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The basis of statistical quantum mechanics
Authors: Dirac P. A. M.
Published: November 2008

In classical mechanics the state of a dynamical system at any particular time can be described by the values of a set of coordinates and their conjugate momenta, thus, if the system has n degrees of freedom, by 2n numbers. In quantum mechanics, on the other hand, we have to describe a state of the system by a wave function involving a set of coordinates, thus by a function of n variables. The quantum description is, therefore, much more complicated than the classical one. Let us consider, however, an ensemble of systems in Gibbs' sense, i.e. not a large number of actual systems which could, perhaps, interact with one another, but a large number of hypothetical systems which are introduced to describe one actual system of which our knowledge is only of a statistical nature. The basis of the quantum treatment of such an ensemble has been given by Neumann. The description obtained by Neumann of an ensemble on the quantum theory is no more complicated than the corresponding classical description. Thus the quantum theory, which appears to such a disadvantage on the score of complication when applied to individual systems, recovers its own when applied to an ensemble. It is the object of the present note to examine this question more closely and to show how complete the analogy is between the quantum and classical treatments of an ensemble.
A note on Ramanujan's arithmetical function τ(n)
Authors: Wilton J. R.
Published: November 2008

The function τ (n), defined by the equation

is considered by Ramanujan* in his memoir “On certain arithmetical functions”

The associated Diriċhlet series

converges when σ = > σ0, for a sufficiently large positive σ0.
Boundary layer growth
Authors: Goldstein S. , Rosenhead L.
Published: November 2008

When relative motion of a viscous incompressible fluid of constant density and an immersed solid body is started impulsively from rest, the initial motion of the fluid is irrotational, without circulation. This is shown by observation, and may be seen in many of the published photographs of fluid flow. The theoretical proof is exactly the same as that given, for inviscid fluids, in treatises on hydrodynamics; for it may be assumed that the viscous stresses remain finite. The fluid in contact with the solid body is, however, at rest relative to the boundary, whilst the adjacent layer of fluid is slipping past the boundary with a velocity determined from the theory of the velocity potential. There is thus initially a surface of slip, or vortex sheet, in the fluid, coincident with the surface of the solid body. In other words, there is a “boundary layer” of zero thickness. The vorticity in the sheet diffuses from the boundary further into the fluid, and is convected by the stream. The boundary layer grows in thickness. (The same results follow from a consideration of the equations for the vorticity components in a viscous incompressible fluid, or of the equation for the circulation in a circuit moving with the fluid.)
Further significance tests
Authors: Jeffreys Harold
Published: November 2008

In a previous paper (afterwards referred to as Paper I) tests have been given for the significance of some quantities found statistically. The results are given in the formP(q| θh)/P(˜q|θh); herehdenotes the previous knowledge and θ the experimental evidence used, whileqis the hypothesis that all the variations outstanding can be attributed to accidental error or random variation, and ˜qthe hypothesis that at least part of them is systematic. It has been supposed in the analysis thatqand ˜qare equally probable on the informationh; but if they are not, the only alteration is that the ratios evaluated now represent

If successive batches of relevant information are available the total effect on the probability ofqcan therefore be got by multiplying the values of

given by the investigations separately. In each case the assumption thatqhas prior probability ½ is really a practical working rule rather than a statement of fact.
The information available in small samples
Authors: Bartlett M. S.
Published: November 2008

When a probability distribution is specified by more than one parameter, the statistical information in a sample on an unknown α will usually depend on the values of the other unknowns. For large samples where the joint distribution of the estimates tends to normality, and the efficiency of the estimation to a maximum, little theoretical difficulty exists. In a discussion(1) of the more general theory of small samples, I have stressed the importance of properties of sufficiency.
The “closest” estimates of statistical parameters
Authors: Pitman E. J. G.
Published: November 2008

A definition of “closer” and “closest” as applied to estimates of statistical parameters is given, and it is shown that we can sometimes prove that estimates properly derived from sufficient statistics are the closest possible.

The scaling of a gamma distribution, and the location and scaling of an exponential distribution and of a rectangular distribution are discussed in detail, and the closest estimates of the parameters obtained.
Determinantal representations of x4+y4+z4
Authors: Edge W. L.
Published: November 2008

A ternary form of degree n can be expressed as a symmetrical determinant, of n rows and columns, whose elements are linear forms; furthermore, not only is such a mode of expression known to be possible, but A. C. Dixon, in 1902, gave* a process by which the determinant can be obtained when the ternary form is given. This process, however, although it admits of such a straightforward theoretical description, cannot be carried through in practice, for a general ternary form, without the introduction of complicated algebraical irrationalities, even if we restrict ourselves to forms of the fourth degree; consequently no application of Dixon's process to an actual example seems to have been published. If then a choice can be made of a quartic form for which the reduction to a symmetrical determinant can be carried out without undue complication, it seems fitting to give some account of it. The following pages are therefore devoted to the study, from this aspect, of the form x4+y4+z4, for which the reduction can be accomplished without introducing any irrationality other than the fourth root of − 1.
Further aspects of the theory of multiple regression
Authors: Bartlett M. S.
Published: November 2008

