bluuebunny/crossref_metadata_2025 huggingface dataset:
The generalized Fermi interaction
Authors: Hoyle F.
Published: November 2008

Konopinski and Uhlenbeck have shown that the empirical β-ray spectra seem to require a modification of Fermi's original expression for the interaction leading to β decay‡. Their expression includes a derivative of the neutrino wave function. The purpose of the present paper is to investigate the most general type of such an interaction term and the possibility of introducing second-order derivatives. We have to deal with the transition§

neutron + neutrino → proton + electron

or proton + anti-neutrino → neutron + positron.

The whole system has the equations of motion

where ψ1 and ψ2 are the wave functions of the initial and final states in configuration space, H1 and H2 are the Hamiltonians of the particles in the initial and final states, and A is a small interaction operator.
The numerical calculation of
Authors: Burnett D.
Published: November 2008

The type of integral considered, is of frequent occurrence in problems in the kinetic theory of gases and in particular is required in the paper which follows, in calculating viscosity. The result is an extension of one given by Berger, and the analysis follows closely the discussion of Gauss's formula for approximate quadrature in Whittaker and Robinson, The calculus of observations; the novelty lies in the use of Sonine polynomials, which are peculiarly well suited for the discussion of this type of integral.
The effect of temperature on the reflexion of X-rays
Authors: Blackman M. , Fowler R. H.
Published: November 2008

1. The reflexion of X-rays from an ideal crystal would be independent of temperature if the particles forming the crystal were at rest. Because of the motion of the particles a temperature effect does exist. It can easily be seen that only the component of the displacement at right angles to the reflecting plane is of importance, since it is the path difference between such planes which matters. The complete theory has been worked out by Debye and by Waller. The result is that, owing to the influence of the temperature, the intensity is multiplied by a factor e−2m, where

Here and μs is the mean square of the component of the displacement (in a direction at right angles to the reflecting planes) due to a normal mode π is the glancing angle of incidence of the X-rays on the reflecting planes, and λ is the wave-length.
Lattice changes associated with the formation of superlattices in alloys
Authors: Wilson A. H.
Published: November 2008

6. Bethe's theory of order and disorder in alloys is generalized so as to include transitions in which the lattice structure changes from cubic to tetragonal. It is shown that, with the assumptions made, there is a discontinuity in the order at the transition temperature and also a latent heat. An expression is derived for the degree of tetragonality as a function of the order.
Note on fluctuations and the shot effect
Authors: Rowland E. N.
Published: November 2008

The present author recently published what he believed to be a proof and extension of Campbell's theorem on fluctuations. More recently a paper has been published containing a criticism of this proof. It therefore seems desirable that some parts of the proof be amplified. Further explanation of the foundations of the theory seems required, in order that it shall not appear that the theory ignores the randomness of the events, and a more detailed account of the derivation of the mean value of functions of pairs of events (i.e. sums of the type ) is necessary.
An apparatus for recording the strength of wireless signals from a Morse sender
Authors: Budden K. G.
Published: November 2008

In many radio investigations of the ionosphere it is desirable to use senders working on different wave-lengths and situated in different positions with respect to the receiver. For this purpose it is very convenient to make use of commercial senders which work more or less continuously at high power. It so happens, however, that these senders usually emit Morse signals consisting of dots and dashes of variable lengths and with variable intervals between them, so that an instrument which records the mean signal will give an indication which depends on the speed of sending and on the spacing and length of the signs. In order to obtain a reading which depends only on the signal amplitude, it is possible to employ a quick moving recorder such as a string galvanometer or a cathode ray oscillograph, which will follow the individual Morse signs. With both these types of recorder the record must be made photographically so that changes in the signal strength are not noticed until the record is developed. This is a serious disadvantage in investigations of short-lived disturbances of the “catastrophic” type.
Zur summabilitätstheorie der Fourierschen und Laplaceschen Reihe
Authors: Fejér Leopold
Published: November 2008

1. Die unendliche Folge der Partialsummen {sn} der Fourierschen Reihe einer integrierbaren und nach 2π periodischen Funktion f(ξ) der reellen Variabeln ξ ist an einer Stelle ξ = x, wo die Funktion stetig ist, (C, 1) limitierbarzu f(x) = s als Grenzwert, d.h. es ist
The penetrating power of cosmic-ray shower particles
Authors: Jánossy L.
Published: November 2008

The absorption curve for cosmic-ray shower particles has been observed in lead and iron with a counter arrangement. Since the observed curves are in good agreement with the absorption curve of cosmic ray electrons calculated by Heitler, it has been concluded that nearly all the showers are of the cascade type.

