bluuebunny/crossref_metadata_2025 huggingface dataset:
Note on a new power-angle diagram for alternators
Authors: Ingram W. H.
Published: November 2008

In an alternating current motor (of simplest type) the internal E.M.F. e, the terminal tension e, the useful electrical power transformed p and the rotor phase angle φ are connected by the known formula

in which

where r is the resistance and l the self-inductance of the armature and ω the angular velocity of the rotor.
The emission of positive ions from hot tungsten
Authors: Moon P. B.
Published: November 2008

Measurements have been made of the positive ion currents emitted by tungsten at temperatures between 3000°K. and 3200°K. The results are in general accordance with the values calculated on the basis of the Saha equation from the rate of evaporation of neutral atoms, the electron work-function and the ionisation potential of tungsten. The “work-function” associated with the ionic evaporation appears to lie between 10 and 11 electron-volts.
An integral which occurs in statistics
Authors: Ingham A. E.
Published: November 2008

1. In this note we give a direct evaluation of the integral

whose value has been inferred from the theory of statistics. Here A = Ap = (αμν) and C = Cp = (Cμν) are real symmetrical matrices, of which A is positive definite; there are ½ p (p + 1) independent variables of integration tμν (1 ≤ μ ≤ ν ≤ p), and tμν is written also as tνμ for symmetry of notation; in the summation ∑ the variables μ, ν run independently from 1 to p; k is a real number. A word of explanation is necessary with regard to the determination of the power |A − iT|−k. Since A is positive definite and T real and symmetric, the roots of the equation
The limits of classical scattering
Authors: Blackett P. M. S.
Published: November 2008

The methods of geometrical optics provide an approximate solution of the equation of wave propagation, if the following condition, given by De Broglie, is satisfied. If V is the phase velocity, λ the wave-length, l the direction of greatest increase of V, and θ the angle between l and V, then the condition is that

that is, the relative change of V over a distance of the order of θ must be small†.
An Experiment on the Stability of Superposed Streams of Fluid
Authors: Taylor G. I.
Published: November 2008

It is well known that the calm which commonly comes on a cold clear night is due to the cooling effect of the ground on the lowest layers of the atmosphere. It frequently happens that the wind at a height of a few hundred feet does not die down at all at night; in fact observations taken at the top of the Eiffel tower show that there is frequently a slight increase in wind at that height. The cooling of the air near the ground makes the lower layers heavier than the higher layers and thus tends to prevent the formation of turbulence at the earth's surface. Since it is only through the medium of turbulence that the higher layers are able to drag the lowest layers over the ground the surface air stops moving.
The Stopping Power of Hydrogen Atoms for α-Particles according to the New Quantum Theory
Authors: Gaunt J. A.
Published: November 2008

An α-particle passing through a gas loses velocity because it gives up energy to the gaseous atoms. If we can calculate the average energy transferred to an atom, the stopping power of the gas follows immediately. The earlier theories, by Thomson and Darwin, took account of only the close collisions, in which the α-particle actually passes through the atom. Bohr, however, in 1913, took into account the transfers of energy to atoms at distances from the track considerably larger than atomic dimensions. His calculation was purely classical, and dealt with an atomic, model in which the electrons were capable of simple harmonic motions. The result was in very satisfactory agreement with experiment. For atoms containing each a single electron, with mass μ, charge ε, and natural period ω, Bohr's formula is

where − dT/dx is the rate of loss of energy by an α-particle whose charge is E, and velocity v; N is the number of atoms per c.c.; and γ = 1·123.
Some chains of theorems derived by successive projection
Authors: Lob H.
Published: November 2008

1. In a paper read to this Society Mr F. P. White has derived a remarkable chain of theorems in [n] which, for n = 2, becomes Clifford's chain. White's method is so elegant that, as he has restricted himself to curves going through two given points, it may be permissible to shew that his line of argument applies equally to curves going through three or more given points, thus producing other chains of theorems analogous to Clifford's. For this purpose it is necessary to consider the successive projections, not of a simplex, but of a figure of n, n − 1, n − 2,…vertices in [n].
On the quintic surface in space of five dimensions
Authors: Bath F.
Published: November 2008

Surfaces of order n in space of n dimensions, for 3 ≤ n ≤ 9, were discussed by Del Pezzo, who showed that the prime sections of such a surface are represented on the plane by cubic curves through (9 − n) base points.
The Classical limit of the space distribution law of a gas in a field of force
Authors: Mott N. F.
Published: November 2008