This paper may be regarded as a sequel to a previous papers(1) in these Proceedings. The vector and matrix notation of that paper used for a statistical sample is systematized somewhat further, so that while a sample S refers as before to the matrix of nm values (a sample of m observations in one variate only being a row vector), we write

for the linear regression formula between the dependent and independent variates into which a sample is supposed partitioned (in place of equation (12) of (1)). More generally, a third submatrix S0 is partitioned off, and its effect eliminated (corresponding to equation (13) of (1)), but without loss of generality we assume that S2 in equation (1) above can always stand for S2.0 if necessary.
The use of minimum χ2 as an approximation to the method of maximum likelihood
Authors: Jeffreys Harold
Published: November 2008

Suppose that we have a set of observed frequencies classified in groups, the number in a typical group being nr, and that we wish to estimate the expectations in the groups according to some law with adjustable parameters. If these expectations are mr we take

Σ denoting summation over the groups, and according to the law in question the probability that one observation should come in the rth group is mr/N. The joint probability, for given mr, that there should be n1 observations in the first group, n2 in the second and so on, is therefore
The shot effect with space charge
Authors: Whittaker J. M.
Published: November 2008

The technical importance of constructing valve amplifiers disturbed by as little background noise as possible has led to a very searching investigation of the causes of this defect. Some causes—bad contacts, changes in the surface of the filament, secondary ions from the grid, and so forth—though technically speaking not trivial, are of minor physical interest; but it has long been recognized that there are two which cannot be dismissed so easily.
The internal conversion electrons of actinon
Authors: Bennett W. E.
Published: November 2008

Actinium emanation was introduced into a cloud chamber for the purpose of making a study of the secondary β-rays. The energies were in agreement with those given by Surugue using the magnetic spectrum method, except that his strong line at 105 kV. was absent. By comparing the numbers of β-rays with the relative intensities of the groups of α-particles, measured by Rosenblum, it was found that the internal conversion coefficients were in general agreement with previous experimental values in the same region of energy. Tracks of Auger electrons associated with the secondary β-rays were found in the photographs. In general, the results from the cloud chamber method agreed with accepted theories and confirmed the results obtained by other methods.
A modernized version of Gibbs' use of the grand canonical ensemble
Authors: Fowler R. H.
Published: November 2008

In the last chapter of his Introduction to statistical mechanics Gibbs introduces the idea of the grand canonical ensemble. He had previously determined the properties of an assembly or a phase containing a given number of systems by averaging the properties of the assembly over an ensemble of examples canonically distributed in phase, keeping the number of systems in the assembly fixed. This means of course constructing what we now call the partition function for the assembly by summing or integrating e−E/kT over the whole of the accessible phase space.
Absorption and coincidence experiments on the radiations from the radioactive sodium, Na24
Authors: Feather N. , Dunworth J. V.
Published: November 2008

The radiations from Na24 have been examined by the absorption and coincidence methods and evidence has been obtained for the conclusion that the energy spectrum of the β-particles is a simple continuous spectrum with an upper limit at 1·40 × 106 e.V., and that the emission of γ-radiation after the particle disintegration takes place almost always as a cascade process in which at least two quanta are involved, and very rarely by a single transition to the ground state.
Mercerian theorems
Authors: Pitt H. R.
Published: November 2008

Suppose that k(x) is of bounded variation in (− ∞, ∞) and that s(x) is bounded and continuous in the same range. Let
The mean value of the fluctuations in pressure and pressure gradient in a turbulent fluid
Authors: Green A. E.
Published: November 2008

1. Taylor has shown (1) that two characteristic lengthsλ and λη may be defined for turbulent fluid motion. The length λ, which is connected with the dissipation of energy, is, for isotropic turbulence, given by

where is the mean rate of dissipation of energy per unit volume and represents the mean square value of any component of velocity. The length λη can be defined in terms of thus

where For isotropic turbulence Taylor assumed that

where B is a constant. Since the turbulence is isotropic,

and so, from (1), (2), (3) and (4) we have
The rates of unimolecular reactions in gases
Authors: Slater N. B.
Published: November 2008

The molecule of a dissociating gas is treated as a dynamical system with n normal modes of vibration with incommensurable frequencies. The assumption that the molecule dissociates when one coordinate attains a critical high value leads to a unimolecular velocity constant expressed as an integral; analytical expressions are derived for bounds to its value. Calculations in a simple case indicate that the velocity constant is approximately of the form

This is in general agreement with experiment if the constant W is identified with the corresponding empirical energy constant.
On the number of real roots of a random algebraic equation. II
Authors: Littlewood J. E. , Offord A. C.
Published: November 2008