Observations of Schmeisser and Bothe which have previously been interpreted as giving experimental evidence for the occurrence of penetrating particles in showers are not in contradiction with the observations given here, so that they also must be interpreted in terms of energetic electron cascades.
Suitability of a diamagnetic crystal for the measurement of magnetic fields
Authors: Stansfield R. G.
Published: November 2008

The magnetic anisotropy of crystals of calcite and sodium nitrate does not alter between room temperature and liquid nitrogen temperature. By using such diamagnetic crystals, it is easy to compare strong magnetic fields to 1 part in 1000. The effect is large enough for the measurement of fields of the order of 100 gauss, but in small fields the accuracy is limited by ferromagnetic contamination.
Note on the slow motion of fluid
Authors: Dean W. R.
Published: November 2008

In the first part of the paper a slow two-dimensional motion of viscous fluid is considered which approximates to a motion of uniform shear past an infinite fixed plane, and differs from this motion because there is a gap in the plane (Fig. 1). A simple expression in finite terms is found for the stream function.
The design of a small differential analyser
Authors: Lennard-Jones J. E. , Wilkes M. V. , Bratt J. B.
Published: November 2008

Many problems in physics, chemistry and other subjects give rise to differential equations which are difficult or impossible to treat mathematically. It has long been possible to obtain numerical solutions by arithmetic methods, and the development of adding and multiplying machines to their present state of perfection has made these much simpler to carry out. The amount of work involved is, however, still very great.
On the generalized helices of Hayden and Sypták in an N-space
Authors: Wong Yung-Chow , Hodge W. V. D.
Published: November 2008

In this paper we are concerned with curves (C) of the following types:

where k1, k2, …, kh−1, kh = 0 (h ≤ n) are the curvatures of (C) relative to the space Vn in which (C) lie. Hayden proved that a curve in a Vn is an (A)2m, h = 2m + 1, if and only if it admits an auto-parallel vector along it which lies in the osculating space of the curve and makes constant angles with the tangent and the principal normals. Independently, Sypták∥ stated without proof that a curve in an Rn is a (B)n if and only if it admits a certain number of fixed R2's having the same angle properties; he also gave to such a curve a set of canonical equations from which many interesting properties follow as immediate consequences.
The influence of radiation damping on the scattering of light and mesons by free particles. I
Authors: Heitler W.
Published: November 2008

1. In the classical theory there is no difficulty in treating the effect of radiation damping on the scattering of light by a free electron in so far as it is a result of the conservation of energy. In the non-relativistic approximation the equation of motion of a free electron under the influence of a light wave is

with the periodic solution

The total energy radiated per second is then

and the total cross-section(1) is

Formula (1) differs from the Thomson formula by the factor 1/(1 +κ2). This factor becomes appreciable for energies ħν ≥ 137mc2.
A new type of disintegration produced by deuterons
Authors: Krishnan R. S. , Banks T. E.
Published: November 2008

The formation of 62Cu, 106Ag and 120Sb by deuteron bombardment of copper, silver and antimony respectively is reported. The energy-yield curves for these reactions were determined and it is concluded that in each case the process involved is taking place, in all probability, through the formation of a ‘compound nucleus’ of the Bohr type.
On Hausdorff's methods of summability
Authors: Rogosinski W. W.
Published: November 2008

The aim of this paper is to give a general theory of the ‘strength’ of Hausdorff methods of summability. These methods are defined by the linear transforms

of a sequence (sk). Here the μk form a given sequence of real or complex numbers and the Δp(μk) denote their differences of order p; i.e. Δ0(μk) = μk and
On the theory of successive radioactive transformations
Authors: Sedgwick W. F.
Published: November 2008

13. The main object of this paper is to supply simple algebraical proofs of the general formulae for the number of particles, at any time t, of the rth product in a series of successive radioactive transformations. This is done both for ‘Case 1’, where the matter is initially all of one kind (paragraphs 3 and 4), including a stable end-product (paragraph 5), and for ‘Case 2’, where the successive products are initially all in existence and in equilibrium (paragraph 6). In paragraph 7 it is remarked that there exists a relation between the amounts of the end-product in ‘Case 1’, supposed to be the (r + 1)th product, and of the preceding rth product in ‘Case 2’, for the same time t, and the necessity of this relation is explained on general grounds. In paragraph 9 numerical values are given for the particular example (paragraph 8) considered by Rutherford in his Newer alchemy (pp. 11, 12), whilst paragraphs 10 and 11 deal with the degree of accuracy attained, and show the need for close accuracy in the experimental data.
Initial stress and elastic instability
Authors: Jeffreys Harold
Published: November 2008

By a well-known theorem of Kirchhoff the strain energy of an elastic solid is less in the equilibrium position than in any other position satisfying the same boundary conditions, and under the same body forces. The theorem contradicts the fact that elastic instability can occur, since two or more positions of equilibrium can then exist, and both cannot have the smaller strain energy. Numerous writers (Bryan (1), Southwell (2), Dean (3)) have explained the apparent discrepancy as due to the neglect of second-order terms in the elastic equations. This is correct so far as it goes, but it does not explain why the usual discussions of elastic instability give the right answers. The elastic constants vary somewhat with stress and in any case will be different according as a stressed or an unstressed state is taken as the standard. If any second-order terms should be included, we might expect that this variation would make a contribution comparable with those hitherto considered. Further, several theories of elasticity now exist that differ in the higher terms (Dean (3), Seth (8), Murnaghan (6)), and it may be asked whether they should lead to the same estimates of the critical loads.
The ‘odd’ number six
Authors: Todd J. A.
Published: November 2008