The Statistical Mechanics which has been developed in accordance with the requirements of the new Quantum Theory is concerned with distribution laws over energy values only—over, that is, the characteristics of Schrödinger's equation. To obtain a space distribution law, even for the Classical limit, some use must be made of the characteristic functions. A formula has been suggested by Fowler, but it has not been shown that this formula gives the Classical law for gases at ordinary temperatures and pressures. In this paper we shall show that this is so, but before doing so we shall sketch the analogous method of obtaining the law, on the Classical theory.
On the hypothesis of the obtuse angle
Authors: Hill M. J. M.
Published: November 2008

Engel and Stäckel in their Theorie der Parallellinien von Euklid bis auf Gauss (1895) call attention to the fact that Saccheri and Lambert have used in their argument the property of the triangle that the external angle is greater than either of the interior and opposite angles (Euc. I. 16), which is not true for triangles of all magnitudes when the hypothesis of the obtuse angle holds good. They do not show how starting from the point of view of Saccheri and Lambert the proofs should be amended so that they would be valid in the Elliptic Plane. Bonola (Non-Euclidean Geometry, translated by Carslaw, 1912, p. 31) in his account of the subject also uses Euc. I. 16. It is the object of this paper to obtain the main results proved by Saccheri and Lambert when the hypothesis of the obtuse angle holds without using Euc. I. 16. It is not suggested that this is a convenient method of procedure but it may perhaps have some historical interest. The key to the procedure here adopted is found in the reversal of the Euclidean order in a certain group of propositions, viz. I. 16, 18, 19, 20.
Periodic solutions of a linear differential equation of the second order with periodic coefficients
Authors: Ince E. L.
Published: November 2008

The equation to be considered is of the type

where p (x) is continuous for all real values of x, even, and periodic. It is no restriction to suppose that the period is π, and this assumption will be made, so that the equation is virtually Hill's equation.
A Contribution to the Theory of Ferromagnetism
Authors: Mahajani G. S.
Published: November 2008

The following gives a brief summary of a paper, which it is hoped to publish in extenso later. It is a contribution to the theory of ferromagnetic crystals, and contains in particular a theoretical explanation of Webster's experimental results.
Finite groups generated by reflections, and their subgroups generated by reflections
Authors: Coxeter H. S. M.
Published: November 2008

In connection with his work on singularities of surfaces, Du Val asked me to enumerate certain subgroups in the symmetry groups of the “pure Archimedean” polytopes n21 (n < 5), namely those subgroups which are generated by reflections. For the sake of completeness, I have enumerated such subgroups of all the discrete groups generated by reflections (including the symmetry groups of the regular polytopes). The work involved being somewhat intricate, several slips would have been overlooked but for the information that Du Val was able to supply from the (apparently remote) theory of surfaces.
On the Theory of Linear Integral Equations
Authors: Smithies F.
Published: November 2008

The object of this paper is to extend the ordinary theory of integral equations of the second kind to certain classes of more general kernels and functions, with special reference to equations of the Volterra type.
An Attempt to Detect the Neutrino
Authors: Nahmias M. E.
Published: November 2008

It has been shown by Chadwick and Lee(1), using a high-pressure ionization chamber, that if one neutrino is emitted by each disintegrating Ra E nucleus, then the neutrinos do not produce more than one pair of ions in 150 km. of air at N.T.P. Calculations based on the wave mechanics show that the ionization due to a neutrino having a magnetic moment of one Bohr magneton would be very easily detectable(2), whereas it has peen estimated that if the neutrino has no magnetic moment at all its encounters with nuclei will be as scarce as one in 1016 km. of water(3). I have investigated the matter again, using two Geiger-Müller counters, instead of an ionization chamber. The counters have the advantage of giving a discharge for a single pair of ions(4). The two counters of 15×5cm. were connected in parallel, and filled with air at 76mm. pressure. They were shielded on all sides by 45 mm. of lead. A source of 7 mg. of Ra (D, E, F) was placed at a distance of 11 cm. from the centres of the two counters, so that the total solid angle subtended by the counters at the source was 0·10, expressed as a fraction of the complete sphere. Assuming that both the disintegrating atoms of Ra D and Ra E give one neutrino, then the number of neutrinos crossing the counters per minute is 3·1 × 109. The average effective length of the path in the counter, reduced to N.T.P., is 0·3 cm., so in one min. the total range in air of all the neutrinos crossing the counters is 9·3 × 108 cm.
Ionization Power of a Neutrino with Magnetic Moment
Authors: Bethe H. A.
Published: November 2008

