bluuebunny/crossref_metadata_2025 huggingface dataset:
A Proof of Miquel's Theorem by Involutions in the Argand Diagram
Authors: Frazer R. A.
Published: November 2008

Miquel's Theorem, as generalised by Clifford, states that the four circles through the intersections of four straight lines, taken in triads, meet in a point; that the five such points derived from five lines, taken in sets of four, lie on a circle; that the six such circles, determined by six lines, meet in a point……. The process continues indefinitely.
Some Calculations relevant to the Quantum Defect in the Extended Ritz Formula
Authors: Hargreaves J.
Published: November 2008

The object of this paper is to make certain calculations which will be required in the estimation of the quantum defect. Hartree has defined certain solutions Gi(σ, ζ), Hi(σ, ζ) of the wave equation for an electron in a Coulomb field, viz.
The Quadratic form for Radial Acceleration, in the Theory of Relativity
Authors: Hargreaves R.
Published: November 2008

§ 1. In the transformation corresponding to constant acceleration, and in the resulting quadratic form, there is an arbitrary constant. In Einstein's formula for central acceleration a mass m defining the acceleration appears, but no arbitrary constant. In seeking to account for, or to repair, the deficiency I retained all constants naturally arising in the integration of the differential equations, and it then appeared that the result could be obtained by transformation. It will be convenient to state the transformation at the outset, and then to examine the differential equations with a view to a clear understanding of the assumptions made in reaching the more general result.
On loci which have two systems of generating spaces
Authors: Richmond H. W. , Bath F.
Published: November 2008

In hypergeometry loci defined by parametric equations

have been frequently discussed from various points of view. Among other properties these loci possess two systems of generating spaces obtained by keeping the α's (or β's) fixed while the β's (or α's) vary, the simplest example being the generators of a quadric in S3. Another method of obtaining loci having these properties is by taking the flat spaces through the sets of a linear series of sets of points on an elliptic curve and the flat spaces through the sets of the residual series (the curve being supposed normal. In this way we shall obtain a configuration of ∞PSP's and ∞qSq's, the dual of which gives a locus included among those represented by (1).
The Thallous Thallic Halides
Authors: Berry A. J.
Published: November 2008

In addition to the simple thallous and thallic chlorides and bromides, a considerable number of intermediate and mixed halides have been described by various investigators. Most of the modern work on these substances has been done by Thomas, by Meyer and by Cushman. There is general agreement as regards the existence of substances of the type TlX3. 3TlX, but considerable divergence exists with regard to the mixed chlorobromides. Compounds of the type TlX3. TlX have also been described, the best known of which appears to be the dibromide TlBr3. TlBr.
On the substitution of Wallis's Postulate of Similarity for Euclid's Postulate of Parallels
Authors: Hill M. J. M.
Published: November 2008

In the year 1663 John Wallis, Savilian Professor of Geometry in the University of Oxford, delivered a lecture in which he claimed that he had proved Euclid's Postulate of Parallels according to the strictest laws of demonstration after Euclid's manner.
Directional Properties of Wireless Receiving Aerials
Authors: Burnett D.
Published: November 2008

Equations have been given by Moullin to determine the current in a receiving antenna in wireless telegraphy and they have been discussed and extended by Colebrooke, who in particular considered the reception of a plane polarised wave with its electric vector vertical by an antenna which is partly vertical and partly horizontal. His result shows that the intensity of the received signal is independent of the orientation of the receiver. In practice, however, the incident wave front is not vertical, its tilt depending on the wave length and on the resistance of the earth, as shown by Zenneck, and the purpose of this paper is to investigate the effect of the orientation of a bent antenna in the reception of this type of wave. An approximate theory of the Beverage antenna is obtained by the same method.
The relation between Riemann's method of summation and Cesàro's
Authors: Verblunsky S.
Published: November 2008

A series is said to be summable by Riemann's method of order k to S, or summable (R, k) to S, if

where k is a positive integer, and where we suppose that

is convergent for values of h in a neighbourhood of the origin.
Extension of a Lemma of Stroh
Authors: Grace J. H.
Published: November 2008

The lemma I propose to discuss may be stated as follows;

If ξ, η, ζ are three quantities whose sum is zero and λ, μ, ν three positive integers (≤n) satisfying the condition

then any polynomial of order n in ξ, η, ζ can be expressed in the form

where P, Q, R are polynomials of orders n − λ, n − μ, n − ν.
A recording gyroscope
Authors: Searle G. F. C.
Published: November 2008

Gyroscopic effects furnish important instances of dynamical actions, but their general discussion is outside the range of most students reading for Part I of the Natural Sciences Tripos. If, however, we limit ourselves to the case in which the axis of the wheel makes a constant angle with the vertical, the theory becomes elementary and simple devices suffice for recording the movements of the revolving wheel. The recording gyroscope was designed to give students at the Cavendish Laboratory, Cambridge, an opportunity of making practical measurements to verify the theoretical results.
On some series of functions, (1)
Authors: Paley R. E. A. C. , Zygmund A.
Published: November 2008