1. A few months ago, in the course of teaching an elementary class. I had occasion to discuss problems of the familiar type, ‘if α, β, γ are the roots of the equation x3 + qx + r = 0, form the equation whose roots are α2 + βγ, β2 + γα, γ2 + αβ'. After I had explained the device of replacing βγ by αβγ/α = −r/α, so that the roots of the new equation are the same rational function of the separate roots of the original equation, I was asked ‘whether such a transformation was always possible’. On investigation the answer to the question proved to be a surprising one.
Point-curve correspondences
Authors: Scott D. B.
Published: November 2008

Attempts have been made to extend to higher dimensional varieties the theory of correspondences developed for algebraic curves. So far efforts have been concentrated on ‘point-point’ correspondences (i.e. between two varieties of the same dimension such that a generic point of one corresponds to a 0-dimensional variety of the other), and even in the case of surfaces important problems, such as the base number for the correspondences, are still unsolved (cf. Hodge(3)). The purpose of these papers is to draw attention to, and study in the simpler cases, another class of correspondences, the ‘point-primal’ type, in which to a generic point of one variety corresponds a primal of the other. These correspondences provide at least as natural a generalization of the theory for curves as point-point correspondences, but have so far only been touched on by Severi(7, 8) in two simple cases. In this paper we give a number of general results for point-primal correspondences (mostly immediate generalizations of Severi's results), and embark on a detailed discussion of such correspondences, which we naturally call point-curve correspondences, between two surfaces, the chief result being the determination of the base in that case. Further results on the transformation of the cycles of a surface by a point-curve correspondence, the correspondences induced between curves of the two surfaces and, in the case where the surfaces coincide, the ‘united curve’ of the correspondence, as well as the question of correspondences of non-zero valency, will be dealt with in a later paper, where we shall also consider the case of correspondences between a curve and a surface discussed in Severi's paper(7).
A method for computing the resonant wave-length of a type of cavity resonator
Authors: Goddard L. S.
Published: November 2008

In this paper, using the results of the preceding paper, I develop a modification of Hahn's method (1) for calculating the electromagnetic field inside a type of axially symmetric resonator. A process of successive approximation is given for solving the equation which leads to the resonant wave-length λ, both in the case where all the resonator parameters are given and we wish to find λ, and also in the case where λ is given and one of the parameters (the outer radius) is to be found. Some numerical results are given.
Excitation of indium 113 by X-rays
Authors: Dunworth J. V. , Pontecorvo B.
Published: November 2008

It is shown that In113* can be obtained by nuclear excitation of indium by X-rays. The cross-section per nucleus for the production of In113* with 2 MeV. X-rays is of the same order as the cross-section for the production of In115*. The decay period of In115*, for which some discrepancy was apparent in the literature, was definitely proved to be 4·5 hr.
On a class of linear integro-differential equations
Authors: Pitt H. R.
Published: November 2008

This paper is a sequel to a previous one(1) with the same title which dealt with the general solution of equations of the type

We consider here the more general equation

where g(x) is a given function. We are interested particularly in the existence and uniqueness of solutions of the latter equation and show how these are related to the closure and completeness properties of sets of functions {xneωnx} derived from the kernels kr(y).
A note on the Hamiltonian theory of quantization. II
Authors: Chang T. S.
Published: November 2008

It is pointed out that the equations of motion for any field obtained by varying a Lagrangian subject to auxiliary conditions are exactly equivalent to a certain set of canonical equations and that the commutation relations between the dynamical variables for the latter equations are Lorentz-invariant. By extending the theory to Lagrangians containing higher derivatives of the field quantities, it is shown that any given set of field equations can be put into the canonical form, though it is not derived from variational principles. The question of Lagrangians with missing momenta is also considered. It is shown that if the Lagrangian is ‘gauge-invariant’, some of the p's must be missing and the corresponding Eulerian equations can be replaced by equations containing no q and then can be replaced by initial conditions. The commutation relations between gauge-invariant quantities are Lorentz-invariant. For Lagrangians which are not gauge-invariant but are such as to have missing momenta, the passage to quantum theory will in general give rise to non-Lorentz-invariant commutation relations. In both cases, the equations of motion can be cast in canonical forms.
An electrical method for the determination of short-period activities and its use in the evaluation of the half-life of 12B
Authors: Jelley J. V. , Paul E. B.
Published: November 2008

Equipment has been designed and constructed for the interruption of the ion beam in a high-voltage accelerating tube and for the detection and measurement of the period of short-lived radioactive elements. This method has been applied to the measurement of the half-life of 12B which has been found to be 27 ± 2 msec. The reaction 15N(n, α)12B has been detected.
Inequalities for the derivatives of a bounded harmonic function
Authors: Verblunsky S.
Published: November 2008

