bluuebunny/crossref_metadata_2025 huggingface dataset:
The Absorption of β-Rays
Authors: Chalmers J. A.
Published: November 2008

Many measurements have been made of the complete absorption curves for β-rays in various substances, and it has been found that, for homogeneous β-rays, the curve is not exponential, and, in particular, when aluminium is the absorber, the curve is linear over a considerable portion of its range (e.g. Schonland, etc.).
The Second Solution of Mathieu's Differential Equation
Authors: Goldstein S.
Published: November 2008

The Mathieu functions of period π and 2π have recently been constructed by the help of analysis similar to that developed by Laplace, Kelvin, Darwin and Hough to find the free tides symmetrical about the axis of a rotating globe. The purpose of this note is to show that a similar construction can be carried out for the second solution of the Mathieu equation, when one solution is periodic in π or 2π, by the help of analysis similar to that used for forced tides. The construction is effected in a form suitable for numerical computation.
Some Applications and Implications of Duane's Quantum Theory of Diffraction
Authors: Williams E. J.
Published: November 2008

Duane's quantum theory of diffraction is applied to the reflexion of electrons by crystals and to the spatial distribution of photoelectrons and fluorescent radiation from a crystal.

Two alternative criteria for coherence are given. According to the second of these there is coherence provided that the momentum imparted to the components of a system during the process concerned is insufficient, owing to quantum restrictions to their motion, to change their energy. Calculations made on this supposition show that in the case of the scattering of radiation by a crystal there is complete coherence for the lower orders, while for higher orders this ceases to be the case and the reflected intensity is reduced as the result of incoherent scattering. The specular reflexion by gases is also considered.

Duane's theory of diffraction has not been placed on a rigorous basis and the solutions which have been proposed here for various problems may be incorrect. Even so the theory has at least served to bring to light several points of theoretical and experimental interest which deserve consideration.
On the strong summability of Fourier series
Authors: Paley R. E. A. C.
Published: November 2008

Let f(t) be a function periodic in 2π, and absolutely integrable in the interval (0, 2π). We write
Wave forms, and a special problem
Authors: Hargreaves R.
Published: November 2008

I think that some light may be thrown on a recent paper of Professor Whittaker's by giving concrete examples. The first example considered belongs to a transformation, which I gave some years ago, whereby a constant acceleration is incorporated in Einstein's form. Opportunity is taken to develop some features of the kinetic form of solution, in which all space coordinates are relative to a source, and the time of the latter is the only time appearing. This is carried as far as the treatment of energy.
Note on the sum of an oscillating series, II
Authors: Verblunsky S.
Published: November 2008

This note is a sequel to a former one, a knowledge of which will be assumed. We first prove a theorem which is analogous to O Theorem I, and is indeed a simple consequence of it. As an easy inference from this theorem we obtain a necessary condition that the series allied with a Fourier series may be summable (C, k) for all k >p, where p is any number greater than 1. We then show that the condition thus obtained is sufficient. We finally show that the same method can be applied to Fourier series. This note thus contains a solution of the Cesàro summability problem for Fourier series and for the allied series, which is the most precise one yet obtained.
Remarks on the scattering of electrons by atomic fields
Authors: Bullard E. C. , Massey H. S. W.
Published: November 2008

It is shown that, by use of the Thomas-Fermi atom, effective scattering cross-sections may be obtained from a single curve for all atoms (provided that the atomic number is not too small). This curve is calculated. The range of validity of such values has been obtained and Born's formula shown to be inaccurate for representing the scattering of electrons with velocities less than 400 volts except by atoms of very low atomic number. For the case of high velocities an approximate expression is obtained for the angular distribution allowing for screening by the extra-nuclear electrons.
Diagonal matrices
Authors: Turnbull H. W.
Published: November 2008

In the following pages I have developed the theory of matrices by resolving them into parallel components arranged diagonally, rather than into the usual rows and columns. This treatment is natural in view of the fundamental fact that the resolution is undestroyed when matrices are formed into products (Theorem 2). It is closely related to the theory of continuants and of continued fractions. Certain features stand out in such a presentation—the distinction between the length and range of a diagonal (§ 4), that between regular and irregular diagonals (§ 6), and the use of equable partition (§ 7). The exact conditions for the existence of an rth root of a given singular matrix are examined in § 9 and summarized under the title, the condition of equability.
Investigations on condensation phenomena in mercury vapour
Authors: Ho P. C.
Published: November 2008

Owing to its physical and chemical properties being greatly different from those of any of the liquids which have hitherto been used in the Wilson cloud chamber, mercury has been used in the experiments described in this paper and the condensation phenomena of its vapour at different temperatures observed. Before constructing the apparatus it was considered necessary to get from theoretical considerations some idea about the magnitude of the critical supersaturation for mercury vapour in equilibrium with a drop carrying unit charge. Assuming that J. J. Thomson's formula.