The number of ions produced by a neutral particle of small mass with a magnetic moment equal to n Bohr magnetons has been calculated (I–III) and found to be 103n2 per km. path in air at N.T.P., being practically independent of the mass and energy of the particle (IV). A comparatively large fraction of the secondary electrons produced have high energy and may therefore be counted if they are produced in the wall of the Geiger counter instead of in the counter itself. This fact should facilitate the experimental detection of the ionization considerably (V).
The Number of Free Electrons in a Metal
Authors: Fröhlich H.
Published: November 2008

The idea of “free” electrons is purely classical, but it is convenient to introduce it also in the quantum theory of metals. In order to calculate the number of free electrons NF from a quantum mechanical model it is necessary to give an appropriate definition.
The Virial Theorem and the Theory of Strong Electrolytes
Authors: Finkelstein B. N.
Published: November 2008

Since the pioneer work of Prof. R. H. Fowler(1) on the statistical foundations of the Debye-Hückel theory of strong electrolytes (2), several attempts have been made to obtain as much information as possible on the question of the equation of state of electrolytes from merely thermodynamical and dimensional considerations.
Note Introductory to the Study of Klein's Group of Order 168
Authors: Baker H. F.
Published: November 2008

The group of order 168 discovered by Klein, which it is now known can be generated by two operations E, of order 7, and ϑ, of order 2, which satisfy the relations E7 = 1, ϑ2 = 1, (Eϑ)3 = 1, (E4ϑ)4 = 1, has a vast literature. But for the most part each author pursues the matter from his own point of view; and it seems it may be useful to present a simplified approach to the theory which takes account of various possible aspects, in particular the geometrical. This is the object of the present note; for most of its contents I have found it necessary to do fresh work, so that the paper is by no means a transcript of what is already available.
The emission of radiation by a quadripole electric moment on the Quantum Mechanics
Authors: Gaunt J. A. , McCrea W. H.
Published: November 2008

The usual method of calculating the probability of a switch between stationary states by the Quantum Mechanics is really equivalent to finding the intensity of the dipole radiation. Other switches of the system not involved in this radiation may be possible but are not taken account of by the calculation. A method of calculating the probability of switches due to radiation by the quadripole moment seems to be supplied by Dirac's recent theory of the interaction between matter and radiation in his paper “The Quantum Theory of Dispersion.” Equations from this paper will be denoted by a D. The results are probably of a purely theoretical interest and the intensities found probably too faint to be observed; but the present considerations do appear to emphasise the need for including the radiation in the exact theoretical treatment of a Quantum Mechanical system.
An extended form of Kronecker's theorem with an application which shows that Burgers' theorem on adiabatic invariants is statistically true for an assembly
Authors: Thomas L. H.
Published: November 2008

The following paper is in two parts.

In Part 1 it is shown that Kronecker's theorem can be extended in the form

If is greater than ηfor all sets of integers l1, l2, … l3 less in absolute value than K/σ and not all zero, l also being an integer, then, for any x1, x2 … x3, an integer q less than l/(ησ8) can be found such that qv1–x1, qv2–x2…qv3–x4 all differ from integers by less than σ;. K, L depend only on s.

It is an immediate corollary that if

is greator than for all sets of integers l1, l2… l3, l less in absolute value than K/σ and not all zero while F(v1, v2… v3), l less in absolute value than K/σ and not all zero while F(v1, v2… v3, v) is periodic period 1, in v1, v2 … v3,v, then T can be found between 0 and such that

Where K, L depend on s, and N on s and the bounds of
On Tidal Waves in Canals having Sinuous Banks
Authors: Banerji A. C. , Varma Rama Shankar
Published: November 2008

Particular cases of the solution of the equation of tidal-wave propagation in canals of varying section have been obtained by various writers. It is believed that the cases which we are going to consider now have not been investigated before.
Note on the Petzval optical condition
Authors: Steward G. C.
Published: November 2008

