J. P.
Published: November 2008

Experiments are described in which observations were made of the motion of electrically charged cloud particles past a sphere. The cloud particles were moving vertically up in an air stream, and there was a vertical electric field. This gave conditions similar to those surrounding a falling rain drop in a thundercloud, and the observations are in accordance with the theory proposed by Wilson to account for the mechanism of thunderclouds.
On the diffusion of gases through metals
Authors: Wang Jwu-Shi
Published: November 2008

The results of experiments on the diffusion of hydrogen through metals from a pressure p on one side to a vacuum on the other show that at high pressures the amount diffusing varies linearly with p½ but that at low pressures it varies more rapidly. The difficulty usually encountered when diffusion from an adsorbed layer into the solid is considered theoretically is that the theory indicates that saturation should be reached. In this paper it is shown that this difficulty is due to the omission of an important process at the surface and that by including this process the experimental results can be explained.
On an equation occurring in Falkner and Skan's approximate treatment of the equations of the boundary layer
Authors: Hartree D. R.
Published: November 2008

The differential analyser has been used to evaluate solutions of the equation

with boundary conditions y = y′ = 0 at x = 0, y′ → 1 as x → ∞, which occurs in Falkner and Skan's approximate treatment of the laminar boundary layer. A numerical iterative method has been used to improve the accuracy of the solutions, and the results show that the accuracy of the machine solutions is about 1 in 1000, or rather better.

It is shown that the conditions are insufficient to specify a unique solution for negative values of β a discussion of this situation is given, and it is shown that for the application to be made of the solution the appropriate condition is that y′ → 1 from below, and as rapidly as possible, as x → ∞. The condition that y′ → 1 from below can be satisfied only for values of β0, greater than a limiting value β0, whose value is approximately − 0·199, and which is related to the point at which the laminar boundary layer breaks away from the boundary.
The fixed part of the canonical system on an algebraic surface
Authors: Du Val Patrick
Published: November 2008

It is familiar that if on an algebraic surface there is an exceptional curve, that is an irreducible rational curve of virtual grade − 1 when no points of it are assigned as base points, and if there is on the surface a canonical system containing some actual curves, so that pg ≥ 1, then the exceptional curve is a fixed constituent of every curve of the canonical system, generally a simple constituent, and in that case has no intersections with the residual constituent. More generally, if there is on the surface a reducible exceptional curve, i.e. a set of curves which can be transformed into the neighbourhoods of a family of simple points (some of which are in the neighbourhoods of others) on a surface birationally equivalent to the given one, then the canonical system has as a fixed constituent of all its curves at least that combination of the curves which corresponds to the sum of the total neighbourhoods of the points, and generally just this combination, in which case this fixed part has no intersection with the residual variable part.
Some properties of mobile and immobile adsorbed films
Authors: Roberts J. K.
Published: November 2008

The essential difference between mobile and immobile adsorbed films of gases on solids when there is repulsive interaction between adsorbed particles is discussed. Ordinary statistical methods can be applied to the equilibrium configurations of the particles in a mobile film but not to immobile films. Films formed by the adsorption of diatomic molecules with dissociation are then considered. It is pointed out that for a mobile (equilibrium) film there are differences between the true heat of adsorption and the apparent heat obtained from the temperature variation of the evaporation rate. A detailed comparison is made by obtaining a formula for the apparent heat of evaporation and comparing it with that for the true heat obtained earlier by Wang, and the physical reasons for the differences are discussed. A model suitable for studying the pseudo-equilibrium states in immobile films is described and formulae and diagrams in connexion with the kinetics of the formation of such films and the variation of heat of adsorption with fraction of surface covered are given.
The emission and absorption of heavy electrons
Authors: Massey H. S. W. , Corben H. C.
Published: November 2008

The quantum theory of the heavy electron field is applied to calculate approximately the cross-sections for “photoelectric” absorption of heavy electrons by a bound nuclear particle and for emission of a heavy electron by a free nuclear particle on collision with a nucleus. The absorption cross-section per nuclear particle is of the order 10−26 cm.2 for heavy electrons with energies up to 108 e.V., but probably decreases rapidly at higher energies. Although the energy loss of a proton, of energy 108 e.V. or higher, due to heavy electron emission is found to be greater than that due to radiation, the cross-section is very small (⋍ 10−29 cm.2) and the phenomenon is unlikely to be observed in a cloud chamber. In the present state of theory it is impossible to decide whether radiative emission of heavy electrons by nuclear particles is capable of contributing appreciably to the heavy electrons observed at sea-level but, if so, the cross-section for the process must be very much greater at high energies than in the non-relativistic energy region which we have investigated.
Methods of construction of Geiger-Mueller counters and their use in coincidence experiments
Authors: Curran S. C. , Petržílka V.
Published: November 2008