where s is the supersaturation of mercury vapour in equilibrium with a drop of mercury of radius a, charge e, density σ and surface tension T, the value of which is assumed here to be independent of the radius of the drop, K the specific inductive capacity of the dielectric surrounding the drop, and R the gas constant for one gramme of weight, all at temperature θ, can be applied to the present problem, this critical supersaturation sm is given by the formula
On certain Cremona transformations in circle-space connected with the Miquel-Clifford configuration
Authors: Rao A. Narasinga
Published: November 2008

The classical configuration connected with the names of Miquel and Clifford associates with a set of n straight lines in a plane a point called their Miquel point when n is even, and a circle, their Clifford circle, when n is odd. When the number of lines n in a set is even, the omission of these, one at a time, gives n subsets whose associated Clifford circles are concurrent at the Miquel point of the original set; when n is odd, the omission of the lines one at a time gives n subsets whose associated Miquel points are concyclic on the Clifford circle of the original set. To start the chain, it is only necessary to define the Miquel point of two lines as their common point.
The postulation of a multiple variety
Authors: Todd J. A.
Published: November 2008

1. The determination of an explicit formula for the postulation of a multiple variety of given characters for primals of sufficiently large order is one which seems to have received attention only in special cases. The postulation of a multiple curve for surfaces in ordinary space is known*, and Roth† has obtained the postulation of a multiple surface in [4], free from singularities, for primals in that space. Apart from these results, and the trivial case of isolated multiple points, the only general results seem to be those of Torelli‡ and Giambelli§ for the case in which the multiple variety is a Vh which is the complete intersection of r − h primals in [r].
The theory of the structure and formation of a certain type of immobile adsorbed film with an application to the adsorption of oxygen on tungsten
Authors: Roberts J. K.
Published: November 2008

The theory is given of the structure and behaviour of immobile adsorbed films on a simple quadratic lattice in which each particle is so large that it occupies more than one site and makes it impossible for adsorption to occur on sites which are neighbours to that on which it is adsorbed. The results are obtained partly by statistical methods and partly by the use of a model of the film.

The final or complete film of this type is not homogeneous and has gaps in it. The sites on the surface can be divided into two groups, the gaps occurring where regions of the surface occupied by particles adsorbed on one group and those occupied by particles on the other group meet. The number of these gaps or “boundary sites” is determined, and the possible configurations of particles in the boundary regions are discussed in detail.

The results are considered in connexion with some experiments on the adsorption of oxygen on tungsten. In this case a second layer is formed over the boundary sites and the amount that can be so adsorbed or the number of “second layer sites” to which the boundary sites give rise is obtained. It is shown that numerically the experimental results for oxygen are in better agreement with the view that the first or very stable layer is a molecular layer of the above type than with the view that it is atomic.
Introduction to the theory of operators
Authors: Steen S. W. P.
Published: November 2008

This paper is a continuation of four others under the same title†. The paragraphs are numbered following on to those of the fourth paper of the series. In § XXIV we show that, if λ(A) is a linear functional, then there exists a resolution Eμ such that λ(A) = ∫μdr(AEμ), and if B = ∫μdEμ is bounded, then λ(A) = τ(AB)‡ for all A, where τ is the trace. This implies that τ(A) is a linear functional, and that the conjugate space ℒ, i.e. the space of the linear functionals, has a subset ℒ′ which is in (1, 1) correspondence with the original set of operators, and that in this correspondence the linear functional τ(A) is associated with the unit operator.
On the capture of interstellar matter by stars
Authors: Atkinson R. d'E.
Published: November 2008

The derivation given by Hoyle and Lyttleton for an accretion formula proposed by them is examined. A number of arguments against its validity are put forward, especially that on the one hand their capture radius depends on the theorem that if the velocity of certain masses of gas after collision is less than the velocity of escape at the point, they will not in fact escape, while on the other hand it is clear (and is now admitted) that the gas cannot in fact move with this velocity at all. It is also shown that since, ex hypothesi, the individual molecules will all, on the average, retain their hyperbolic velocities, there is not the compelling reason for their capture that there appeared to be in Hoyle and Lyttleton's argument, where only the mean radial velocity of the centre of gravity of the mass was considered. Further, it seems improbable that the temperature of the interstellar matter can be low enough for the initial assumptions of their theory to hold.
The variational equation of relativistic dynamics
Authors: Mathisson Myron
Published: November 2008

If Tαβ is the energy-tensor, then, for any vector field ξα, the equation

follows from the energy equation

Suppose first that the physical system considered is complete, i.e. that the energytensor vanishes beyond some world tube Z. Let L be a time-like world line running inside the tube. We integrate both sides of (a) over a portion of Z, and we transform an integral over a four-dimensional region into a linear integral over L. We obtain the variational equation

in which the m's are tensors characteristic of the physical system. An essential feature of the m's is that they are symmetrical in their two last superscripts. Equation (c) has to be satisfied by every field ξα, provided that the ξ's vanish, with all their derivatives, at the ends of the integration path.
The Measure of Plane Sets
Authors: Moran P. A. P.
Published: November 2008