If h(r, θ) is harmonic in the unit circle | r | < 1 and satisfies the condition | h | ≤ 1, then there is a function u(ø) which satisfies | u | ≤ 1 such that

and conversely. Hence, any properties of such harmonic functions should be deducible from equation (1). A number of such properties have been proved by Koebe (Math. Z. 6 (1920), 52–84, 69), using Schwarz's lemma and the geometry of simple conformal transformations. They can be deduced from (1) together with an elementary lemma on the rearrangement of a function (Lemma 1 below). As, however, students of this subject will regard Koebe's method as the one best adapted to establish his theorems, we shall illustrate the alternative method by considering two new problems, namely to find max ∂h/∂r, max ∂h/∂θ, where the maximum in each case is taken for all harmonic functions h which satisfy
Linear difference-differential equations
Authors: Wright E. M.
Published: November 2008

The general linear difference-differential equation takes the form

where x is a real variable, ν(x) and Aμν(x) are known functions and
The breakdown of the hodograph transformation for irrotational compressible fluid flow in two dimensions
Authors: Craggs J. W.
Published: November 2008

It has been noticed by various authors that the use of the hodograph transformation for problems of compressible fluid flow leads to flows having singularities where the acceleration becomes infinite, streamlines have cusps and the solution of the equations of motion in the hodograph plane ceases to represent any possible physical flow. In this paper we investigate the nature of the singularities which can occur in the representation of one plane upon another and apply our results to the hodograph transformation. Finally, we discuss the significance for practical purposes of such singularities and in particular their connexion with the occurrence of shock waves.
The disintegration of boron by slow neutrons. II
Authors: Gilbert C. W.
Published: November 2008

A further study has been made of the disintegration of boron by slow neutrons, B10 (n, α) Li, in the cloud chamber. The total ranges of the α-particles and the lithium particles were measured and two groups found. 91·4 % of the disintegrations form the main group, which was previously measured, and 8·6 % a long-range group corresponding to the lithium being left in the ground state. The results of the previous measurements are corrected for the variation of stopping power with velocity and the ranges in standard air of the particles of the two groups are now given as 7·7 mm. α-particle and 4·8 mm. lithium particle, total 12·5 mm. for the main group, and 9·3 mm. α-particle and 5·6 mm. lithium particle, total 14·9 mm. for the long-range group.

These results, taken together with those of other workers, make it certain that in about 92 % of disintegrations the lithium is left in an excited state of about 0·47 MeV. energy and in the remaining 8 % in the ground state. The energies calculated from this scheme by using the known masses do not agree with those deduced from the measured ranges and the range-energy relation for a-particles. It is suggested that the rangeenergy relation is at fault and that these measurements provide two experimental points to which a range-energy relation should be fitted.
On iterative solutions of algebraic equations
Authors: Domb C.
Published: November 2008

1. In a recent paper (1) Hartree has given a systematic classification of iterative processes, and has shown how to construct processes of any order for the solution of equations of the form

where p is a positive or negative integer. It is the purpose of the present note to describe general methods of generating iterative processes, and in particular to show how to construct processes of any order for the solution of algebraic equations.
The cardinal series in Hilbert space
Authors: Weston J. D.
Published: November 2008

The name ‘cardinal function’ was given to

by E. T. Whittaker (1), who considered it as a ‘smooth’ approximation to a function f(x), having the same values as f(x) at the points a + rw (r = 0, ± 1, ± 2, …). It has since been extensively studied (2), mainly from the point of view of interpolation theory. Hardy (3), however, observed that the functions νr(t) defined by

form a normal orthogonal set on the interval (−∞, ∞), for r = 0, ± 1, ± 2, …. This fact suggests a discussion of the cardinal series from the point of view of mean-square approximation.
On fourfolds with canonical curve sections
Authors: Roth L.
Published: November 2008

In a recent note the writer has examined the varieties whose generic curve sections are canonical curves of genus p, of general character, and whose surface sections contain only complete intersections with primals; following Fano's classification, we call these varieties of the first species. Such varieties are all rational provided that r > 3 and p > 6. In the present paper we consider their representations on linear spaces for the case r = 4, from which, in conjunction with the previous results, we conclude that fourfolds of the first species exist if, and only if, p ≤ 10; this agrees with the conjecture made by Fano in the case r = 3. It will be seen that the representation of these varieties on [4] provides interesting illustrations of Semple's formulae for composite surfaces in higher space.
A stable ninety-nine channel pulse amplitude analyser for slow counting
Authors: Wilkinson D. H.
Published: November 2008

1.1. It is often necessary in nuclear physics research to know the distribution in amplitude of a series of electrical impulses. An instrument which reveals this distribution directly is called a ‘pulse-amplitude analyser’, or, in the jargon of the late war, when the problem of building such an instrument was first seriously considered, a ‘kick sorter’. Many kick sorters have been described or discussed ((1)—(16)), and many new types are at present under development.
Sets of uniqueness for trigonometric series and integrals
Authors: Henstock R.
Published: November 2008