The technique adopted in some experiments employing Geiger-Mueller counters is described. The behaviour of a counter in which the extinction of the discharge is external is contrasted with behaviour of a counter in which extinction is internal and dependent on the addition to the gas of some organic vapour such as alcohol. It is shown that the internally extinguished type of counter may be used with a resistance of the order 50,000 ohms in series with it while still retaining all its useful features.

Methods of construction of internally extinguished counters are described, particular attention being paid to the question of developing a thin-walled type suitable for experiments employing the method of coincidences. The effect of using different wall materials and gases is investigated with a view to finding those most suitable.

The measurement of the quantum energy of hard γ-radiation by the method of coincidences is discussed and details of a reliable two-stage amplifying circuit with a resolving time T of 10−6 sec. are given. Various factors which may influence the value of T are noted.
Canonical systems on a reducible variety
Authors: Macpherson R. E.
Published: November 2008

The object of this note is to extend the proof, for the general canonical varieties of any dimension k < d on a Vd whose existence was established by Dr Todd in a recent paper, of the property of adjunction given by the relation

where S is any non-singular variety of dimension d−1 lying on Vd, and Xk[S] denotes the canonical system of dimension k on S. This formula was proved by Todd in the case when Vd is a general non-singular variety, and S is non-singular and belongs to a sufficiently general linear system. Using this result, I show how to remove the latter restriction. (1·1) cannot be taken, without further proof, as defining canonical systems on virtual or isolated varieties, as is done by B. Segre, since the canonical systems on an effective isolated variety are already well defined.
The probability distribution of the extent of a random chain
Authors: Daniels H. E. , Smithies F.
Published: November 2008

1. Introduction and summary. A chain of N links is allowed to assume a random configuration in space. The extent of the chain in any direction is defined as the shortest distance between a pair of planes perpendicular to that direction, such that the chain is contained entirely between them. In the present paper the probability distribution of the extent is discussed, starting with a chain in one dimension for which formulae are derived for the probability and mean extent for all values of N. The limiting forms for large N are then considered. The results are extended to the case of a chain in three dimensions, and it is shown that the extents in two directions at right angles tend to be independently distributed when N is large. It is assumed that the links are infinitely thin, so that a point in space may be occupied by the chain any number of times.
On the theory of the liquid state
Authors: Fürth R.
Published: November 2008

It is assumed that the viscosity of a liquid is due to the transfer of momentum by the irregular movement of the holes (see part I of this paper). A formula for the absolute value of the viscosity and its dependence on the temperature can be derived by the theory and is in agreement with its experiments.
The postulation of a multiple curve
Authors: Segre B.
Published: November 2008

1. The postulation of a multiple curve for primals of sufficiently large order in space of any number of dimensions has been obtained recently by J. A. Todd, by a simple and elegant degeneration argument which, however, is not deemed to be a conclusive proof by the author himself. And, indeed, in order to make sure of the unconditional validity of such an argument, one should ascertain whether

(i) the postulation θk of an irreducible non-singular curve ϲ, of order c and genus p, for the primals of sufficiently large order n of [r + 2] (r ≥ 1), required to go through it with multiplicity k (≥ 1), is a function of k, c, p, n, r only;

(ii) it is possible, by means of a continuous variation of ϲ, to reduce this curve to connected polygon ϲ′ having the same virtual characters as ϲ, in such a way that each intermediate position of ϲ is still irreducible and non-singular;

(iii) the postulation θk of ϲ equals the similarly defined postulation of ϲ′.
On the number of terms in a simple algebraic form
Authors: Littlewood D. E.
Published: November 2008

In a note in the April 1942 issue of these Proceedings, Hodge gives a formula for the number of independent terms in what he calls a k-connex, but while conjecturing its general validity he proves it only for very restricted cases. Its truth is here demonstrated in the general case.
On the stability of crystal lattices VII. Long-wave and short-wave stability for the face-centred cubic lattice
Authors: Power S. C.
Published: November 2008

It is shown that the theorem stated in Born's paper, and proved for the case of a linear lattice of N equal particles under certain restrictions concerning the forces between the particles, that macroscopic stability (stability for long waves) implies microscopic stability, may be extended to three dimensions for the particular case of a face-centred cubic lattice, where the effects of all neighbours, other than the first twelve neighbours, are neglected.