We consider bounded sets in a plane. If X is such a set, we denote by Pθ(X) the projection of X on the line y = x tan θ, where x and y belong to some fixed coordinate system. By f(θ, X) we denote the measure of Pθ(X), taking this, in general, as an outer Lebesgue measure. The least upper bound of f (θ, X) for all θ we denote by M. We write sm X for the outer two-dimensional Lebesgue measure of X. Then G. Szekeres(1) has proved that if X consists of a finite number of continua,

Béla v. Sz. Nagy(2) has obtained a stronger inequality, and it is the purpose of this paper to show that these results hold for more general classes of sets.
Some manifolds generated by normal rational curves
Authors: Jones R. R.
Published: November 2008

In a paper ‘The projective generation of curves and surfaces in space of four dimensions’, F. P. White considers the representation of prime sections of the general Bordiga surface F6 in [4] by quartic curves in the plane through ten fundamental points. We shall make use of several results from the same section of that paper, and, for convenience, list them as follows. From a point P′ of F6 can be drawn an ∞1 of trisecant lines lying on a cubic conical sheet. The curve of intersection of these trisecants and F6 is a curve C8, of order eight and with a double point at P′, which meets each of the ten lines on F6 in two points. In the plane, the representation of this curve is a curve C7, of order seven and with eleven double points, namely, the ten fundamental points and the point P which corresponds to P′. C7 is hyperelliptic, containing an involution of points Q, R corresponding to the points Q′, R′ in which a trisecant from P′ meets F6. The envelope of joints of corresponding points Q, R in the plane is a conic; this conic will continually be referred to as Σ.
The focal form of the quadric in n dimensions
Authors: Green H. Gwynedd
Published: November 2008

The object of this note is to show that the equation of the quadric in n dimensions can be transformed to a ‘focus-directrix’ form which is exactly similar to those for two and three dimensions: this form is extended throughout by the introduction of a constant k. A short discussion of confocal quadrics is appended.
The effect of a hole on certain stress distributions in aeolotropic and isotropic plates
Authors: Holgate S.
Published: November 2008

1. The problems of stress distributions in an infinite plate containing a circular hole were solved in a general manner by Bickley(1) for isotropic materials. An alternative method, using complex stress functions, was later given by Green(5) and extended so as to apply to aeolotropic materials. In the present paper Green's method is employed to determine the stress distributions that arise when the stresses are produced by isolated forces acting at points on, or near to, the edge of the circular hole, and though some of the solutions are cumbersome they are all obtained in finite form.
A Note on Some Integrals Involving Bessel Functions
Authors: Fox C.
Published: November 2008

The object of this note is to prove the following results, all of which hold when |a| < 1.

(2) If r is any positive integer other than zero, the
Note on the Number of High Velocity β-rays
Authors: Cave H. M.
Published: November 2008

Recently Yovanovitch and d'Espine using a magnetic spectrum method of low resolving power have found a band of very fast β-rays in the spectrum of radium B + C, having energies up to 7·6 × 106 volts. As early as 1911 Danysz found some evidence of the existence of such high velocity β-rays but they have not been observed by other investigators. A moderate number of such rays, if spread over a considerable range of velocities, might escape detection in most experimental arrangements for the study of β-rays, on account of their low absorption and low ionizing power. As it is important to know how many of these fast β-rays there are on the average per disintegration, the following simple experiment was performed by the writer in an attempt to determine this.
On the parametric representation of bounded functions
Authors: Verblunsky S.
Published: November 2008

Let a1, a2, … be a sequence of variables. Write

We use the notation Bn = Bn (a1, …, an) to indicate the variables on which Bn depends.
On a problem of Chowla and some related problems
Authors: Erdös Paul
Published: November 2008

Let d(m) denote the number of divisors of the integer m. Chowla has conjectured that the integers for which d(m + 1) > d(m) have density ½. In this paper I prove and generalize this conjecture. I prove in § 1 a corresponding result for a general class of functions f(m), and in § 2 the result for d(m) which is not included among the f(m). I employ the method used in my paper: “On the density of some sequences of numbers.”
The theory of the mean square variation of a function formed by adding known functions with random phases, and applications to the theories of the shot effect and of light
Authors: Rowland E. N.
Published: November 2008

In the study of random events and associated fluctuations such as occur in the shot effect, a theorem first stated and discussed by Dr N. R. Campbell can often be employed. It applies on any occasion when there occur at random a number of events whose effects are additive. Let us suppose that a single event occurring at time tr causes at time t an effect f(t − tr) in some part of the observed system, and that the effects of different events are additive, so that the total effect or output is ϑ(t), given by

We may suppose that the same events cause another set of effects g(t − tr) with output ϑ(t), where

Both the functions are assumed to be bounded and integrable in the Riemann sense, as are all the functions studied in physics.
An infinite integral involving Bessel functions and parabolic cylinder functions
Authors: Varma R. S.
Published: November 2008