Let c0, c1,…, cn,…be a sequence of real constants, and ƒ0(x), ƒ1(x),…, ƒn(x),… a sequence of functions defined, for example, in the interval (0, 1). In this paper we shall investigate some of the properties of the series

which may be obtained from the standard series

by interchanging the signs of the terms in a quite arbitrary way.
On the Annihilation of Electrons and Protons
Authors: Dirac P. A. M.
Published: November 2008

An electron, according to relativity quantum theory, has two different kinds of states of motion, those for which the kinetic energy is positive and those for which it is negative. Only the former, of course, can correspond to actual electrons as observed in the laboratory. The latter, however, must also have a physical meaning, since the theory predicts that transitions will take place from one kind to the other. It has recently been proposed that one should assume that nearly all the possible states of negative energy are occupied, with just one electron in each state in accordance with Pauli's exclusion principle, and that the unoccupied states or ‘holes’ in the negative-energy distribution should be regarded as protons. According to these ideas, when an electron of positive energy makes a transition into one of the unoccupied negative-energy states, we have an electron and proton disappearing simultaneously, their energy being emitted in the form of electromagnetic radiation. The object of the present paper is to calculate the frequency of occurrence of these processes of annihilation of electrons and protons.
The deflection of a cantilever bar rotated under end load
Authors: Hunter W.
Published: November 2008

In the Wöhler rotating bar fatigue test, in order to examine the behaviour of a specimen under alternating stress, a load is suspended from the end of a cantilever bar, and the bar is then rotated, the load remaining stationary. It is evident that the rotation will set up additional stresses in the bar, so that the limits between which the stress at any point will alternate are not the same as the stress at the corresponding point of a statically deflected bar.
The gramophone horn
Authors: Larmor Joseph
Published: November 2008

This paper illustrates the problems of general linear propagation mainly by analysis of the conditions necessary for effective resonance, which are now of great technical importance. Another theory is also broached, that of the relation of the arterial pulse to the blood-flow, first explored with adequate hydraulic instincts by Young in 1808.
On the excitation of a lithium atom by collision with a slow α-particle
Authors: Frame J. W.
Published: November 2008

It has been shown by Mott that it is legitimate in calculating the probability of excitation of an atom by close collision with a slow α-particle to treat the α-particle as a moving centre of force. This method gives for the probabilityPof excitation

where

and
Paramagnetic hysteresis
Authors: Temperley H. N. V.
Published: November 2008

Ewing's original theory of hysteresis, though not admissible for metals, seems to apply to paramagnetic salts. The peculiar form of the interaction between dipoles implies that a certain fraction of them may be incapable of following small changes in the external field. The agreement with experiment is satisfactory, considering the rough nature of the theory.

In conclusion, I should like to thank Messrs E. S. Shire and H. M. Barkla for kindly discussing their results with me before publication, and also Prof. R. H. Fowler, Dr J. D. Cockcroft and Mr A. H. Wilson for helpful discussions. Also I should like to thank the Provost and Fellows of King's College for the award of a studentship, during the tenure of which this work was carried out.
Note on Dr Atkinson's paper
Authors: Hoyle F. , Lyttleton R. A.
Published: November 2008

The foregoing paper by Dr Atkinson relates to the new formula for the rate of accretion of mass by a star that has recently been proposed by us. The derivation of this formula was made by means of a dynamical argument, certain physical aspects having to be neglected.
A note on Morley's trisector theorem
Authors: Lob H.
Published: November 2008

Two theorems by the late Prof. F. Morley have aroused special interest, one being the chain of circle properties discovered by De Longchamps and independently by Pesci, Grace and by Morley himself, and the other the remarkable property that if the angles of any triangle are trisected, then the trisecting lines meet in pairs to form an equilateral triangle. In a previous paper‡ the present writer has shown that the circles making up the chain may be derived from a rational normal Cn in [n] by the following process.
The electrostatic dipole moment of a nucleus in the meson theory
Authors: Ma S. T.
Published: November 2008

The interaction between an external electromagnetic field and a nucleus, including its exchange meson field, has recently been investigated by several authors (1). The interaction between a vector potential A and a nucleus has been found to be expressible in terms of the electric and magnetic multipole moments of the latter. It is the object of this note to discuss the corresponding interaction with a scalar potential V, and its connexion with previous results.
On Pulses whose Travel Times are not True Minima
Authors: Jeffreys Harold
Published: November 2008

In the construction of the seismological tables it has been found that the times of several pulses at a given distance show a well-marked maximum frequency for each, about which the bulk of the observations are spread with a standard deviation of 1·5−3 sec. For others there was only a vague concentration, spread over about half a minute (Jeffreys and Bullen, 1936–40). The theoretical times of the latter type could all be calculated from those of the former by using the principle of stationary time for small variations of the path, and the observed times might be either shorter or longer than the theoretical ones. There was a complete association between the type of distribution of the residuals and the nature of the path. Whenever the time as calculated was a minimum for all small variations of the path, the distribution of residuals was of the former type; when it was stationary without being either a true minimum or a maximum it was of the latter.
Notes on the two-centre problem in wave mechanics
Authors: Marrack P. E.
Published: November 2008