I take this opportunity of expressing my sincere thanks to Prof. Born for much valuable advice.
An extension of a theorem of Mehler's on Hermite polynomials
Authors: Kibble W. F.
Published: November 2008

It was shown by Mehler (1866) that

where Hk(x) denotes the Hermite polynomial

(Hermite, 1864a, b), which can be expressed in terms of Weber's parabolic cylinder function (Whittaker, 1903). The series is convergent if | ρ | < 1, and divergent if | ρ | > 1. If ρ = 1 and x = y = 0 the series is divergent, and Hille's work (1938) shows that it will therefore be divergent for all real or complex values, except possibly real positive values, of x and y.
On some cases of the reflexion of surface waves by an inclined plane barrier
Authors: Dean W. R.
Published: November 2008

1. The reflexion of a simple harmonic train of surface waves by a rigid plane barrier inclined at an angle α to the undisturbed free surface of the water is considered in the particular case in which 2sα = π, where s is an integer. The solution if s = 2 has been given by Gourret(1), while the solution if s = 1 is well known; the solution for any integral value of s is given here.
The large-sample theory of sequential tests
Authors: Bartlett M. S.
Published: November 2008

1. The general theory underlying sampling inspection methods introduced during the war under the name of sequential analysis, applicable in cases where the sampling units can be taken serially, has mainly been developed by Wald and is now published (5). The original purpose of the present investigation was to exhibit the main structure of the distributional theory relating to the size of sample required by noting its relationship with the classical ‘random-walk’ problem. Thus in Part I of this paper are derived the distribution and characteristic functions of the absorption time for one-dimensional random-walk theory with constant drift and absorbing barriers. Attention is confined to the asymptotic case of numerous independent displacements, for which it is well known that the unrestricted total displacement becomes Gaussian; the motion may also be treated as continuous. Since these results were obtained Wald has independently given an alternative and more general discussion of this problem((4); see also Tweedie (3)), but the extension here of the method of images used by Chandrasekhar (1) still appears of interest, and has advantages over alternative direct methods of solution for the distribution function involving Fourier expansion (cf. 2).
A theory of the critical temperatures of the normal paraffins
Authors: Corner J.
Published: November 2008

A simple model of independent curled-up molecules is examined theoretically. The critical temperature is found to be of the right order of magnitude but does not increase sufficiently rapidly with the size of the molecule. The Onnes constant and the critical density show the discrepancies expected from the theoretical equation of state, and the former also has an incorrect dependence on molecular weight. In the critical region, therefore, the paraffins cannot be regarded as an assembly of independent molecules, curled up in the way expected at high temperatures and low densities.
The radiations emitted by and its formation in the deuteron bombardment of uranium
Authors: Feather N. , Krishnan R. S.
Published: November 2008

The production of by deuteron bombardment of uranium has been established and the radiations of this product have been examined by the absorption method. The normal mode of disintegration involves β-particles having an upper energy limit of 1·20 ± 0·02 MeV. and a low-energy γ-ray of 76 ± 3 keV. quantum energy. Evidence has been found for electrons of less energy (about 0·1 per disintegration) and more penetrating γ-rays of relatively low intensity. Quantum radiations corresponding broadly to the L X-radiations, to be expected to follow the internal absorption of the low-energy γ-ray, have also been detected.

The excitation function for the process has been studied, and the conclusion reached that the Oppenheimer-Phillips mechanism is responsible for the larger part of the yield.
The problem of random intervals on a line
Authors: Domb C.
Published: November 2008

Suppose that events occur at random points on a line from t = −∞ to +∞, the probability of an event occurring between t and t + dt being λdt. If we select any interval of the line, say the interval [0, y], there will be a finite probability that it contains 0, 1, 2,…,r,…events; in fact, it is not difficult to show that these probabilities form a Poisson distribution, the probability that the interval contains r events being (see e.g. (1)). Consider the case when each event consists of an interval of length α (an event being characterized by its first point). What is the probability that the covered portion of the interval [0, y] lies between x and x + dx?
On the number of representations of an integer as the sum of three r-free integers
Authors: Mirsky L.
Published: November 2008

Let r, s be two fixed integers greater than 1. A positive integer will be called r-free if it is not divisible by the rth power of any prime.