The object of this paper is to evaluate an infinite integral involving Bessel functions and parabolic cylinder functions. The following two lemmas are required:

Lemma 1.

provided that R(m) > 0.
Regular canonical surfaces of genus three and four
Authors: Maxwell Edwin A.
Published: November 2008

The object of this paper is to make more precise the results given by Enriques-Campedelli for canonical surfaces, that is, surfaces whose prime sections are the canonical curves. We consider the case of regular surfaces in space of three dimensions (pg = pn = 4), and of regular multiple planes (pg = pn = 3).
Electronic energy bands in metallic copper and silver
Authors: Tibbs S. R.
Published: November 2008

The present discussion of the conduction levels of copper and silver was undertaken in view of the experimental investigations of the absorption and emission bands of metals, in the soft X-ray region, which have been carried out in this laboratory and elsewhere*. It is a well-known result of the electron theory of metals that, for cubic crystals, of lattice constant a, if k is the wave vector of an electron moving in the crystal lattice, there are discontinuities in the energy E of the electron for values of k such that

(n1, n2, n3) being the Miller indices of the planes for which Bragg reflexion of the electron waves takes place. If we draw the energy E as a function of k for a particular direction in the lattice, then we obtain, using the usual notation, curves such as those shown in fig. 1, d being the half-distance between the corresponding planes for which Bragg reflexion takes place.
Further experiments on the slow neutrons transmitted by cadmium: influence of the temperature of the slowing down medium
Authors: Westcott C. H. , Sosnowski L.
Published: November 2008

The paper is concerned with the influence on the effects produced by slow neutrons, of the temperature of the medium in which they are slowed down, attention being focussed on those neutrons which pass freely through cadmium. Results obtained by Arsenjewa-Heil, Heil and Westcott (10) had appeared to show that the groups A and B responsible for activating silver had energies low enough to be affected by temperature. In the present work, a confirmation of the change for group B has been obtained, and a similar effect has been observed with a boron detector. It is suggested that the explanation of the results is that cadmium transmits neutrons of energies down to about 0·2 e. V., and that “group B” is inhomogeneous. During the course of the work, a study of the conditions under which reproducible results could be obtained was undertaken.
The properties of a perfect Einstein-Bose gas at low temperatures
Authors: Fowler R. H. , Jones H.
Published: November 2008

In a recent letter to Nature F. London(3) has called attention to a discontinuity in the derivative dCV/dT of the specific heat with respect to temperature which arises in a perfect Einstein-Bose gas at low temperatures. When discussing the properties of a perfect gas satisfying the Bose statistics, Einstein(1) remarked that at low temperatures something resembling a condensed phase should appear. The temperature at which this condensation should begin is given by the equation

where n is the number of particles of mass m per unit volume and the other symbols have their usual significance.
Momentum transfer and heat flow in liquid helium II
Authors: Allen J. F. , Reekie J.
Published: November 2008

It has been found by one of the authors (1) in collaboration with Dr H. Jones that a flow of heat in liquid He ii is accompanied by what seems to be a transfer of momentum. The effect can be seen when the channel through which the heat and liquid flow consists of a smooth-walled glass capillary, such as shown in Fig. 1a. Due to the high thermal conductivity of He ii, a considerable part of the heat put into the reservoir is carried down through the capillary to the bath. When a steady heat flow exists, a flow of liquid takes place in the opposite direction, and the level of the liquid in the reservoir is seen to be higher than that in the bath. Smooth capillaries, however, produce a rise in level of only 1 or 2 cm. at most, since the viscosity of the liquid is small and hydrostatic pressure pulls the accumulated liquid in the reservoir back through the capillary. When the heat flow is large, violent surging is observed in the reservoir, but there is no further rise in level.
Zur Summabilitätstheorie der Fourierschen Doppelreihe
Authors: Grünwald Géza
Published: November 2008

Es sei f(ξ) eine L-integrierbare und nach 2π periodische Funktion der reellen Variabeln ξ und es sei {sn} die Folge der Partialsummen ihrer Fourierschen Reihe an einer Stelle ξ = x, wo die Funktion stetig ist. Dann ist nach einem klassischen Satze von Fejér

und nach Hardy und Littlewood

Vor kurzem hat L. Fejér gefunden, dass der Hardy-Littlewoodsche Satz (2) Spezialfall eines “gewöhnlichen” Summabilitätssatzes wie (1) ist, der sich aber auf die Fouriersche Reihe einer Funktion zweier Variabeln bezieht. Dieser Satz ist der folgende.
Quantum electrodynamics. II
Authors: Hoyle F.
Published: November 2008

It is shown in this paper and the preceding one that two separate forms of theory can be developed in which a “finite size” is attributed to a charged particle by means of its interaction with the radiation field. The region attributed in this way to the particle is four dimensional and is determined in such a manner that the usual difficulties with relativistic invariance do not arise.