It is well known that, if a trigonometric series

converges to zero in 0 ≤ x < 2π then all the coefficients are zero. To generalize this property of the series, sets of uniqueness have been defined. A point-set E in 0 ≤ x < 2π is a set of uniqueness if every series (0·1), converging to zero in [0, 2π) − E, has zero coefficients. Otherwise E is a set of multiplicity. For example, every enumerable set is a set of uniqueness. An account of the theory may be found in Zygmund (2), chapter 11, pp. 267 et seq.
The extremal values of Legendre polynomials and of certain related functions
Authors: Cooper R.
Published: November 2008

1. The tabulated values of the Legendre polynomials suggest that the right-hand minimum of Pn(x) changes monotonically as n increases. Let xr, n be the value of x which gives the rth extreme value to the left of 1 of Pn(x). Then we can show that

where jr is the rth pösitive zero of J1(z), and that after some term the sequence Pn(xr, n) is monotonic with the moduli of the terms decreasing. We cannot, however, show that the sequence is monotonic from the place at which its terms become significant.
Unsteady laminar flow of gas near an infinite flat plate
Authors: Illingworth C. R.
Published: November 2008

Boundary-layer equations for the unsteady flow near an effectively infinite flat plate set into motion in its own plane are subjected to von Mises's transformation. Solutions are obtained for the flows in which gravity is neglected, the Prandtl number σ is arbitrary, and the plate has a constant temperature and a velocity that is either uniform or, with dissipation neglected, non-uniform. Explicit solutions are obtained for the case in which the viscosity μr varies directly as the absolute temperature Tr. Solutions are also obtained for the diffusion of a plane vortex sheet in a gas, and for the boundary layer near a uniformly accelerated plate of constant temperature when gravity is not neglected. For the non-uniform motion of a heat-insulated plate, dissipation not being negligible, a solution is obtained when σ is 1 and μr ∝ Tr. The relative importance of free convection due to gravity and forced convection due to viscosity is discussed, and a solution is obtained, with μr ∝ Tr, for the free convection current set up near a plate that is at rest in a gas at a temperature different from that of the plate, dissipation being neglected.
A theorem on multiple integrals
Authors: Hammersley J. M. , Moran P. A. P.
Published: November 2008

Suppose thatdυ anddυ′ are two volume elements situated at pointsPand P′ respectively in a three-dimensional right circular cylinder, thatyis the distancePP′, thatz(y) is a given function ofy, and that we wish to evaluate the sixfold integral

taken over all pairs of pointsP, P′within the cylinder. We observe thatz(y) is a function ofyonly; so that the sixfold integral can be expressed as a single integral

that is to say a weighted mean ofz(y) over the relevant values ofy,where the weight function is evidently given by
Statistics of the simple cubic lattice
Authors: Wakefield A. J.
Published: November 2008

Both the Ising theory of ferromagnetism and the theory of regular solutions are concerned with systems arranged on a lattice and make the assumption that each system interacts only with its nearest neighbours. Mathematically, there is a close parallel between the two problems (see, for instance, Rushbrooke (1)). In the first half of this present paper the partition functions for these two problems are examined in some detail. Power series expansions of the partition function of the Ising model, valid for low and high temperatures, are obtained. The terms obtained in the power series have been analysed and approximate numerical results obtained. It is hoped to publish these in a second paper.
On a class of unirational varieties
Authors: Roth L.
Published: November 2008

A theorem of Castelnuovo, which has played a considerable part in the general theory of surfaces, states that any surface which contains a net of elliptic curves is either rational or elliptic scrollar; more precisely, in the first case it is proved that the surface is unirational, and that its unirational representation is obtained by adjoining the irrationality on which depends the determination of one of its points, while the rest of the conclusion follows from Castelnuovo's theorem on the rationality of plane involutions. A somewhat similar result holds for surfaces which contain a net of hyperelliptic curves: thus, it is shown by Castelnuovo (loc. cit.) that, if the characteristic series of the net is not compounded of a g½ the surface is either rational or hyperelliptic scrollar.
Relativistic wave-functions and K-capture for a modified coulomb field
Authors: Malcolm I. , Strachan C.
Published: November 2008

In the theory of beta-decay, as is well known, when the electron is described by relativistic wave-functions, e.g. in a Coulomb field, a difficulty arises when these become infinite at the centre of force, r = 0. The wave-functions are then evaluated at the nuclear radius, r = R, and in many formulae there appears (1) a rather sensitive dependence on R. For the capture of an orbital electron by a nucleus the results of calculation depend strongly on the behaviour of the wave-function of the bound electron near the nucleus. The nature of the potential energy near the nucleus will thus affect such quantities as the ratio of K-capture to positron emission. Wannier (2) has discussed a similar problem for the Schrödinger equation, but the singular behaviour of the solutions of the Dirac equation for the ground state makes it desirable to have the solution for the relativistic case. In the following, relativistic wave-functions are derived and used for a potential energy which is Coulombian outside the nucleus and constant inside. Similar problems have been considered by others, Breit et al. (3), Racah (4), in application to the calculation of hyperfine structure, and also by Broch (5).
The inhomogeneous minimum of binary quadratic, ternary cubic and quaternary quartic forms
Authors: Cassels J. W. S.
Published: November 2008

In this paper I prove the following results.