In a series of papers ((l)–(5)) Evelyn and Linfoot considered the problem of determining an asymptotic formula for the number Qr, s(n) of representations of a large positive integer n as the sum of s r-free integers; for s ≥ 4 their results were subsequently sharpened by Barham and Estermann(6).
Expansions in terms of sets of functions with complex eigenvalues
Authors: Peierls R.
Published: November 2008

In the following I discuss the properties, in particular the completeness of the set of eigenfunctions, of an eigenvalue problem which differs from the well-known Sturm-Liouville problem by the boundary condition being of a rather unusual type.

The problem arises in the theory of nuclear collisions, and for our present purpose we take it in the simplified form

where 0 ≤ x ≤ 1. V(x) is a given real function, which we assume to be integrable and to remain between the bounds ± M, and W is an eigenvalue. The eigenfunction ψ(x) is subject to the boundary conditions

and
On perturbation problems associated with finite boundaries
Authors: Wassermann G. D.
Published: November 2008

The methods of a previous paper (9) are improved and extended. It is assumed that the eigenfunctions and eigenvalues of an eigenvalue problem given by an elliptic differential equation are known subject to given boundary conditions on a finite boundary. It is shown how the corresponding quantities can be obtained for a similar problem in which the original differential equation, boundary and boundary conditions are simultaneously perturbed. The introduction of a surface displacement vector allows of a Taylor expansion of all quantities and a subsequent separation of orders. The problem of finding the perturbed eigenfunctions for each order then reduces to the solution of an inhomogeneous differential equation subject to known boundary conditions. These equations are solved by a variational method. An application of Green's theorem at each stage enables us to find the perturbed eigenvalues. The method is applied to a problem of which an exact solution is known and good agreement is obtained.

The author is greatly indebted to Dr H. Fröhlich for many interesting discussions and some valuable suggestions.
On a theorem in the additive theory of numbers due to evelyn and linfoot
Authors: Mirsky L.
Published: November 2008

Some years ago Evelyn and Linfoot found an asymptotic formula, with estimation of remainder, for the number of representations of a large integernas the sum ofs r-free integers. Fors≥ 4 their formula was subsequently sharpened by Barham and Estermann, and fors= 3 recently by me.
A note on the division values of ℘(u)
Authors: Cassels J. W. S.
Published: November 2008

Letx= ℘′(u) andy= ½℘(u), where the Weierstrass ℘-function has invariantsg2= 4A, g3= 4Band differentiation with respect to the parameteruis denoted by a dash, so that
A note on the algebra of S-functions
Authors: Todd J. A.
Published: November 2008

1. The particular set of symmetric polynomials known as S-functions has recently been shown to be of importance in a variety of algebraic problems. Many of the applications of these functions depend upon the properties of the operation which Littlewood terms ‘new multiplication’, by which, from two given S-functions {μ}, {ν} of respective degrees m and n, is constructed a symmetric function {μ} ⊗ {ν} of degree mn. Littlewood has devoted a considerable part of his paper (2) to explaining various methods by which this function can be expressed in terms of S-functions of degree mn. None of these methods is really simple (in the sense that the rule for expressing the ordinary product {μ} {ν} in terms of S-functions of degree m + n is simple); and, indeed, it would be unreasonable to expect any simple rule of general validity for writing the resulting expression down since, for instance, a knowledge of the explicit expression for {μ} ⊗ {n} would yield immediately a knowledge of all the linearly independent concomitants, of degree n, of an algebraic form of type {μ} in an arbitrary number of variables. Nevertheless, the evaluation of {μ} ⊗ {ν} in particular cases is often necessary. Of the methods suggested by Littlewood for performing this evaluation the one which seems to be normally the simplest (his ‘third method’) involves a process which is not shown to be free from ambiguity, and which can actually be shown by examples to give, in certain cases, alternative solutions, the choice between which must be made by other considerations. It is therefore perhaps worth while putting on record a quite different method of procedure, which, apart from any intrinsic interest which it may possess, seems to be quite practicable for most of the actual evaluations performed by Littlewood.
The Kummer quartic and the tetrahedroids based on the Maschke forms
Authors: Edge W. L.
Published: November 2008