The advantage of such a theory becomes clear when the theory is applied to those problems in which the usual calculations give infinite results. The problem of the method of successive approximations is considered and satisfactory results are obtained provided that the space dimensions of the finite region are of the order of the classical radius of the electron, when the electron is at rest.

It may be noted explicitly that the difficulty that has been associated with the emission of low energy quanta by “Bremsstrahlung” will not arise in the present formulation of the electromagnetic interaction between field and particles. This case is interesting since an infinity arises here which is not analogous to the self energy infinities, but occurs in the direct calculation of a physical process and not in a virtual transition.

The theory seems satisfactory so far as low energy processes (< 137 mc2) are concerned and the real test of its applicability may be expected to arise in discussing processes of high energy. It is hoped to treat these in a later paper.
Continuants and precontinuants
Authors: Bennett G. T.
Published: November 2008

1. It is here expedient to ignore the continued fractions from which they are usually derived and to define the simple continuants determined by the parameters a1, a2, a3, …, an, … as given by the chain of difference equations

together with the initial values u0 = 0, u1 = 1: and these data lead successively to the functions

and
Introduction to the theory of operators. IV Linear functionals
Authors: Steen S. W. P.
Published: November 2008

This paper is a continuation of three others under the same title. The paragraphs are numbered following on to those of the third paper.
The evolution of the stars
Authors: Hoyle F. , Lyttleton R. A.
Published: November 2008

Difficulties associated with the evolution of stars by radiation alone are briefly discussed. It is clear that some other process is also affecting the stars and it is shown that the stars are capable of adding to their mass by the process of accretion of the cosmical cloud. The gravitation of a moving star causes additional collisions of the atoms of the interstellar matter and the motions become randomized to such an extent that the star probably captures all material passing within the distance at which the velocity of the star relative to the cloud is the parabolic velocity. This rate of accretion of mass of a star is 4πγ2ρM2/ν3, and is accordingly of great importance for stars of low velocity. Stars of high velocity are least affected by accretion and therefore in general remain of low mass, while stars of low velocity must attain great mass. The periods of time involved in bringing about appreciable changes in the mass of a star are of the order of 5 × 1010 years and are in agreement with independent estimates of the time scale, as deduced, for example, from the companion of Sirius. The evolution of the components and orbits of binary stars are consequences of the accretion process. The more massive component increases in mass more rapidly than the less massive component in the case of wide pairs, and may therefore in general continue to emit more ergs per gram. The orbit evolves in such a way that the total angular momentum remains constant. For equal masses the separation is proportional to the inverse cube of the mass, and the period to the inverse fifth power, so that great changes of separation and period occur. The evolution of the stars is governed almost entirely by their velocities relative to the cosmical cloud. In the case of double stars the evolution takes the form of decreasing period and decreasing separation. Such features as galactic concentration and the correlation between spectral type and velocity are direct results of accretion.
A critical examination of the Benedicks homogeneous thermoelectric effect
Authors: Ch'eng Henry L. C. , Band William
Published: November 2008

Although the existence of an electromotive force in apparently homogeneous circuits under asymmetrical temperature distributions has been questioned by only a minority of workers in this field, all attempts at explaining the effect have been little more than speculations. These speculations may be divided roughly into two types, first(2), those which agree with the contention of Benedicks that the E.M.F. is an essentially new effect not explicable in terms of the classical theory of Kelvin, and secondly(3), those which contend that the effect is essentially spurious, to be explained away as due either to hidden sources of error, or to unsuspected heterogeneities in the circuit. The somewhat fantastic claim of Benedicks' school that the effect was an inverse of the Thomson effect, which proved entirely false upon examination in the light of Sommerfeld's work(4), did much to increase scepticism. Benedicks' later claim to have found yet another inverse of his original effect(5), the “electro-thermal” effect, has also been refuted(6), and this does not inspire confidence in the genuine nature of the original effect.
The quantum theory of radiation damping
Authors: Wilson A. H.
Published: November 2008

A general method is given of calculating the effect of damping on the collision cross-sections for problems involving free electrons and mesons. The result is expressed in the form of an integral equation which can only be solved if certain simplifying assumptions are made. It is shown that radiation damping has a negligible effect on the scattering of light by free electrons, so that the Klein-Nishina formula is unchanged by the inclusion of damping effects. The effect of damping for mesons, on the other hand, is extremely large and reduces the cross-sections considerably. The main problems considered are the nuclear scattering of mesons and the energy radiated by mesons during collisions. The revised cross-sections are much more reasonable than those calculated previously, but on account of the inadequacy of the data no detailed comparison with the experimental results is possible.
On certain expansions of the solutions of the general Lamé equation
Authors: Erdélyi A.
Published: November 2008

1. In this paper I shall deal with the solutions of the Lamé equation

when n and h are arbitrary complex or real parameters and k is any number in the complex plane cut along the real axis from 1 to ∞ and from −1 to −∞. Since the coefficients of (1) are periodic functions of am(x, k), we conclude ](5), § 19·4] that there is a solution of (1), y0(x), which has a trigonometric expansion of the form

where θ is a certain constant, the characteristic exponent, which depends on h, k and n. Unless θ is an integer, y0(x) and y0(−x) are two distinct solutions of the Lamé equation.