The paper deals with new solutions of the differential equations of the two-centre problem which are expressed in terms of the confluent hypergeometric function. By means of this function a solution of the λ-equation (1) is obtained which enables the separation constant A to be found for small values of the parameter p in a rapidly convergent series, which for the special case of m = 0 is still more rapidly convergent. The solution for the μ-equation (2) in the general case is of the same character as regards rapidity of convergence as solutions previously obtained, but when m = 0 it again possesses a higher rate of convergence as compared with solutions given by other investigators.
On the shearing motion of fluid past a projection
Authors: Dean W. R.
Published: November 2008

1. In this paper, a continuation of an earlier paper(1), we consider the two-dimensional motion of incompressible viscous liquid past a projection, 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 the area in the z-plane (Fig. 1) can be represented conformally on a circle in the ζ-plane by a rational function of ζ; this function contains a parameter a (0 < a ≤ 1), and by varying a the shape of the projection can be varied. Since a rational function is concerned in the conformal transformation a method lately developed by N. Muschelišvili(2) can be used in solving the biharmonic equation for the stream function, though the method actually used differs in some points of detail from that originally proposed by Muschelišvili and appears to be somewhat simpler.
Configuration of Quadrivalent Atoms
Authors: Lowry T. M.
Published: November 2008

Two alternative views have been expressed in regard to the configuration of quadrivalent atoms. On the one hand le Bel and van't Hoff assigned to quadrivalent carbon a tetrahedral configuration, which has since been confirmed by the X-ray analysis of the diamond. On the other hand, Werner in 1893 adopted an octahedral configuration for radicals of the type MA6, e.g. in

and then suggested that “the molecules [MA4]X2 are incomplete molecules [MA6]X2. The radicals [MA4] result from the octahedrally-conceived radicals [MA6] by loss of two groups A, but with no function-change of the acid residue…. They behave as if the bivalent metallic atom in the centre of the octahedron could no longer bind all six of the groups A and lost two of them leaving behind the fragment [MA4]” (p. 303).
Polarities for the nodes of a Segre cubic primal in space of four dimensions
Authors: Baker H. F.
Published: November 2008

The present note was inspired by the desire to see whether the exposition of the theory of the Segre cubic primal of ten nodes was simplified by using only five coordinates instead of the six redundant coordinates introduced by Stéphanos and Castelnuovo. But the simple remark that the ten nodes may be separated into two simplexes which are polars of one another in regard to a quadric—which may or may not be novel—suggests the comparison of the theory of five associated lines in space of four dimensions with familiar properties of eight associated points in three dimensions; especially as it appears (§ 8) that the ten nodes do not form an associated set. With the repetition, for the sake of clearness, of several results which are familiar in general terms, there seems enough novelty to make the note of some utility. The form found for the equation of the Segre primal in five coordinates (§ 5) seems also noticeable.
A symmetrical configuration of n + 1 rational normal curves in [2n]
Authors: Babbage D. W.
Published: November 2008

This paper is concerned with the extension to [2n] of a well-known symmetrical configuration in [4], namely, that of three rational quartic curves, with six points in common, having the property that a trisecant plane of any one which meets a second also meets the third.
A study of the properties of hydrogen films on tungsten by the method of contact potentials
Authors: Bosworth R. C. L.
Published: November 2008

In a study of the properties of hydrogen on tungsten by the method of contact potentials the following points have been established:

(1) The contact potential of a 92% covered surface of hydrogen on tungsten against bare tungsten is 1·04 V., and the Richardson constants for such a surface are, approximately,

A = 30, b = 5·60 V.

(2) A film of deuterium is 20 m V. positive relative to a similar hydrogen film.

(3) Over the range of temperatures and pressures used by Bryce in a study of the production of atomic hydrogen the films are nearly saturated, so that the production of atomic hydrogen is primarily due to a molecule striking a bare tungsten atom, one atom being adsorbed and the other going into the gas phase.

(4)The condensation coefficient for hydrogen molecules on cold tungsten is 0·01.