Theorem 1. Let f(x, y) = ax2 + bxy + cy2be an indefinite form. Then there exist real (x0, y0) such that

for all (x, y) ≡ (x0, y0) (mod 1). If f represents 0 there are indenumerably many such incongruent (x0, y0).
A superconducting galvanometer
Authors: Pippard A. B. , Pullan G. T.
Published: November 2008

A superconducting tangent galvanometer suitable for the measurement of thermoelectric voltages in circuits of resistance about 10−7 ohm is described. The inductance has been kept low by the use of only a single turn in each of the deflecting coils, but a high sensitivity has been attained by reducing the controlling field to about 0·01 gauss. The theory of operation of such a galvanometer in circuits of low resistance is discussed; the actual behaviour of the instrument is in satisfactory agreement with the theory. The galvanometer will detect a current of 10−5 A. corresponding to an e.m.f. of 10−12 V.
On the set of distances between points of a general metric space
Authors: Besicovitch A. S. , Taylor S. J.
Published: November 2008

The well-known Steinhaus theorem (2) with respect to the set of distances of linear sets of positive Lebesgue measure has been generalized to the case of linearly measurable subsets of rectifiable curves in the Euclidean plane by Besicovitch and Miller (1). An extension of the theorem to rectifiable curves in Euclidean n-space is immediate. Prof. A. P. Morse has suggested the problem as to whether or not the theorem still remains true in a general metric space. By defining a particular curve ℒ in a metric space we prove that the answer to this question is in the negative.
The stability of hexagonal lattices with a simple law of force
Authors: Nabarro F. R. N. , Varley J. H. O.
Published: November 2008

A hexagonal lattice bound by central forces between nearest neighbours is always close-packed. If there is in addition an energy which, like the Fermi energy of free electrons, depends only on the volume, a close-packed equilibrium structure is still possible, but there may be another hexagonal structure of lower energy. A numerical example is given in which this occurs, the lattice spacing, binding energy and elastic constants being comparable with those of Zn, and the law of force showing no obvious peculiarities.
Some generalized Sheffer functions
Authors: Rose Alan
Published: November 2008

It was shown by Sheffer that all functions of the two-valued propositional calculus can be denned in terms of a single primitive. This result has been extended by Post (3) and Webb (5) to the functionally complete m-valued propositional calculus. We shall consider here similar extensions for the extended propositional calculus and for the (functionally incomplete) systems of Łukasiewicz (1).
Differentiation of some classes of set functions
Authors: Hayes Charles A.
Published: November 2008

If A is a subset of a given metric space , and F is a function which correlates with each point x of A a family of non-vacuous subsets of , having the property that there exist members of the family of arbitrarily small diameter, shrinking down upon x, then F is said to be a blanket with domain A. It is well known that if a blanket is endowed with suitable covering properties, then it may be used for the purpose of differentiating one set function with respect to another. It is our intention to consider, in § 3, a type of blanket whose covering families are permitted to overlap in a controlled fashion, and to prove that these may be used for the differentiation of certain set functions with respect to a given one. The paper concludes with the construction, in § 4, of an example which shows that the results achieved are essentially the best possible under the hypotheses.
Transformations depending on sets of associated points
Authors: Room T. G.
Published: November 2008

This paper falls into three sections: (1) a system of birational transformations of the projective plane determined by plane cubic curves of a pencil (with nine associated base points), (2) some one-many transformations determined by the pencil, and (3) a system of birational transformations of three-dimensional projective space determined by the elliptic quartic curves through eight associated points (base of a net of quadric surfaces).
Some new asymptotic expansions for Bessel functions of large orders
Authors: Olver F. W. J.
Published: November 2008

During the course of recent work (6) on the zeros of the Bessel functions Jn(x) and Yn(x), it became evident that the theory of the asymptotic expansion of Bessel functions whose arguments and orders are of comparable magnitudes was incomplete. The existing expansions for large orders are those of Debye and Meissel, detailed derivations of both of which are given by Watson ((8), pp. 237–48).
The classification of multipole radiation
Authors: Jeffreys Bertha
Published: November 2008

A detailed account is given of the classification of the electromagnetic fields arising from a vibrating charge-current distribution. The work owes much to that of Berestetzky and Fierz, but differs from their treatments in that the exact expansions are taken as a starting point. Hence the general forms of the potentials and fields and their approximate forms are obtained directly. Formulae appropriate to large distances are given in §§2·1 and 3·1. The relation to the approach of Condon and Shortley (4) is discussed.
Gravitation and cosmic expansion in conformal space-time
Authors: Gürsey Feza , Bondi H.
Published: November 2008