An equation with real coefficients and given degree n being selected at random, about how many real roots may it be expected to have? The present series of papers is concerned with this question and matters arising out of it. The results we have arrived at were stated without proof in our paper I (with the same general title), which contains also some introductory remarks to which we may refer the interested reader. Here we summarize as follows.
Base conditions and covariant systems in an algebraic threefold
Authors: Semple J. G.
Published: November 2008

If F is a free linear system of surfaces in an algebraic threefold V which is either non-singular or possesses only normal singularities, then F has Jacobian and adjoint surfaces, J2(F) and A2(F), and Jacobian and adjoint curve systems, J1(F) and A1(F), such that

where X2, X2 are the canonical systems of surfaces and curves on V, and X1(F) is the canonical system of curves of F. The imposition of base elements (points or curves) Ei, of assigned multiplicities λi, on F defines a system F1 which we may represent formally by the equation

and it is natural to enquire how the Jacobian systems of F1 differ from those of F, and how we may define adjoint systems A2(F1) and A1(F1) which cut on F1 its canonical curves and sets respectively.
Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions
Authors: Rankin R. A.
Published: November 2008

In this and the succeeding paper I solve two problems suggested by Prof. Hardy, namely (1) that of proving that Ramanujan's function

has no zeros on the line and (2) that of finding an asymptotic formula

where A is a constant. I also prove similar results concerning the coefficients of general modular forms. I am indebted to Prof. Hardy and Mr Ingham for various suggestions, and in particular to Mr Ingham's paper, “A note on Riemann's ζ-function and Dirichlet's L-functions”.
The effect of interstellar matter on climatic variation
Authors: Hoyle F. , Lyttleton R. A.
Published: November 2008

The effect of interstellar matter on the sun's radiation is considered with a view to explaining changes in terrestrial climate. It appears that a star in passing through a nebulous cloud will capture an amount of material which by the energy of its fall to the solar surface can bring about considerable changes in the quantity of radiation emitted. The quantity of matter gathered in by the star depends directly on the density of the cloud and inversely on the cube of its velocity relative to the cloud. Thus vastly different effects on the solar radiation can be brought about under fairly narrow ranges of density and relative velocity (ranges that are in accordance with astronomical evidence). In this way the process is able to explain the small changes in the solar radiation that are necessary to produce an ice age and, under conditions less likely to have taken place frequently, the high increase in radiation required for the Carboniferous Epoch. Despite the large effects that the mechanism can bring about, it is shown that the mass of the sun does not undergo appreciable change and hence reverts to its former luminosity once the cloud has been traversed.
An apparent inconsistency arising from tests of significance based on fiducial distributions of unknown parameters
Authors: Yates F.
Published: November 2008

It has been observed that the Behrens and Fisher test of the difference of the means of two samples gives a smaller percentage of significant results than might be expected on the analogy of the ordinary t test with a pooled estimate of variance. The cause of this apparent anomaly is explained, and it is shown that the criticisms of the test to which the anomaly has given rise have their origin in (a) neglect of the relevant information provided by the estimated values of the variances, and (b) an insufficient appreciation of the fiducial basis of all tests of significance (including the ordinary t test) on small samples.

It is pointed out that Sukhatme's table (constructed for the Behrens and Fisher test) also provides a test for the weighted mean of the means of two sets of observations, concerning whose relative accuracy no prior knowledge is available.
Photoemission in Geiger-Müller counters
Authors: Curran S. C. , Strothers J. E.
Published: November 2008

Nunn May (1) has developed the theory of the mode of action of the Geiger-Müller counter in terms of the photoemission produced in the electron avalanches constituting the discharge of the tube. An avalanche of electrons in the counter produces excitation and ionization of atoms and molecules in the neighbourhood of the positive central wire: the photons liberated by these in their turn release from the walls of the counter the photoelectrons which initate the following avalanche. Direct experimental evidence that such a sequence of events may occur is to be found in the work of Greiner (2).
The fractional dimensional theory of continued fractions
Authors: Good I. J.
Published: November 2008

The notion of fractional dimensions is one which is now well known. The object of the present paper is the investigation of the dimensional numbers of sets of points which, when expressed as continued fractions, obey some simple restriction as to their partial quotients. The sets considered are naturally of linear measure zero. Those properties of the partial quotients which hold for almost all continued fractions make up the subject called by Khintchine ‘the measure theory of continued fractions’.
The influence of radiation damping on the scattering of mesons
Authors: Heitler W. , Peng H. W.
Published: November 2008

I. It is shown that a general argument can be given for omitting the diverging parts of any quantized field interacting with a particle. The field equations thus freed from all singularities still contain the reaction of the field on the particle to the same extent as the reaction can be derived classically from the conservation of energy. The new field equations can be solved generally in terms of a simultaneous set of integral equations and do not give rise to any fundamental difficulties whatsoever.