(5) The effective dipole moment of each hydrogen atom on the surface is −0·42 Debye unit and is independent of the fraction of the surface covered.
Note on an identity connected with diophantine approximation
Authors: Davenport H.
Published: November 2008

In discussing Hecke's function

where {t} = t − [t] − ½ Hardy and Littlewood* made use of the identity

Here

θ is an irrational number between 0 and 1, and pν/qν is the νth convergent to θ (beginning with p0 = 0, q0 = 1). The proof of (1) given by Hardy and Littlewood is of an analytic character.
The Hankel transform of Whittaker's function Wk, m(z)
Authors: Erdélyi A.
Published: November 2008

The Hankel transform of a function x−½νf(x) can be determined by the help of the following rule, obtained for ν = 0, 1 by B. van der Pol and K. Niessen(1), for positive integral values of ν by K. Niessen(2), and for arbitrary values of ν, [R(ν) > − 1], by F. Tricomi(3). If*

F(p) ≒ f(x),

then
The conductivity of thin metallic films according to the electron theory of metals
Authors: Fuchs K.
Published: November 2008

The conductivity of thin films of the alkali metals has recently been measured in the H. W. Wills Physical Laboratory, Bristol*. It was found that as the thickness of the film is decreased to that of a few atomic layers the conductivity drops below that of the bulk metal. In the papers quoted the hypothesis was put forward that this effect is due to the shortening of the mean free paths of the conduction electrons of the metal by collisions with the boundaries of the film. The experimental results were compared with a formula derived on the basis of this hypothesis. This formula was, however, obtained subject to a number of simplifying assumptions, and it is the first purpose of this paper to obtain a more accurate formula. I also compare this formula with experiment, and make certain deductions about the surfaces of thin films.
The moment of the fluid forces acting on a plate which is gliding on the surface of a stream
Authors: Green A. E.
Published: November 2008

13. Formulae have been obtained for the moment of the fluid forces which act on a plane plate which is gliding on the surface of a stream of finite or infinite depth. It has been shown that the formulae for the lift and moment forces acting on a plate which is gliding on a stream of infinite depth may be obtained by a limiting process from the formulae for the general case of gliding on a stream of finite depth. Also, by another limiting process, the lift and moment forces for the Rayleigh flow past a flat plate in an infinite stream have been obtained from the lift and moment forces for gliding on a stream of infinite depth.

Numerical work has been carried out for the case when the angle of incidence of the plate to the stream is 5° and this has shown the variation of the lift and moment forces and the position of the centre of pressure with a range of values of the length of the plate, the depth of the stream and the height of the trailing edge of the plate above the bed of the stream.
Self-consistent field for molecular hydrogen
Authors: Coulson C. A.
Published: November 2008

The method of the self-consistent field is applied to a discussion of the energy and wave-functions of the ground state of the hydrogen molecule; an analytical expansion of the wave-function is given in terms of spheroidal coordinates, and the distribution of charge is determined. Two simpler, though less accurate, wave-functions for the molecule are also included, and the possibility of using the method for more complex molecules is discussed.
The effect of departures from ideal conditions other than non-normality on the t and z tests of significance
Authors: Daniels H. E.
Published: November 2008

It is shown that slight departures from independence and equal weight of the sample observations introduce a bias into Fisher's z, but that the distributions of the variances “within and between groups”, and of Fisher's z would appear to maintain their form to the first order, although the extent to which the approximations hold in practice has not been considered.
The total energy of the γ-radiation emitted from the active deposit of actinium
Authors: Kara-Michailova E.
Published: November 2008

The disintegrations by which Ac B passes into the inactive AcPb are accompanied by a γ-radiation very weak compared with the intense γ-emission in the case of Ra or Th-active deposit. The analysis of the secondary β-ray spectrum of actinium-active deposit has revealed the existence of at least five γ-rays (1) (see Table I), of which the ray with energy 0·349 × 106 e.V. definitely belongs to the disintegration Ac C—C″ and is associated with the fine-structure of α-particles of Ac C. According to the measurements of Surugue the two rays of 0·4038 × 106 and 0·4257 × 106 e.V. energy are to be attributed to the disintegration AcB—C, whereas the origin of the 0·829 × 106 e.V. ray is less definite. The fit with experiments is best if this ray is assumed to be emitted from Ac B—C; but it may also (within experimental error) be attributed to the disintegration Ac C″—Pb. Experiments on the absorption coefficient of the γ-radiation of RaAc and its disintegration products prove that the 0·829 × 106 e.V. ray is the hardest γ-ray emitted by the active deposit of actinium(2). The value for the absorption coefficient between 4·6 and 10·6 cm. of lead was found to be μ/ρ = 0·76, in good agreement with the value found in previous experiments at smaller absorptions in aluminium (3). There is, on the whole, excellent agreement concerning the energies of the γ-components as measured according to different methods by different observers.
Time effects in the magnetic cooling method
Authors: Temperley H. N. V.
Published: November 2008

In consequence of the magnetic and exchange interactions between the ions of a paramagnetic salt, a new mechanism of interaction between ion and lattice is possible, involving the simultaneous transition of two or more ions with the emission or absorption of only one elastic quantum. This new mechanism seems to be capable of removing the discrepancy between theory and experiment for iron alum, and considerably reduces the discrepancy for caesium titanium alum.
The heating of iron ammonium alum by alternating magnetic fields at very low temperatures
Authors: Shire E. S. , Barkla H. M.
Published: November 2008