It is easy to obtain the system of recurrence relations

for the coefficients cr. θ is determined, mod 1, by the condition that this system of recurrence relations should have a solution {cr} for which

k′ being the principal value of (1−k2)½
On the product of three non-homogeneous linear forms
Authors: Davenport H.
Published: November 2008

Let ξ, η, ζ be linear forms inu, v, wwith real coefficients and determinant Δ ≠ 0. A conjecture of Minkowski, which was subsequently proved by Remak, tells us that for any real numbersa, b, cthere exist integral values ofu, v, wfor which

and the constant ⅛ on the right is best possible.
Optical theory of the echelette grating
Authors: Conn G. K. T.
Published: November 2008

The optical properties of reflecting echelette gratings, used in the infra-red region of the spectrum, are discussed theoretically. Expressions are derived for various important characteristics including the wave-length relationship, the dispersion and the resolving power of spectrometers using such gratings. The intensity distribution is discussed and illustrated by reference to a particular spectrometer fitted with a grating of 1200 lines to the inch. The relative importance of other factors such as the variation of available energy with wave-length, the dispersion of the preliminary optical system, etc., is considered and tabulated. Particular consideration is given to the properties of such gratings when used in much higher orders of interference than are common at present. With large aperture spectrometers, this possibility of using comparatively coarse gratings with a high order of interference is of great practical importance.
Demagnetizing coefficients and hysteresis losses of rectangular iron-silicon strips
Authors: Sondheimer E. H.
Published: November 2008

The hysteresis losses and demagnetizing coefficients of rectangular Fe-Si strips of various dimensions have been investigated. The hysteresis loss is approximately independent of the value of N, in agreement with a simple theoretical argument. The value of the demagnetizing coefficient is in fair agreement with the value calculated for the inscribed ellipsoid.
On a finite group of order 576
Authors: Hamill C. M.
Published: November 2008

Bagnera(1) shows that when a certain primitive finite group of order 576, G576, is represented as a group of collineations in three-dimensional space, it contains twenty-four harmonic inversions with respect to points and planes, and he shows how to calculate the coordinates of the points and planes of the inversions. The purpose of this note is to discuss the configuration of these points, which we shall call vertices, to show that the whole group is generated by the twenty-four harmonic inversions, termed projections after Prof. Baker (2) and Dr J. A. Todd (4), and to enumerate the conjugate sets of the group.
A coincidence method of measuring a flux of fast neutrons
Authors: Kinsey B. B. , Cohen S. G. , Dainty J.
Published: November 2008

A method is described in which the rate at which fast neutrons cross unit area is measured by counting protons projected into a small solid angle in the forwards direction from thin and thick layers of polythene. The protons are detected by triple coincidences between three proportional counters mounted coaxially behind the hydrogenous layers. The method is applicable to neutrons of energies from 1 MeV. upwards, and can be used in the presence of intense γ-rays. The flux of these neutrons is calculated in terms of the rate of detection of the protons, the solid angle for proton collection, the mass per unit area of the polythene layer, and the neutron-proton-scattering cross-section. A study of the behaviour of proportional co-axial counters, used in this manner, has been made. A determination of the angular distribution of fast neutrons produced by a deuteron-deuterium source has been made by our coincidence method, and the results compared with those obtained by the ionization chamber method by Bretscher and French. Absolute values agree to within 10%.
Rectilinear plastic flow of a Bingham solid
Authors: Oldroyd J. G.
Published: November 2008

1. The general problem. In the first three papers under the same main title (1, 2, 3), attention has been confined almost entirely to a discussion of steady flow. The equations to be solved in order to determine velocity distributions in non-steady rectilinear plastic flow were given in § 1 of the third paper (3). On the assumptions of isotropy and incompressibility, the conditions of the general problem are as follows.
Thermal conductivity of copper and german silver at liquid helium temperatures
Authors: Allen J. F. , Mendoza E.
Published: November 2008

A method is described for measuring thermal conductivities of metals at liquid helium temperatures. It is shown that the method of determining the temperature gradient by directly measuring the temperature at two points along a rod is probably free from systematic error, but that the end of a heated rod in direct contact with a helium bath is not at the temperature of the surface of the bath, even in the He ii region. A description is also given of a technique for using mercury as a low-temperature solder, tight to He ii. The results for copper are in reasonable agreement with Makinson's theory of metallic conduction. Tentative extrapolation to very low temperatures in the presence of fairly large magnetic fields shows that pure copper is a reasonably good material for thermal transfer in demagnetization experiments. The results for german silver seem to indicate that only the lattice conductivity is important since it follows a T3 law, and that scattering takes place with a mean free path of the order of 10−3 cm.
A theorem on positive harmonic functions
Authors: Verblunsky S.
Published: November 2008