II. The new field equations are applied to the multiple processes of the meson theory which occur, for instance, when a meson collides with a nuclear particle, the meson splitting up into a number n of secondary mesons. The cross-sections γn are worked out for n = 1, 2, 3, 4. γn has a maximum at some energy εn which increases with n. For ε ≫ εn, γn decreases like ε−2n−4 except for n = 1, when γ1 ∞ ε−2. γ1 (ordinary scattering) is larger than all the other γ's at all energies, but the probability of high multiplicities is comparable with that for low multiplicities or even greater in some energy regions.
Relativistic dynamics of a spinning magnetic particle
Authors: Mathisson Myron
Published: November 2008

The author's general variational method is applied to the case of a particle for which second moments are important but third and higher moments are negligible. Equations of motion are obtained for the angular momentum and for the centre of mass, equations (12·35) and (12·41), with arbitrary external forces X.

The angular forces are then calculated for a charged particle with electric and magnetic moments moving in a general electromagnetic field, on the assumption that the effect of a certain part of the energy tensor, Tiii of (15·17), is negligible. This leads to the equations of angular motion, (17·13), from which it is inferred that, in order that the magnitude of the angular momentum may be integrable, the angular momentum, electric and magnetic moments must all be parallel in a frame of reference in which the particle is instantaneously at rest.

The linear forces are then calculated for the case of no electric moment, leading to the equations for linear motion (18·10). From these it is inferred that, in order that the mass may be integrable, the ratio of the magnetic moment to the angular momentum must be constant.
A general form of the covering principle and relative differentiation of additive functions
Authors: Besicovitch A. S.
Published: November 2008

The Vitali covering principle is a powerful method in a wide class of problems of the theory of functions of a real variable and of the theory of sets.
On multiple curves. I
Authors: Hodge W. V. D.
Published: November 2008

Let γ be an irreducible variety of dimension d, and let In denote an involution of order n on Γ. The image of In is a variety C, also irreducible and of dimension d, and Γ is said to be a multiple of C. Among the possible multiples of a given variety C there is a class of particular interest; this arises when the involution In possesses the following properties:

(1) it is unramified, that is, in every set of the involution the n points are all distinct;

(2) it is generated by an Abelian group G of order n of birational transformations of Γ into itself. Given any point P of Γ the n transformations of G transform P into n points P1 = P, P2, …, Pn which constitute a set of In. The two conditions together imply that the function field κ of Γ is an unramified Abelian Galois extension of the function field k of C, and this shows that the study of the particular class of multiple varieties—which we may call normal multiple varieties when it is necessary to give a name to them—is analogous to the study of a well-known branch of the theory of numbers.
A new type of boundary-value problem in hyperbolic equations
Authors: Chandrasekhar S.
Published: November 2008

In developing the theory of radiative transfer in expanding atmospheres (Chandrasekhar, 1945 a, b) the author has recently encountered certain novel types of boundary-value problems in hyperbolic equations which appear to merit consideration for their own sake. As related to the equation

the boundary-value problems which occur are of the following general type

i.e. along AB in Fig. 1. Along AD (x = 0 and 0 ≤ y ≤ l2) and BC (x = l1, 0 ≤ y ≤ l2) we are further given that

and

where φ(y) and ψ(y) are two known functions. The problem is to solve equation (1) in the rectangular strip ABCD satisfying the stated boundary conditions.
On the motion of a Gaussian wave-packet in a parabolic potential field
Authors: Coulson C. A. , Rushbrooke G. S.
Published: November 2008

The paper deals with the motion of a Gaussian wave-packet in a parabolic potential field, contrasting the quantum-mechanical behaviour of the packet with that which follows from pure classical mechanics.

The behaviour is entirely classical only if the initial phase of the space-wave-function is linear in the space coordinate; in which case the wave-packet is initially a ‘best-possible’ packet according to Heisenberg's uncertainty relationship.

The paper concludes with a brief reference to the similar problem for a free particle.

Note added in Proof, 16 February 1946. Through the courtesy of Mr M. S. Bartlett, we have been privileged to see the manuscripts of some work by Mr J. E. Moyal and himself, in which the motion of a probability wave-packet is discussed in more general terms than our own: and in which it appears that the motion of our wave-packet, which we have described as non-classical, is indeed actually classical provided that a certain correlation exists between momentum and position in the p, q-distribution function. With this knowledge we now find that classical and quantal predictions agree precisely provided that the initial classical probability function is given by

this is a particular case of their more general results.
Functions of a statistical variate with given means, with special reference to Laplacian distributions
Authors: Tweedie M. C. K.
Published: November 2008

A method is described in general terms for finding the function of a variate of which the mean is a given function of a parameter of the population. This can sometimes be used for finding unbiased estimates and for finding the moments and moment-generating functions of a statistic when another statistic based on the same observations has a constant value. It is always available when the latter statistic is a ‘sufficient statistic’ for estimating the parameter, which requires the frequency function to be of a certain form. A number of examples are given.
A note on Talbot's bands
Authors: Shoenberg D.
Published: November 2008