The irreversible heating of iron ammonium alum by an alternating field is shown to be proportional to the frequency of the field. The heating is therefore similar to that arising from hysteresis in ferromagnetics, but no remanence could be found by a ballistic method at the lowest temperature obtained, about 0·075° K. The rate of heating was much increased by the superposition of a steady magnetic field of a few hundred gauss.
A note on functions regular and bounded in the unit circle and small at a set of points near the circumference of the circle
Authors: Maitland B. J.
Published: November 2008

The following theorem was proved by Milloux:

Theorem A. Suppose that f (z) is regular and that | f(z) | < 1 in the unit circle. Suppose also that the set of points in the circle | z | ≤ r′ < 1 at which

cannot be enclosed in circles, the sum of whose radii is equal to 2eh−1. Then

where K is a numerical constant.
On Lorentz invariance in the quantum theory
Authors: Dirac P. A. M. , Peierls R. , Pryce M. H. L.
Published: November 2008

In a recent paper, Eddington raises an objection against the customary use of the Lorentz transformation in quantum mechanics, as for instance when applied to the theory of the hydrogen atom or the behaviour of a degenerate gas. This objection seems to us to be mainly based on a misunderstanding, and our purpose here is to show that the practice of theoretical physicists on this point is quite consistent. The issue is a little confused because Eddington's system of mechanics is in many important respects completely different from quantum mechanics, and although Eddington's objection is to an alleged illogical practice in quantum mechanics he occasionally makes use of concepts which have no place there. Such arguments will not have any bearing on the question whether or not the practice in quantum mechanics is logically consistent—although they may have bearing on which of the two systems describes physical phenomena better.
Relations between concentrated sets and sets possessing property C
Authors: Besicovitch A. S.
Published: November 2008

1. Definition 1. A linear set E is said to possess the property C if, to any sequence of positive numbers {ln} (n = 1, 2, …), there corresponds a set of intervals, of length not greater than l1, l2, …, which includes all the points of E.
On the stability of crystal lattices IX. Covariant theory of lattice deformations and the stability of some hexagonal lattices
Authors: Born M.
Published: November 2008

The theory of lattice deformations is presented in a new form, using the tensor calculus. The case of central forces is worked out in detail, and the results are applied to some simple hexagonal lattices. It is shown that the Bravais hexagonal lattice is unstable but the close-packed hexagonal lattice stable. The elastic constants of this lattice are calculated.
On multiple curves. II
Authors: Hodge W. V. D.
Published: November 2008

In this note I consider the Abelian integrals of the first kind on an algebraic curve Γ which is a normal multiple of a curve C, as defined in Note I*.
The biaxial surfaces, and the equivalence of binary forms
Authors: Segre B.
Published: November 2008

The problem of the equivalence of binary forms is of great importance, both historically and intrinsically, and is also significant for the problems of the so-called canonical and automorphic forms. It consists in deciding whether two given forms are equivalent, i.e. transformable one into the other linearly, and, when they are, in finding all the linear substitutions transforming one form into the other.
Additive functions of intervals and Hausdorff measure
Authors: Moran P. A. P.
Published: November 2008

Consider bounded sets of points in a Euclidean space Rq of q dimensions. Let h(t) be a continuous increasing function, positive for t>0, and such that h(0) = 0. Then the Hausdroff measure h–mE of a set E in Rq, relative to the function h(t), is defined as follows. Let ε be a small positive number and suppose E is covered by a finite or enumerably infinite sequence of convex sets {Ui} (open or closed) of diameters di less than or equal to ε. Write h–mεE = greatest lower bound for any such sequence {Ui}. Then h–mεE is non-decreasing as ε tends to zero. We define
Radiating electron in a magnetic field
Authors: Eliezer C. Jayaratnam
Published: November 2008

The paper considers the application of Dirac's classical theory of radiating electrons to examine the motion of an electron in a uniform magnetic field. The non-relativistic equations of motion are solved exactly. It is shown that for the particular case when the motion is confined to a plane, the physical motion is such that the electron describes an equiangular spiral with velocity which steadily decreases as

exp {negative constant × t};

and in the non-physical motion the electron spirals outwards with velocity which steadily increases as

exp {positive constant × t},

and ultimately the electron escapes to infinity. The relativistic equations of motion are solved approximately as a series in ascending powers of the field. It is shown that in the physical motion the velocity begins to decrease, and hence after a time the motion is given correctly by the non-relativistic solution.