1. The quadrics of space are linearly dependent on ten among them; any ten linearly independent quadrics may be chosen to constitute the base, but it is customary in analytical work to select the four squares and the six products of pairs of four planes which are the faces of some tetrahedron of reference. This choice is adequate enough for many purposes, but it gives an unsymmetrical twist to the work; whereas four quadrics of the base are repeated planes the other six are plane pairs, and the curve common to two of the ten base quadrics can be a skew quadrilateral, a repeated line-pair, or a line counted four times. This radical defect can be mitigated by taking as base a certain set of ten non-singular quadrics, namely, the set of ten fundamental quadrics which is so prominent a feature of Klein's figure of six mutually apolar linear complexes. Every pair of such a set of ten quadrics has its two members related to one another in precisely the same way; their common curve is a skew quadrilateral and they are their own polar reciprocals with respect to each other.
On the iteration of a continuous mapping of a compact space into itself
Authors: Friedlander F. G.
Published: November 2008

1. The questions considered in this note are suggested by the elementary topology of the trajectories of systems of non-linear differential equations. Such a system may be assumed in the form

and the values of the dependent variables x1, x2, …, xn at ‘time’ t can be represented by a point P(t) in a ‘phase space’ . As t varies, P(t) describes a curve in , which is a trajectory of (1). Now it often happens that contains a subspace E (usually of lower dimension) with the following properties: (i) by considering the trajectories generated by points P(t) which are, for t = 0, in E, all the trajectories of (1) are obtained; (ii) if P(0) is in E, then P(t) is not in E for 0 < t < c, where c is a constant independent of P(0) in E; (iii) if P(0) is in E, then the trajectory meets E again for some finite t at a point P(T) (T is not necessarily the same for all points of E). By considering P(T) as the image of P(O), a mapping of E into itself is defined which is associated with the system (1), and the topology of the trajectories of (1) can be studied conveniently by discussing this mapping. When the functions fi in (1) satisfy the continuity and Lipschitz conditions of the classical existence-and-uniqueness theorem, the mapping is one-one and continuous. The study of this ‘transformation theory’, initiated by Poincaré, has been developed chiefly by G. D. Birkhoff(l,2). His results have been applied to problems of ‘non-linear mechanics’ by N. Levinson(3).
On generalized subharmonic functions
Authors: Bonsall F. F.
Published: November 2008

In a recent paper (1) I studied a class of generalized convex functions of a single real variable which I called sub-(L) functions. Given an ordinary linear differential equation of the second order L(y) = 0, a function f(x) is sub-(L) in (a, b) if it is majorized there by the solutions of the equation. More precisely, for every x1, x2 in (a, b),f(x) ≤ F12(x) in (x1, x2), where F12 is that solution of L(y) = 0 (supposed unique) which takes the values f(xi) at xi. It was found that sub-(L) functions are characterized in a manner closely analogous to ordinary convex functions.
Stochastic processes relating to particles distributed in a continuous infinity of states
Authors: Ramakrishnan Alladi
Published: November 2008

Many stochastic problems arise in physics where we have to deal with a stochastic variable representing the number of particles distributed in a continuous infinity of states characterized by a parameterE, and this distribution varies with another parametert(which may be continuous or discrete; iftrepresents time or thickness it is of course continuous). This variation occurs because of transitions characteristic of the stochastic process under consideration. If theE-space were discrete and the states represented byE1,E2, …, then it would be possible to define a function

representing the probability that there are ν1particles inE1, ν2particles inE2, …, att. The variation of π withtis governed by the transitions defined for the process; ν1, ν2, … are thus stochastic variables, and it is possible to study the moments or the distribution function of the sum of such stochastic variables

with the help of the π function which yields also the correlation between the stochastic variables νi.
A reduction for the matrix equation AB = εBA
Authors: Drazin M. P.
Published: November 2008

1. It is well known that, if two n × n matrices A, B commute, then there is a non-singular matrix P such that P−1AP, P−2BP are both triangular (i.e. have all their subdiagonal elements zero). This result has been generalized, and, in particular, has been shown to hold even if the commutator K = AB − BA is not zero, provided that K is properly nilpotent in the polynomial ring generated by A, B.
The frequency goodness of fit test for probability chains
Authors: Bartlett M. S.
Published: November 2008