A simple formula is obtained for the visibility of Talbot's bands. This formula shows that the bands occur only if the glass plate is inserted at one particular side of the telescope objective, and that when they occur, their visibility is proportional to the plate thickness up to a certain optimum thickness, and then decreases linearly, vanishing at and beyond twice the optimum thickness.
Simple progressive solutions of the wave equation
Authors: Friedlander F. G.
Published: November 2008

The object of this paper is to determine all the solutions of the wave equation

which are of the simple form

where F denotes an arbitrary function. It will be shown that, in addition to the obvious cases of plane or spherical progressive waves, such solutions exist only when the wave fronts

are certain algebraic surfaces of the fourth order, the cyclides of Dupin. These include, as degenerate cases, the sphere, the plane, the cylinder, the cone, and the torus.
The effect of a fixed vertical barrier on surface waves in deep water
Authors: Ursell F.
Published: November 2008

In this paper the two-dimensional reflection of surface waves from a vertical barrier in deep water is studied theoretically.

It can be shown that when the normal velocity is prescribed at each point of an infinite vertical plane extending from the surface, the motion on each side of the plane is completely determined, apart from a motion consisting of simple standing waves. In the cases considered here the normal velocity is prescribed on a part of the vertical plane and is taken to be unknown elsewhere. From the condition of continuity of the motion above and below the barrier an integral equation for the normal velocity can be derived, which is of a simple type, in the case of deep water. We begin by considering in detail the reflection from a fixed vertical barrier extending from depth a to some point above the mean surface.
Two-dimensional plastic flow of a Bingham solid
Authors: Oldroyd J. G.
Published: November 2008

The rheological equations of state for an isotropic Bingham solid may be written in tensor form:

where

The notation is explained fully in a previous paper (1). The frame of reference is essentially Eulerian. Primes denote deviatoric components of the stress, strain and rate of strain tensors, pik, εik and eik (for example, ). The dilatation is denoted by Δ. The rigidity and bulk moduli μ and κ are, if not constants, essentially scalar functions of the stress tensor. The variable viscosity η involves two constants, the yield value ϑ and the reciprocal mobility η1.
A family of cubical graphs
Authors: Tutte W. T.
Published: November 2008

We begin with some definitions.

A cubical graph is a simplicial 1-complex in which each 0-simplex is incident with just three 1-simplexes.
On the self-accelerating electron
Authors: Ashauer S.
Published: November 2008

After solving the equations of motion of Dirac's self-accelerating electron, a physical picture of it is formed by plotting graphically the surfaces of constant scalar potential when the electron has built up a velocity close to the velocity of light.
Symmetric energy-momentum tensors in relativistic field theories
Authors: de Wet J. S.
Published: November 2008

In relativistic field theories derived by a variation principle from a Lagrangian, the problem arises of finding a symmetric tensor of rank 2 which has vanishing divergence in virtue of the field equations and is such that taken over a space-like section is equal to the corresponding integral of the so-called canonical energy-momentum tensor. It is well known that the latter condition is satisfied if the difference between the two tensors is the divergence of an antisymmetric tensor of rank 3.
Kolmogoroff's theory of locally isotropic turbulence
Authors: Batchelor G. K.
Published: November 2008

A new and fruitful theory of turbulent motion was published in 1941 by A. N. Kolmogoroff. It does not seem to be as widely known outside the U.S.S.R. as its importance warrants, and the present paper therefore describes the theory in some detail before presenting a number of extensions and making a comparison of experimental results with some of the theoretical predictions.

Kolmogoroff's basic notion is that at high Reynolds number all kinds of turbulent motion, of arbitrary mean-flow characteristics, show a similar structure if attention is confined to the smallest eddies. The motion due to these eddies of limited size is conceived to be isotropic and statistically steady. Within this range of eddies we recognize two limiting processes. The influence of viscosity on the larger eddies of the range is negligible if the Reynolds number is large enough, so that their motion is determined entirely by the amount of energy which they are continually passing on to smaller eddies. This quantity of energy is the local mean energy dissipation due to turbulence. On the other hand, the smaller eddies of the range dissipate through the action of viscosity a considerable proportion of the energy which they receive, and the motion of the very smallest eddies is entirely laminar. The analytical expression of this physical picture is that the motion due to eddies less than a certain limiting size in an arbitrary field of turbulence is determined uniquely by two quantities, the viscosity and the local mean energy dissipation per unit mass of the fluid.