I take this opportunity to express my deepest gratitude to Prof. Dirac, who suggested this problem, for his patient guidance and supervision.
On the closure of
Authors: Fuchs W. H. J.
Published: November 2008

1. A set of functions {øν (t)} (ν = 1, 2, …) is said to be closed L2 in (a, b), if

implies f (t) = 0 p.p. in (a, b). It is well known that a set of functions is closed L2, if and only if every function of L2 (a, b) can be approximated in the mean square sense as closely as desired by finite linear combinations of the øν (t).
Collective electron assemblies in a metal with overlapping energy bands
Authors: Band William
Published: November 2008

The success of Stoner's extensions of the theory of collective electron ferromagnetism(1) opens up new possibilities of considerable importance. The background of the theory can be found in standard works on the theory of metals(2) and statistical mechanics(3) and will not be given here.
The combustion of wood. Part I
Authors: Bamford C. H. , Crank J. , Malan D. H.
Published: November 2008

The combustion of wood presents an interesting problem in non-steady heat flow. When wood is heated, the temperature distribution at a given time may be calculated by means of the well-known conduction equation together with the relevant boundary conditions, provided that the temperature is nowhere sufficiently high to cause appreciable thermal decomposition. When this condition does not apply, the calculation becomes much more complicated, since, as has been recognized for a considerable time, the decomposition is exothermic. The general problem, therefore, is to calculate temperatures and rates of decomposition inside a mass of material, the thermal breakdown of which is accompanied by a heat change, given an initial set of conditions, and a known rate of supply of heat to the surface. The theoretical part of this paper aims at solving this problem for the case of sheets of wood heated in a comparatively simple manner. The treatment is, however, general and may be applied to other materials which undergo thermal changes without melting, if the relevant thermal and other constants are inserted.
A note on the representation of functions by absolutely convergent fourier integrals
Authors: Pitt H. R.
Published: November 2008

1. We write L for the class of integrable functions in (− ∞, ∞), V for the class of functions of bounded variation, and define A, A to be the classes of functions F(x) which may be expressed in the forms

respectively.
The theory of neutral particles
Authors: Couteur K. J. Le
Published: November 2008

Several authors (1, 2, 3) have discussed the formulation of quantum electrodynamics in the configuration space of the light quanta; they conclude that no meaning can be given to the position of a light quantum, but it is not made clear whether this situation is due to the vanishing of the charge or of the rest mass. However, since the previous discussion (2, 3), various neutral particles of non-vanishing rest mass have been postulated, and it is of interest to consider what meaning can be given to the position of such particles. We shall at first consider integral spin and Bose statistics; the case of spin ½ and Fermi statistics is simpler and is briefly discussed in the last section.
Convection of a fluid in a porous medium
Authors: Lapwood E. R.
Published: November 2008

It is shown that under certain conditions convective flow may occur in fluid which permeates a porous stratum and is subject to a vertical temperature gradient, on the assumption that the flow obeys Darcy's law. The criterion for marginal stability is obtained for three sets of boundary conditions, and the motion described. If such convection occurs in a stratum through which a bore-hole passes, the usual method of calculation of the heat flow must be modified, but in general the correction will not be large.
The ‘thick-film chamber’ method for the measurement of fast neutron flux
Authors: Allen K. W. , Wilkinson D. H.
Published: November 2008

Several methods have been developed in recent years in the Cavendish Laboratory for the absolute measurement of fast neutron flux. All depend on observing the effects of elastic collisions between neutrons and light nuclei. The methods fall into two categories according as individual recoil nuclei are counted (1, 2), or the total ionization current due to the recoils is measured (3). In order completely to interpret the experimental results, it is necessary to know the cross-section for scattering and the angular distribution of the recoil nuclei. The total number of recoil nuclei and their energy distribution are then determined for a known incident neutron spectrum. For precise neutron flux measurements, recoil protons are invariably studied, as the neutron-proton scattering cross-section has been measured over a wide range of energies (4, 5), and the angular distribution of the recoils is isotropic in the centre of gravity space for neutrons of energy less than about 10 MeV. (6, 7). This makes the reduction of the experimental results particularly simple and certain.
On the quantum-mechanical treatment of the optics of crystal lattices
Authors: Huang Kun
Published: November 2008

In Ewald's theory of crystal optics, based on classical principles, an optical wave through a crystal lattice of polarizable atoms (idealized as isotropic oscillators) is a self-sustaining system of vibrations so constituted that, on the one hand, the electric moment of each atom is caused to oscillate by the electromagnetic field and, on the other hand, the electromagnetic field is itself the resultant field due to the superposition of the dipole waves produced by the lattice atoms. Born extended the theory to the case of movable lattice ions and showed that in an optical wave the electromagnetic field is so coupled to the lattice vibrations that each lattice ion vibrates in phase with the local field which, as in Ewald's case, is itself produced by the vibrating ions. In Born's original theory, the motion of the lattice particles had to be treated by classical mechanics. It is shown that the results of the quantum-mechanical treatment of the lattice motion agrees exactly with the classical theory. Not only is the induced current at each lattice point in phase with the local field, but the magnitude of the current is also identical with the classical value, completely independent of whichever vibrational state the crystal might be in.
Random associations on a lattice
Authors: Moran P. A. P.
Published: November 2008

P. 323, line 3 from bottom:

for [8mn−7m−7n+n] read [8mn−7m−7n+4].

P. 328, lines 16 and 18:

for (6mn−6m−6n+l) read (6mn−6m−6n+4).

line 17,

in formula ending … +13m+13n−9) read …+13m+13n−8).