1. Let z = reiθ, and let h(z) denote a (regular) positive harmonic function in the unit circle r < 1. Then h(r) (1−r) and h(r)/(1 − r) tend to limits as r → 1. The first limit is finite; the second may be infinite. Such properties of h can be obtained in a straightforward way by using the fact that we can write

where α(phgr) is non-decreasing in the closed interval (− π, π). Another method is to write

where h* is a harmonic function conjugate to h. Then the function

has the property | f | < 1 in the unit circle. Such functions have been studied by Julia, Wolff, Carathéodory and others.
Note on a theorem of R. Baer
Authors: Higman G.
Published: November 2008

It is trivial that a group all of whose elements except the identity have order two is Abelian; and F. Levi and B. L. van der Waerden(1) have shown that a group all of whose elements except the identity have order three has class less than or equal to three. On the other hand, R. Baer(3) has shown that if the fact that all the elements of a group have orders dividing n implies a limitation on the class of the group, then n is a prime. The object of the present note is to extend this result by showing that if M is a fixed integer there are at most a finite number of prime powers n other than primes, such that the fact that all the elements of a group have orders dividing n implies a limitation on the class of its Mth derived group.
The interaction of point particles with charged fields
Authors: Le Couteur K. J.
Published: November 2008

1. The method introduced by Dirac (1) in his classical theory of the interaction of an electron with an electromagnetic field has since been extended to the interaction of point particles with more general wave fields, such as occur in the neutral meson theory. Bhabha (2, 3) considered the scattering of neutral mesons by nucleons and gave arguments to show that typically quantum-mechanical effects should be unimportant if the energy of the incident meson is less than the rest energy of a nucleon, so that a classical theory with damping is appropriate to such problems. Now only the charged mesons are observable, and therefore a discussion of the classical theory of the interaction of a nucleon with a charge-bearing field is desirable. Apart from a paper by Fierz (4) which omits the dipole-like coupling terms, there is little published work on this subject.
On the classical equations of motion of radiating electrons
Authors: Ashauer S.
Published: November 2008

This paper deals with some details of the application of the Lorentz-Dirac equations of motion of an electron to two simple cases, (a) with no incident field, and (b) with an incident pulse of radiation. In case (a), the field-energy distribution in the self-accelerating motion of the electron when the electron has built up a velocity close to the velocity of light is considered. Numerical and graphical methods are used to form a picture showing how the field energy tends to be concentrated in a disk for this state of motion. A discontinuity of the advanced field variables of the self-accelerating electron is also examined. In case (b), the general solution of the equations of motion for the velocity of the electron is worked out accurately. In the physical solution, this brings out the effects of radiation damping when the pulse has great intensity. Some aspects of the non-physical solutions are pointed out.
Further results for the counterfeit coin problems
Authors: Hammersley J. M.
Published: November 2008

The counterfeit coin problems consist in detecting and locating the anomalous weights of counterfeit coins amongst a set of genuine coins. Suppose that altogether there are n coins, of which at least n − k are known to be genuine. We do not know which coins are genuine; but we do know that all genuine coins have the same (unknown) weight. The weights of the remaining k coins are unspecified and possibly all different.
On the product of three non-homogeneous linear forms
Authors: Clarke L. E.
Published: November 2008

Let ξ, η, ζ be linear forms in u, v, w with real coefficients and determinant Δ ≠ 0. Then there exists a number ℳ such that, corresponding to any real numbers a, b, c, there exist rational integers u, v, w for which
A new sequence of minima in the geometry of numbers
Authors: Macbeath A. M.
Published: November 2008

The problem of evaluating the non-homogeneous critical determinant of the plane region |y − x2| ≤ 1 is solved as a particular case of a result concerning the non-homogeneous properties of convex regions proved in the author's thesis (not yet published). The search for an alternative method of proof led to a proof of the existence of an infinite sequence of minima analogous to the Markoff chain. This is the subject of the present paper.
On the recurrent solutions of a class of non-linear differential equations
Authors: Friedlander F. G.
Published: November 2008

1. This paper is concerned with certain properties of the solutions of differential equations of the type

where f, g have continuous partial derivatives up to the second order satisfying Lipschitz conditions in some bounded domain and are periodic in t with period 2π, ω is a positive constant and k is a small parameter. If f = 0, (1·1) is equivalent to

an equation representing the forced vibrations of a quasi-linear system of one degree of freedom.
A regression problem
Authors: Lindley D. V.
Published: November 2008