A simple theory of gravitation is formulated in conformal Riemannian space-time. The metric is determined by a scalar function which satisfies a linear equation. A conclusion in favour of Einstein's general tensor theory is drawn from a discussion of the corrections to the Newtonian theory for purely gravitational phenomena. Finally the theory is applied to the cosmological problem and especially to the possibility of a steady-state universe. The velocity-distance law is shown to be compatible with a constant uniform distribution of matter without the need of artificial assumptions.
Some results concerning the local analytic branches of an algebraic variety
Authors: Northcott D. G.
Published: November 2008

The recent progress of modern algebra in analysing, from the algebraic standpoint, the foundations of algebraic geometry, has been marked by the rapid development of what may be called ‘analytic algebra’. By this we mean the topological theories of Noetherian rings that arise when one uses ideals to define neighbourhoods; this includes, for instance, the theory of power-series rings and of local rings. In the present paper some applications are made of this kind of algebra to some problems connected with the notion of a branch of a variety at a point.
On the evaluation of the confluent hypergeometric function
Authors: Slater L. J.
Published: November 2008

This paper contains a table of the confluent hypergeometric function over the range a = − 1·0(0·1) + 1·0, b = 0·1(0·1)1·0, x= 1·0(1·0)10·0, and the expansions in converging factors by means of which the accuracy of the asymptotic expansions for higher values of x can be improved.
A measureless one-dimensional set
Authors: Eggleston H. G.
Published: November 2008

It has been known for some time that there are sets in Euclidean space which are of infinite measure in a certain Hausdorff dimension, say α, and yet which contain no subsets that are of finite positive a measure. The properties of such a set, say X, could be developed in much the same way as those of sets which are of positive finite α measure, if there existed a Caratheodory outer measure Γ, defined over the subsets of X, which was such that ∞ > Γ(X) > 0, and for any subset Y of X, Λα (Y) = 0 implied Γ (Y) = O. The object of this note is to show that there are sets for which no such outer measure exists. It is shown that a set defined by Sierpinski (7) is one-dimensional and is such that any Caratheodory outer measure defined over its subsets either takes infinite values or is identically zero or is not zero for all subsets that consist of a single point. It has been remarked by Prof. Besicovitch that these properties imply that the set is measurable with respect to any Caratheodory outer measure defined over subsets of the plane, and in fact any subset is measurable with respect to any such outer measure.
On axially symmetric crack and punch problems for a medium with transverse isotropy
Authors: Payne L. E. , Green A. E.
Published: November 2008

1. Introduction. The problem of determining the distribution of stress in a medium with transverse isotropy (one axis of elastic symmetry) has recently been considered by Elliott (l, 2) and Shield (3). They have obtained solutions to certain special problems both with and without rotational symmetry. Elliott (2) has employed the method of dual integral equations to solve a number of axially symmetric crack and punch problems. (This crack problem is not to be confused with the torsional crack problem treated by Weinstein (4)). Shield (3) has reduced the general crack and punch problems for a medium with transverse isotropy to corresponding isotropic problems as formulated by Green (5), and has treated a few special examples. For the axially symmetric isotropic crack and punch problems the author (6) has shown that Green's formulation may be simplified, and has obtained in each case a simple usable solution of the general axially symmetric problem without the need for dual integral equations. As one might expect these simplifications may also be realized in the case of a medium with transverse isotropy. In this paper the author, guided by the work of Green (5) and Shield (3), presents a straightforward method for solving the general axially symmetric crack and punch problems for media with transverse isotropy. This method permits one to write down immediately in an appropriate system of curvilinear coordinates the solution to the general axially symmetric problem, whereas the method given by Shield requires an individual treatment of each problem. In fact it would appear to be extremely difficult if not impossible to obtain the complete solution to the general axially symmetric crack or punch problem by the methods suggested by Shield. Since the formulation given in this paper differs from that given by Shield, his equations for transforming the problem in transverse isotropy to an isotropic problem must be altered slightly. The solutions to the general axially symmetric problems for a medium with transverse isotropy then follow immediately from the solutions to the corresponding isotropic problems in (6). We shall therefore find it necessary to repeat some of the arguments and results of (6).
The evaluation of a diffraction integral
Authors: Hartree D. R.
Published: November 2008

The calculation of the diffraction pattern for an out-of-focus line source viewed through an axially symmetrical lens involves the evaluation of the integral

for a constant value of β (related to the numerical aperture of the lens) and a range of values ofxandy. For small values ofxandythis can best be done by quadrature; for larger values ofxoryor both it is more practicable to use numerical evaluation of an appropriate ordinary differential equation. It is shown that the variation of

withxfory= 0, and withyforx= const., can both be expressed by second-order ordinary differential equations of a convenient form for numerical integration; the evaluation ofI1(x,y) from solutions of these equations is straightforward. For comparison with experiment, results are required in the form ofI1(x,y) and a derived quantity as functions ofy, at relatively widely spaced values ofx; numerical work for β = 0.72 has been carried out on the EDSAC and has been organized to produce the results in the required form directly, without interpolation.