Vol. 43 (1947), pp. 321–8.
An inequality in the theory of arithmetic on algebraic varieties
Authors: Northcott D. G.
Published: November 2008

The inequality which is proved here was established under different conditions in a paper entitled ‘Periodic points on an algebraic variety', which is to appear in the Annals of Mathematics. It is hoped that the new formulation will be more useful for applications. The present investigations use the rather sophisticated techniques of the ‘Theory of distributions’ which was developed by Weil (1). As the content of this theory may be unfamiliar to the reader, Weil's results are described at some length, though of course, his proofs are not given. The one result which we need and which is not given explicitly by Weil, is proved in detail.
A further inequality in the theory of arithmetic on algebraic varieties
Authors: Northcott D. G.
Published: November 2008

This paper is the sequel to one entitled ‘An Inequality in the Theory of Arithmetic on Algebraic Varieties’ (1). For the definition of the complexity of an algebraic point P—here denoted by C[P]—we refer the reader to that paper, where he will also find a short account of as much of the theory of ‘distributions’ as will be used here. For full details of the latter, the reader should consult A. Weil's article (2). Both this paper and the preceding one are the result of an attempt to generalize and simplify a few of the many fruitful ideas which are embodied in Weil's thesis.
The tabulation of Bessel functions for large argument
Authors: Hartree D. R.
Published: November 2008

By the use of an auxiliary independent variable, as well as of auxiliary functions, it may be possible to obtain tables which are very compact and in which, nevertheless, linear interpolation is adequate for most practical purposes. In the case of the Bessel functions of order zero, it is shown that by the use of 1/x2 as auxiliary independent variable, and appropriate auxiliary functions, linear interpolation in two tables of forty-one entries each is sufficient to give x½12J0(x) and x½12Y0(x) with an uncertainty of a unit in the 7th decimal, from x = 5 to ∞, and two tables of only three values each are sufficient to cover the range x = 22·5 to ∞ in the same way. Two possible forms of auxiliary functions for this purpose are considered.
A sextic primal in five dimensions
Authors: Hartley E. M.
Published: November 2008

In a recent tract (Baker (1)) there is described in considerable detail a configuration of forty-five points which are nodes of a quartic primal in four dimensions. The geometry of this primal is very fascinating; among its interesting properties is the fact that a number of well-known geometrical configurations, which usually arise as unrelated phenomena, here all appear in connexion with the one figure. The interest of the primal, of course, lies chiefly in the large number of collineations which leave it invariant. The group G* of these collineations is considered in a paper by Burkhardt (2), in which are given explicit expressions for five algebraically independent functions of the five variables, which are left invariant by the operations of the group. The simplest of these invariants is of the fourth order, and when equated to zero represents the quartic primal which is the subject of Baker's tract.
On quadrature formulae
Authors: Reiz Anders
Published: November 2008

1. It is frequently required to find the numerical value of the definite integralIt is, however, often found that even if the analytical expression off(x) is given, it cannot be integrated in terms of known elementary functions. The elliptic integrals are perhaps the best known examples of functions of this type; and more common are cases wheref(x) is not even defined by an analytical expression but merely specified by a table of numerical values. In the two cases mentioned one has to resort to a numerical integration procedure in which the integral is evaluated in terms of the values off(x) at a finite number of the argumentsx1,x2, …,xn. There are essentially two different types of quadrature formulae available, those which depend on the use of finite differences and the tabulation off(x) at equidistant points, and those which depend on the values off(x) at suitably selected pointsxi. These latter, so called mean-value methods, will be discussed in the present paper. For a recent treatment of such methods see, for example, Beard (5).
The Parseval formulae for monotonic functions. III
Authors: Edmonds Sheila M.
Published: November 2008

This sequence of papers is concerned with a group of theorems on Fourier transforms, and with the corresponding results for Fourier series. The transform theorems deal with the equations

when two of the functions concerned are monotonic; and the series theorems, with the equations

when two of the functions or sequences of coefficients are monotonic. A general survey of the problem was given in two earlier papers, which we shall call M.F. (I) and M.F. (II), and the results obtained were tabulated in M.F. (II). There is thus no need to restate the position in detail here. We shall merely recall as briefly as possible what has been proved already, so as to show the scope of the present paper.
On the theory of grand partition functions
Authors: Münster Arnold
Published: November 2008

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

Das Schwarzsche Lemma der klassischen Funktionentheorie einer Veränderlichen kann als vollständige Beschreibung der inneren Abbildungen des Einheitskreises (mit Fixpunkt O) aufgefaßt werden. Ist nämlich f(z) eine in |z| ≤ 1 reguläre Funktion mit |f(z)| ≤ 1 und f(0) = 0, so muß |f′(0)| ≤ 1 sein, und es stellt diese Ungleichung bekanntlich den ersten Schritt zur Lösung des Koeffizientenproblems der im Einheitskreis gleichmäßig beschränkten Funktionen dar. Die Aussage

kann andererseits so gedeutet werden, daß jede innere Abbildung des Einheitskreises mit Fixpunkt O auch innere Abbildung jedes konzentrischen Kreises |z| ≤ |ρ| < 1 ist.
Notes on gas flow through a nozzle
Authors: Binnie A. M.
Published: November 2008

In Fart 1 the oscillations are examined which may develop in supersonic flow through the divergent part of a nozzle. Hooker's theory is found to be confirmed by observations on a steam nozzle.