1. Introduction. Following recent developments in the theory of stochastic processes, a beginning has been made by various authors with the associated problems of sampling and statistical inference as these are concerned with dependent and ordered observations, they are distinct from and usually more difficult than the corresponding problems for independent and unordered observations.
A comparison of heisenberg's spectrum of turbulence with experiment
Authors: Proudman Ian , Batchelor G. K.
Published: November 2008

In this paper, the theoretical double and triple velocity correlation functions, f(r), g(r) and h(r), which correspond to Heisenberg's spectrum of isotropic turbulence, are obtained numerically for two Reynolds numbers. One set of these correlations is for the limiting case of infinite Reynolds number. In addition, a method is developed for deriving the approximate form of the double correlations for any Reynolds number, which is not too small, from the corresponding correlations for infinite Reynolds number. These theoretical correlations are then compared with the results of experiment.
The structure of the turbulent boundary layer
Authors: Townsend A. A.
Published: November 2008

The nature of the turbulent motion in a boundary layer with zero longitudinal pressure gradient has been investigated with the techniques of hot-wire anemometry which have been developed for the study of shear flow in wakes. Measurements have been made of the intensities of the turbulent velocity components, the turbulent shear stresses, the rates of transport of turbulent energy by diffusive movements, the intensities and flattening factors of the down-stream spatial derivatives of the three velocity components, and spectra of the down-stream component of the velocity fluctuation, at traverses through the boundary layer at three stations where the Reynolds numbers (based on the displacement thickness and free stream velocity) were respectively 3630, 4360 and 5080. Over the range of measurement, which did not include the laminar sublayer, the turbulent motion was similar at all three stations, and could be described in terms of universal functions. By considering the turbulent energy balance, it is shown that, except in the outer part of the layer where the turbulence resembles strongly the turbulence in a wake, there is a strong flow of energy directed toward the wall and transported by the action of turbulent pressure gradients. It is concluded that, most probably in contact with the laminar sublayer, there must be a ‘dissipative layer’ within which most of the turbulent energy dissipation takes place, and that the bulk of the eddies are, in a sense, attached to the wall and have very high rates of shear in that region. In agreement with this view of the structure of the turbulence, length scales derived from the apparent eddy viscosity and the local turbulent intensity are found to be comparable with distance from the wall. Length scales in the direction of the mean stream are much larger, and it is believed that the typical eddy is very elongated in this direction, and has its vorticity directed roughly parallel to the direction of maximum positive mean rate of strain.
On direct decompositions. I
Authors: Goldie A. W.
Published: November 2008

It is known that the refinement theorems for direct decompositions, now classical for group theory, are true, under suitable chain conditions, for two large classes of algebras. These are (1) the algebras whose congruences commute and (2) the algebras with a binary operation of a particular type generated by what we shall call a decomposition operator. On the other hand, there exist finite algebras for which the refinement theorems are not true. The problem of characterizing algebras for which the theorems are true arises at once. It is difficult, because the structures of algebras of classes (1) and (2) can be, to a large extent, incompatible. To illustrate this, we mention that any algebra of (2) has a naturally defined centre which is an Abelian group under the binary operation.
Sequential tests for composite hypotheses
Authors: Cox D. R.
Published: November 2008

A method is given for obtaining sequential tests in the presence of nuisance parameters. It is assumed that a jointly sufficient set of estimators exists for the unknown parameters. A number of special tests are described and their properties discussed.
Isolated minima of the product of n linear forms
Authors: Barnes E. S.
Published: November 2008

Let

be n linear forms with real coefficients and determinant Δ = ∥ aij∥ ≠ 0; and denote by M(X) the lower bound of | X1X2 … Xn| over all integer sets (u) ≠ (0). It is well known that γn, the upper bound of M(X)/|Δ| over all sets of forms Xi, is finite, and the value of γn has been determined when n = 2 and n = 3.
The solution of the Schrödinger equation for finite systems, with special reference to the motion of electrons in Coulomb electric fields and uniform magnetic fields
Authors: Dingle R. B.
Published: November 2008

A number of methods are formulated for solving the Schrödinger equation for systems of finite extent. The methods are developed in detail for the particular case of an electron moving in a Coulomb field (e.g. hydrogen-like atom), with a boundary consisting of a sphere of given radius. In the second part of the paper these results are transformed into those for an electron moving in a uniform magnetic field, the boundary of the system being cylindrical.