The mathematical method of describing the motion due to eddies of a particular size is to construct correlations between the differences of parallel-velocity components at two points at an appropriate distance apart. Kinematical results analogous to those for turbulence which is isotropic in the ordinary sense are obtained, and then the scalar functions occurring in the expressions for the correlations are determined by dimensional analysis. The consequences of the theory in the case of turbulence which possesses ordinary isotropy are analysed and various predictions are made. One of these, namely that dimensionless ratios of moments of the probability distribution of the rate of extension of the fluid in any direction are universal constants, is confirmed by recent experiments, so far as the second and third moments are concerned. In several other cases it can be said that relations predicted by the theory have the correct form, but further experiments at Reynolds numbers higher than those hitherto used will be required before the theory can be regarded as fully confirmed. If valid, Kolmogoroff's theory of locally isotropic turbulence will provide a powerful tool for the analysis of problems of non-uniform turbulent flow, and for the determination of statistical characteristics of space and time derivatives of quantities influenced by the turbulence.
The measurement of double and triple correlation derivatives in isotropic turbulence
Authors: Townsend A. A.
Published: November 2008

By electrical analysis of the output from a hot-wire anemometer, it is possible to measure rapidly and accurately all the quantities appearing in the theoretical equations for the decay of isotropic turbulence. The technique for these measurements is described, and possible extensions and limitations of the method are discussed. Measurements have been made of the second and fourth derivatives of the longitudinal double-velocity correlation at the origin, the third derivative of the longitudinal triple-velocity correlation at the origin, the statistical distribution in time of the velocity fluctuations, and the integral scale of turbulence.
The lattice properties of asymmetric hyperbolic regions
Authors: Cassels J. W. S.
Published: November 2008

In this paper, all numbers are real and all radicals are positive.

Let f(x, y) = ax2 + bxy + cy2 be an indefinite quadratic form, and let d = Δ2 = b2 − 4ac, where Δ > 0. A well-known theorem of Minkowski states that, if (x0, y0) is any pair of numbers, then there exists a pair (x, y), x ≡ x0 (mod 1), y ≡ y0 (mod 1), say (x, y) ≡ (x0, y0) (mod 1), such that

for k ≥ ¼, and Davenport has shown how this result may be improved if we know a value assumed by f(x, y) for coprime integral values, (x, y) = (m, n) ≠ (0, 0). In this paper, we discuss the more general inequality

where R and S are constants, and use the method developed in the first paper of this series to obtain a sharper and more general theorem than Davenport's. We give an application to the theory of real Euclidean quadratic fields and to a problem in Diophantine approximation discussed by Khintchine.
On the reflexion of surface waves by a submerged circular cylinder
Authors: Dean W. R.
Published: November 2008

1. In a recent paper (1) the reflexion of surface waves by a submerged plane barrier has been considered, and it has been shown that there is a reflected wave of finite amplitude at a great distance from the barrier, so that the coefficient of reflexion (the ratio of the amplitudes, at a great distance from the barrier, of the reflected and incident waves) is finite. In this paper it is shown that in the case of reflexion by a submerged circular cylinder, the coefficient of reflexion is zero, with the result that the only effect of the obstacle at a great distance is that there is a phase-difference between the incident and transmitted waves, their amplitudes being the same. Only one case has been worked out in detail numerically, and the method employed is such that the condition that the section of the cylinder must be a stream-line has been satisfied only to an approximation; it is, however, shown that the coefficient of reflexion is zero in the general case.
Non-convex regions in three and more dimensions
Authors: Macbeath A. M.
Published: November 2008

Blichfeldt (2) gave a method of proving the existence of lattice points in certain regions. His method depends on the following lemma (which he stated in a somewhat different form):

Lemma 1. Let K′ be a closed region of volume V. Let K be another region so related to K′ that the difference of any two points in K′ is a point in K. Let Λ be any lattice whose determinant is positive and does not exceed V.
Notes on iterative processes
Authors: Hartree D. R.
Published: November 2008

If ηn is the error in the result of repeating an iterative process n times, and ηn+1 is the iterative process is called Kth order. It is shown that if for a given equation there is an iterative process of the (K + 1)th order, the iterative process of the Kth order is not unique, and conversely if an iterative process of the Kth order is not unique, it is generally possible to construct from two of the Kth order processes a process of the (K + 1)th order. As an example, three second-order processes for a square root are exhibited, and a third-order process is derived from two of them.

Iterative processes for positive and negative integer roots are given, of kinds suitable for use on machines in which division is a relatively slow process and one to be used sparingly. It is shown how a second-order process can be derived from the results of two repetitions of a first-order process. The extension of this iterative process for the solution of differential and integral equations is a development which is urgently required.
The probability distribution of gene-differences in relation to selection, mutation, and random extinction
Authors: Skellam J. G.
Published: November 2008

In order to determine the mathematical form of the distribution of gene frequencies in a population, R. A. Fisher (1, 2) applied two different but complementary methods. First, differential equations were set up for certain cases and solved. An extension of these results arose from the expression by Wright (10,11,12) of the conditions as an integral equation. But strictly, neither integral nor differential equations can represent the conditions exactly, since integration is substituted for summation, and differentials for minimum steps (1/2n) in gene frequency. The second method employed by Fisher was exact, even though the solutions involved new functions. The properties of generating functions were utilized to yield functional equations, and the special case of rare mutations was dealt with in detail. In the present paper this powerful method is extended to the general case where mutation rates are not necessarily small.
On the steady motion of viscous liquid in a corner
Authors: Dean W. R. , Montagnon P. E.
Published: November 2008