During the Oxford Conference of the Econometric Society in September 1936, Ragnar Frisch proposed a problem in regression theory. A partial solution was found in 1938 by Miss H. V. Allen (1). A more complete solution was given by C. R. Rao (6) in 1947, and in the same year the present author (5) obtained a solution as a particular case of a more general result. These last two papers contained a flaw, and a correct solution was provided by Miss E. Fix (2). This last solution still leaves a part of the problem unanswered, and in the present paper a result of P. Lévy's (4), is used to complete the solution. At the same time further generalizations of the problem are considered and, in the cases of most practical importance, complete solutions are obtained. It is advisable, both from the point of view of rigour and simplicity of analysis, to use a general definition of the conditional expectation of a random variable. Accordingly, the paper begins with a summary of the relevant definitions. These notions were introduced by Kolmogoroff (3). It has been thought worth while giving the definitions here, in forms which are slightly different from Kolmogoroff's and seem more suitable for applications, in order to explain the notation and nomenclature used. The relevant consequences of these definitions are also stated in the form in which they are used.
Non-homogeneous binary cubic forms
Authors: Bambah R. P.
Published: November 2008

1. Let f(x1, x2, …, xn) be a homogeneous form with real coefficients in n variables x1, x2, …, xn. Let a1, a2, …, an be n real numbers. Define mf(a1, …, an) to be the lower bound of | f(x1 + a1, …, xn + an) | for integers x1, …, xn. Let mf be the upper bound of mf(a1, …, an) for all choices of a1, …, an. For many forms f it is known that there exist estimates for mf in terms of the invariants alone of f. On the other hand, it follows from a theorem of Macbeath* that no such estimates exist if the region

has a finite volume. However, for such forms there may be simple estimates for mf dependent on the coefficients of f; for example, Chalk has conjectured that:

If f(x,y) is reduced binary cubic form with negative discriminant, then for any real a, b there exist integers x, y such that
A formalization of the C-O propositional calculus
Authors: Rose Alan
Published: November 2008

In this paper we formalize the two-valued propositional calculus with implication and the logical constant 0 (denoting a false statement), as primitives. We can define the logical constant 1 (denoting a true statement), by

and we can define negation by NP = df. CP0.
Finite rotation groups and crystal classes in four dimensions
Authors: Hurley A. C.
Published: November 2008

The groups of symmetries of three-dimensional lattices have been known for some time. They consist of finite rotation groups, the crystal classes, and infinite discrete motion groups, which include both rotations and translations. The general theory of the corresponding groups in higher dimensional Euclidean spaces has also been developed. This theory includes a demonstration that in Euclidean space of n dimensions the number of motion groups is finite, and leads to a method† for calculating the motion groups, the first step being to determine the crystal classes. The explicit calculation of the various groups by the general method is not simple, and has so far been confined to the case of two and three dimensions. In the special case of the crystal classes in four dimensions, however, we may make use of the results of a paper by Goursat‡. In this paper Goursat sets up a correspondence between the finite rotation groups in four-dimensional Euclidean space, and a set of groups each of which is formed by associating two of Klein's groups of linear non-homogeneous substitutions in one variable. Using this result he is able to evaluate explicitly all the proper and improper finite four-dimensional rotation groups which include the element −I, where I is the four-rowed unit matrix.
Superconductivity of tin isotopes
Authors: Lock J. M. , Pippard A. B. , Shoenberg D.
Published: November 2008

Detailed measurements have been made of the superconducting transition temperatures and critical magnetic fields of the tin isotopes of mass 116, 120 and 124. The transition temperature varies with isotopic mass according to the law Tc ∞ M−n, with n = 0·462 ± 0·014, a result very similar to that already found in mercury. The critical field curves of isotopes 116 and 124 are geometrically similar, in the sense that both many be represented by the same equation, Hc/Ho = f(T/Tc), with the same ratio Ho/Tc for both. It is deduced that the electronic specific heat in normal tin varies only slowly, if at all, with isotopic mass. The variation of Tc and Ho with M is very close to that predicted by Fröhlich and Bardeen.
Core-preserving transformations of a vector space
Authors: Bonsall F. F.
Published: November 2008

In the classical theory (3) due to Knopp, Agnew and others, the core K(x) of a sequence x = {ξn} of complex numbers is defined by where En(x) is the smallest closed convex set containing all ξk with k ≥ n. A matrix transformation T is said to be a core-preserving transformation if

holds for all sequences x. T is core-preserving for bounded sequences if (1·1) holds for all bounded sequences x. It is readily proved that K(x) is the set of complex numbers ζ such that

for all complex numbers α (). Now is a sub-additive, positive-homogeneous real-valued functional defined on the vector space of bounded complex sequences. This suggests the construction of an abstract theory on the following lines.
A problem concerning three-dimensional convex bodies
Authors: Rankin R. A.
Published: November 2008

1. The problem considered in this paper arose during an investigation of what Chabauty has called the ‘anomaly’ of convex bodies.

Throughout the paper denotes a closed bounded convex body in three-dimensional Euclidean space , which is symmetric in the origin O and which contains O as an interior point. Such a body determines uniquely a distance-function f(x, y, z) which is defined and finite for each point (x, y, z) of and possesses the following properties