Comparison of results obtained by using different Gauss quadrature formulae and different interval lengths show some interesting features of the errors of high-order integration formulae, and show that care is necessary in estimating these errors from the differences between quadratures carried out with intervals of different lengths.
Remarks on the isomorphism problem in theories of construction of finite groups
Authors: Taunt D. R.
Published: November 2008

In most theories for the construction of finite groups with given properties a major difficulty is the ‘isomorphism problem’, which consists of specifying how one representative of each class of isomorphic groups may be selected from the totality of groups constructed by the process laid down. To do this we need a practical criterion for the isomorphism of two constructed groups. The main object of the present paper is to establish such a criterion in a particular case, which in spite of its simplicity is important because it gives a method for the construction ofA-groups (i.e. soluble groups whose Sylow subgroups are all Abelian). All soluble groups of cube-free order are included in this class of groups, and to exemplify the application of the criterion we summarize that part of our unpublished dissertation (7) which deals in detail with the groups of order 22.32.52.
The solution of equations by iteration
Authors: Coppel W. A.
Published: November 2008

The method of solving equations by iteration is very old and is discussed in many well-known books. But conditions for its validity have never been properly formulated. In the first place, it is necessary to know that the method will not carry us outside the domain of definition of our functions, and that the ‘approximations’ will not converge to something which is not a solution of the equations. These difficulties are easily forestalled; it is more difficult to ensure that the process really will converge. Our object will be to find the most general conditions under which we can set off with the certainty that we will ultimately arrive at a root, in the case of one equation with one unknown.
Temporally ordered graphs in quantum field theory
Authors: Polkinghorne J. C.
Published: November 2008

The S matrix is expressed in terms of momentum-space operators and its Cayley transform exhibited. A new graphical formalism called O-graphs is developed in which the vertices are temporally ordered and the lines represent real particles. This is used to prove a conjecture of Gupta concerning the relation of real and virtual processes to the Cayley transform. The transformation operator for a semi-infinite time interval is also considered and some of its properties demonstrated. In conclusion, another graphical formalism with temporally ordered vertices is developed and renormalized.
Approximate wave functions, with exchange for Mn+2
Authors: Hartree D. R.
Published: November 2008

There has recently been some interest in atomic wave functions for the Mn+2 ion in two different physical contexts, namely neutron diffraction (Shull, Straussen and Wollan (12), and Halpern (3)) and luminescence in solids (Williams (14)). In the neutron diffraction context the particular substance concerned is MnF2, for which the nuclear scattering is small so that the main contribution to the diffraction of slow neutrons is through the magnetic scattering by the incomplete (3d)5 group of the Mn+2 ion.
A note on the P-homomorphism in homotopy groups of spheres
Authors: Hilton P. J.
Published: November 2008

The homomorphism of the title is a homomorphism of πr(Sn) into πr+n−1(Sn) given by

where ι generates πn(Sn). Various results on the P-homomorphism were proved in (4); in particular it was shown that, if ηn generates πn+1(Sn), n ≥ 2, then
A note on the finite plane-strain equations for isotropic incompressible materials
Authors: Adkins J. E.
Published: November 2008

For elastic deformations beyond the range of the classical infinitesimal theory of elasticity, the governing differential equations are non-linear in form, and orthodox methods of solution are not usually applicable. Simplifying features appear, however, when a restriction is imposed either upon the form of the deformation, or upon the form of strain-energy function employed to define the elastic properties of the material. Thus in the problems of torsion and flexure considered by Rivlin (4, 5, 6) it is possible to avoid introducing partial differential equations into the analysis, while in the theory of finite plane strain developed by Adkins, Green and Shield (1) the reduction in the number of dependent and independent variables involved introduces some measure of simplicity. Some further simplification is achieved when the strain-energy function can be considered as a linear function of the strain invariants as postulated by Mooney(2) for incompressible materials. In the present paper the plane-strain equations for a Mooney material are reduced to symmetrical forms which do not involve the stress components, and some special solutions of these equations are derived.
Thermal conduction in the intermediate state of superconductors
Authors: Laredo S. J. , Pippard A. B.
Published: November 2008

The maximum in thermal resistivity exhibited by superconductors in the intermediate state is found to be particularly marked in tin at 0-5° K., at which temperature heat transport in the pure superconducting state is almost entirely by phonons. A simple theory of the effect is developed, based on the idea that phonon free paths are considerably shorter in the intermediate than in the superconducting state, owing to the high scattering caused by normal regions. Theory and experiment are in satisfactory agreement.
Note on Whitham's ‘The propagation of weak spherical shocks in stars’
Authors: Jones D. S.
Published: November 2008

In his paper on ‘The propagation of weak spherical shocks in stars’ Whitham(1) uses as the basis of his non-linear theory the solution of the linear equation

where the suffixes denote partial derivatives and A, B, C are functions of r only. In obtaining his solution he assumes (i) that

where ξ = t – ∫dr/A and fn(ξ) is the nth integral of f with respect to ξ, and (ii) that the function f may be identified with the corresponding function which occurs when gravity is neglected, i.e. when A becomes the constant A0, B = −2/r and C = 0. Having assumed that (2) exists, Whitham is able to show that it will hold asymptotically for Aξ/r ≪ 1 and A′ = O (A/r).