1. In a steady two-dimensional motion of viscous liquid in the sharp corner formed by the rigid straight boundaries θ = 0, α, where r, θ are plane polar coordinates, it is found that, near enough to the corner, the most important term in the stream-function is of the form rmf(θ). The index m is evaluated in §§ 2–4 for values of α between 360 and 90°, and is found to be complex if α is less than about 146°; the limiting form of the stream-function when α is small is considered in § 5.
Note on the geometrical optics of diffracted wave fronts
Authors: Friedlander F. G.
Published: November 2008

It is well known that a solution of the wave equation

for which u = ∂u/∂t = 0 initially outside a surface S0, vanishes at time t in the exterior of a surface St parallel to, and at normal distance ct from S0, so that the wave fronts of disturbances represented by the solutions of the wave equation obey the laws of geometrical optics. Analogous results hold for the solutions of any linear hyperbolic second-order partial differential equation with boundary value conditions of the ‘Cauchy’ type. But the wave fronts of solutions of problems in which some of the boundary conditions are of the type representing reflexion do not seem to have been treated, and in particular the case of diffraction, when there is a ‘shadow’, does not seem to have been considered at all.
The exact transition probabilities of quantum-mechanical oscillators calculated by the phase-space method
Authors: Bartlett M. S. , Moyal J. E.
Published: November 2008

The ‘phase-space’ method in quantum theory is used to derive exact expressions for the transition probabilities of a perturbed oscillator. Comparison with the approximate results obtained by perturbation methods shows that the latter must be multiplied by an exponential factor exp (− ∊/ℏω), where ∊ is the non-fluctuating part of the work done by the perturbing forces; as long as ∊ is small, exp (− ∊/ℏω) ˜ 1 and only dipole transitions have an appreciable probability. As the perturbation energy increases, however, this is no longer true, and multipole transitions become progressively more probable, the most probable ones being those for which the change in energy is approximately equal to the work done by the perturbing forces.
Grouping corrections and maximum likelihood equations
Authors: Lindley D. V.
Published: November 2008

1. Any mention of the word ‘grouping’ immediately brings to a statistician's mind the Sheppard corrections. These are usually used to make inferences about the underlying ungrouped population from observations made on the grouped population, but it is important to realize that, as stated and proved, they have nothing to do with sampling or inference and are merely expressions for the moments of one population in terms of the moments of another population derived from it. They can only be used for the inference problem when allied to the method of moments. This method, as formulated by K. Pearson, consists in taking for θ* the estimate of the population parameter θ, the same function of the sample moments mi that θ is of the population moments μi, each mi being an estimate of the corresponding μi. If the population is grouped the mi are estimates of the , the grouped population moments, so we require θ as a function of the to apply Pearson's method. This can be done since θ is known as a function of the µi and the µi are known as functions of the by the corrections. Use of the Sheppard corrections with any other inference method, even when this method, when applied to the continuous population, yields an estimate which is a sample moment, so far as I am aware, has not been examined except for the normal curve.
Surface waves on deep water in the presence of a submerged circular cylinder. I
Authors: Ursell F.
Published: November 2008

A method is given for the calculation of the surface waves of small amplitude generated on deep water by a normal velocity distribution of period 2π/σ prescribed over a submerged circular cylinder. The method of solution involves a system of linear equations in an infinite number of unknowns; this system always possesses a solution. The unknowns may be obtained as power series in a parameter Ka, convergent for sufficiently small values of the parameter. When the parameter is not small, the equations can be solved by infinite determinants. It is shown that the reflexion coefficient of waves incident on a fixed circular cylinder vanishes, as was first shown by Dean. The pulsations of a submerged cylinder are discussed when the normal velocity is the same at all points of the cylinder at any given time.
The distribution of stress in the neighbourhood of a flat elliptical crack in an elastic solid
Authors: Green A. E. , Sneddon I. N.
Published: November 2008

This paper is mainly concerned with the distribution of stress near a flat elliptical crack in a body of infinite extent under a uniform tension at infinity perpendicular to the plane of the crack. After an analytical solution of this problem was found the authors received a copy of a paper by Sadowsky and Sternberg (3) in which they solved the more general problem of the stress concentration around a tri-axial ellipsoidal cavity in an elastic body of infinite extent, the body at infinity being in a uniform state of stress whose principal axes are parallel to the axes of the cavity. The method of solution adopted by the present writers for the special case of the elliptical crack is somewhat different from the more general work of Sadowsky and Sternberg, and is, moreover, surprisingly simple, so it seems to be of value to present this solution here.
Probability distributions and orthogonal parameters
Authors: Huzurbazar V. S.
Published: November 2008