In Part 2 it is shown that Taylor's approximate theory of flow between circular arcs can also analyse the throat conditions when the velocity of approach to the throat is supersonic. Limiting symmetrical flow occurs when the velocity at the centre of the throat just exceeds the local velocity of sound, no matter what the radii of the arcs and the ratio of the specific heats of the gas may be. The unique asymmetrical case is merely the reverse of Taylor's solution for the flow when the velocity of approach is subsonic.

A numerical comparison has been made of the velocities across the throat as calculated by Taylor's method and by Fox and Southwell's iterative process when the velocity of approach is subsonic. The agreement is good for limiting symmetrical flow but not so satisfactory for the unique asymmetrical case.
A new method for the evaluation of zeros of Bessel functions and of other solutions of second-order differential equations
Authors: Olver F. W. J.
Published: November 2008

The zeros of solutions of the general second-order homogeneous linear differential equation are shown to satisfy a certain non-linear differential equation. The method here proposed for their determination is the numerical integration of this differential equation. It has the advantage of being independent of tabulated values of the actual functions whose zeros are being sought. As an example of the application of the method the Bessel functions Jn(x), Yn(x) are considered. Numerical techniques for integrating the differential equation for the zeros of these Bessel functions are described in detail.
The quadratic form positive definite on a linear manifold
Authors: Afriat S. N.
Published: November 2008

1. Introduction. Necessary and sufficient conditions are established for a real quadratic form to be positive definite on a linear manifold, in a real vector space, explicit in terms of the dual Grassmann coordinates for the manifold.
On the reciprocal of a segment of a generalized hilbert matrix
Authors: Collar A. R.
Published: November 2008

1. Introductory. The present paper is mainly concerned with some forms for the reciprocal of the square matrix u of which the typical element is

and k is not an integer within the range ± (n − 1) inclusive but is otherwise arbitrary; thus
On the use of the characteristic functional in the analysis of some stochastic processes occurring in physics and biology
Authors: Bartlett M. S. , Kendall David G.
Published: November 2008

1.Introduction.In a recent contribution to theseProceedingsAlladi Rama-krishnan (23) has discussed a number of problems which arise when the development of a cascade shower of cosmic rays is considered from the standpoint of the theory of stochastic processes. As has often been remarked (see, for example, the important study by Niels Arley (1)), there is a close formal analogy between such physical phenomena and the growth of biological populations. In particular, if the distance of penetration (t) is identified with the time, and the energy (E) of a particle in the shower is replaced by the age (x) of an individual in the population, there emerges an obvious analogy between stochastic fluctuations in the energy spectrum of the shower and similar fluctuations in the age distribution of the population.
On a class of determinantal primals and their multiple loci
Authors: Goddard L. S.
Published: November 2008

In this paper I investigate some geometrical properties of a system of primals which arose a few years ago in the study of a purely algebraic problem: to parametrize completely the group of automorphic transformations of a given bilinear form. This problem is classical, and there exists a large literature on the subject, but the algebraists never succeeded in finding a complete parametrization. Indeed, the trend was to move away from those transformations not covered by the known parametrization; and Weyl, for example, writing about the orthogonal group in his book on the Classical Groups remarks ‘unfortunately Cayley's parametric representation leaves out some of the orthogonal matrices, and a good deal of our efforts will be spent in rendering these exceptions ineffective’. In another paper I shall show how to solve this problem of complete parametrization, via a geometrical approach; but here I confine my attention to some preliminary geometrical results.
Quadratic forms positive definite on a linear manifold
Authors: Goddard L. S.
Published: November 2008

1. In a recent paper(1), Afriat has given necessary and sufficient conditions for a real quadratic form to be positive definite on a linear manifold, in terms of the dual Grassmannian coordinates of the manifold. Considerable matrix manipulations were used in Afriat's method, but most of these may be avoided by the method of the present paper, which depends on some well-known properties of the Grassmannian coordinates. We first show that the conditions may be expressed as a set of inequalities which are quadratic in the Grassmannian coordinates of the manifold. Then, by a standard theorem, these may be transformed into Afriat's conditions on the dual coordinates.
Some generalized order-disorder transformations
Authors: Potts R. B.
Published: November 2008

In considering the statistics of the ‘no-field’ square Ising lattice in which each unit is capable of two configurations and only nearest neighbours interact, Kramers and Wannier (3) were able to deduce an inversion transformation under which the partition function of the lattice is invariant when the temperature is transformed from a low to a high (‘inverted’) value. The important property of this inversion transformation is that its fixed point gives the transition point of the lattice.