Let f(x, αi) be the probability density function of a distribution depending on n parameters αi(i = 1,2, …, n). Then following Jeffreys(1) we shall say that the parameters αi are orthogonal if
The distribution of the integers N for which {θN} < φ
Authors: Slater N. B.
Published: November 2008

Let ν(M) denote the number of integers N in the range (0 ≤ N ≤ M) for which

where {y} denotes the fractional part of y. It is a convenient but trivial restriction to assume always that θ and φ lie between 0 and 1. There is a well-known result, which will be called ‘Weyl's Theorem’, that if θ is irrational,
Conduction of heat in a solid with a power law of heat transfer at its surface
Authors: Jaeger J. C.
Published: November 2008

In many technical problems on conduction of heat involving convection, radiation, or evaporation at the surface of a body, the flux of heat at the surface is known empirically as a function of the surface temperature with reasonable accuracy. The thermal properties of the body also vary with the temperature, but in many cases the nature of this variation is completely unknown, and in others it is slight over the range of temperature involved. Thus it seems worth while studying problems on conduction of heat in a medium with constant thermal properties and with heat transfer at its surface a given function of the surface temperature. Mathematically such problems occupy an interesting position between the classical linear theory and the general case in which both the differential equation and the boundary conditions are non-linear.
A problem on topological transformations of the plane. II
Authors: Besicovitch A. S.
Published: November 2008

In the paper under the above title I have given an example of a homoeomorphism of a plane under which the positive half-orbit† of a certain point is everywhere dense on the plane, and I have made a statement that the same is true for every point of the plane, except the origin.
Slow motion of viscous liquid in a semi-infinite channel
Authors: Dean W. R.
Published: November 2008

1. A slow two-dimensional steady motion of liquid caused by a pressure gradient in a semi-infinite channel is considered. The medium is bounded by two parallel semi-infinite planes represented in Fig. 1 by the straight lines AB, DE. The stream-function ψ is a biharmonic function of x, y which exactly satisfies the condition that AB, DE must be stream-lines, but the condition that there must be no velocity of slip on these boundaries is satisfied only approximately, and the calculated velocity of slip gives a measure of the accuracy of the solution.
Statistical mechanics and partitions into non-integral powers of integers
Authors: Agarwala B. K. , Auluck F. C.
Published: November 2008

1. The problem of the partition of numbers, first investigated in detail by Hardy and Ramanujan (1), has in recent years assumed importance on account of its application by Bohr and Kalckar (2) in evaluating the density of energy levels in heavy nuclei. A ‘physical approach’ to the partition theory has been made by Auluck and Kothari (3), who have studied the properties of quantal statistical assemblies corresponding to the partition functions familiar in the theory of numbers. The thermodynamical approach to the partition theory, apart from its intrinsic interest, draws attention to aspects and generalizations of the partition problem that would, otherwise, perhaps go unnoticed. Thus we are led to consider restricted partitions such as: partitions where the summands are repeated not more than a specified number of times; partitions where the summands are all different; partitions into summands which must not be less than a specified value; partitions into a prescribed number of summands, and so on. The generalization that seemed to us to be the most interesting is the extension of the partition concept to include partitions into non-integral powers of integers.
On the representation of rational sections of the Grassmannian of lines of five dimensions
Authors: Hall R.
Published: November 2008

The Grassmannian, of lines of S5 and also its sections by general spaces of 13, 12, 11 and 10 dimensions were shown to be rational by Severi. The section by a general S9 is, by a result of Fano, birationally equivalent to a non-singular cubic primal of S4 and hence is probably not rational.
Random division of an interval
Authors: Mauldon J. G.
Published: November 2008

We consider n − 1 points xj (j = 1, …, n − 1) selected independently at random from the interval (0, 1), the distribution of any xj being the rectangular distribution

The points xj divide the interval (0,1) into n intervals. Let λ be the sum of the lengths of the k largest of these n intervals. It is the purpose of this paper to give the exact sampling distribution of λ for any given n and k (see equation (5)). Fisher (1) in 1929 solved the problem for k = 1. We first investigate the distribution of μ = 1 − λ.
Trapping modes in the theory of surface waves
Authors: Ursell F.
Published: November 2008

It is shown that a mass of fluid bounded by fixed surfaces and by a free surface of infinite extent may be capable of vibrating under gravity in a mode (called a trapping mode) containing finite total energy. Trapping modes appear to be peculiar to the theory of surface waves; it is known that there are no trapping modes in the theory of sound. Two trapping modes are constructed: (1) a mode on a sloping beach in a semi-infinite canal of finite width, (2) a mode near a submerged circular cylinder in an infinite canal of finite width. The existence of trapping modes shows that in general a radiation condition for the waves at infinity is insufficient for uniqueness.