As a preliminary to an investigation of certain diffraction patterns I was led to consider, in some detail, the geometrical aberrations of a symmetrical optical system; and it appeared convenient then to classify the aberrations in orders according as they depend upon various powers of certain small quantities and to exhibit them as coefficients in the expansion of an ‘ Aberration Function.’ If aberrations of the first order only are considered, it becomes evident that one of them stands, in some sense, apart from the rest; I refer to the so-called ‘Petzval’ condition for flatness of field. It is of interest to notice that this condition was known to Coddington and to Airy before the time of Petzval—known at least as far as concerns systems of thin lenses. In the usual notation the condition is ΣΚ/μμ′ = 0; it is therefore independent of the positions of the object and pupil planes and in this respect it stands alone among the first order aberrations. But an increasing number of similar aberrations of higher orders will be found and it is of interest to examine these and to investigate their geometrical meaning. In the following note is given a proof of the Petzval condition, differing from that usually given and falling more into line with the general theory, and indicating also a general method of examining aberrations of this peculiar type.
On a Type of Automatically Interrupted Triode Oscillations
Authors: Ratcliffe J. A. , Vedy L. G.
Published: November 2008

A type of automatically interrupted triode oscillations is described which depends on the interaction between an oscillating triode circuit and. a circuit containing a non-linear resistance and a time-constant device. The theory of the circuit is developed and tested experimentally by means of oscillograms. The theory of the rise and decay of currents in a circuit containing an inductance and a non-linear resistance is dealt with in an appendix.
On the Effect of Continued Small Additions of Poisonous Substances on the Velocity of Gaseous Catalytic Reaction in Closed Vessels
Authors: Constable F. Hurn
Published: November 2008

The Langmuir-Frenkel theory of surface action has been shown to involve a simple relation between the partial pressure of the catalyst poison, and the reduction in reaction velocity which results. Very interesting observations have been made by Beebe showing that the heat of adsorption of carbon monoxide on a reduced copper catalyst falls considerably and reaches a limiting value as the surface approaches saturation. Pease and Stewart have shown that the presence of less than one percent, of the total carbon monoxide that a given copper catalyst can adsorb will reduce the reaction velocity of the combination of hydrogen and ethylene at the surface of the copper by 88 per cent.
Limiting forms of the frequency distribution of the largest or smallest member of a sample
Authors: Fisher R. A. , Tippett L. H. C.
Published: November 2008

The limiting distribution, when n is large, of the greatest or least of a sample of n, must satisfy a functional equation which limits its form to one of two main types. Of these one has, apart from size and position, a single parameter h, while the other is the limit to which it tends when h tends to zero.

The appropriate limiting distribution in any case may be found from the manner in which the probability of exceeding any value x tends to zero as x is increased. For the normal distribution the limiting distribution has h = 0.

From the normal distribution the limiting distribution is approached with extreme slowness; the final series of forms passed through as the ultimate form is approached may be represented by the series of limiting distributions in which h tends to zero in a definite manner as n increases to infinity.

Numerical values are given for the comparison of the actual with the penultimate distributions for samples of 60 to 1000, and of the penultimate with the ultimate distributions for larger samples.
A dynamical method for the determination of Young's modulus by stretching
Authors: Sharman C. F.
Published: November 2008

There are two general methods of measuring the elastic constants of bodies; one involves a study of the static deformation produced by the appropriate kind of stress, and the other a measurement of the period of oscillation of a system of known inertia under the elastic forces.
The Specific Heat of Carbon Dioxide—Correction
Authors: McCrea W. H.
Published: November 2008

In a recent paper in these Proceedings the writer suggested the possibility of a transition from one molecular form to another in CO2. The suggestion is embodied in the equation (10) and the resulting specific heats for low temperatures given. He greatly regrets that it was not till after those results were published that he found they gave a high and altogether impossible maximum in the specific heat curve for higher temperatures before it returns to the neighbourhood of the unmodified curve Cv′.
A Theory of Hartree's Atomic Fields
Authors: Gaunt J. A.
Published: November 2008

In a recent communication to this Society Dr Hartree has put forward a method for calculating the field of anatom containing many electrons. Each orbit—to borrow a metaphor from the old quantum theory—is related to a wave-function Ψ which obeys Schrödinger's equation. The potential energy used in this equation is due partly to the field of the nucleus, and partly to the fields of the electrons in the other orbits. The latter are calculated upon Schrödinger's interpretation of the wave-function, that |Ψ|2 is the density of charge, measured in electronic charges per unit volume. It is not the purpose of this paper to discuss the practical methods of obtaining wave-functions which reproduce the fields from which they are derived; but to relate these wave-functions and their energy parameters to those of the accepted theory.
On the periods of a free-free bar immersed in water
Authors: Moullin E. B. , Browne A. D.
Published: November 2008

If a bar is vibrating under water it will set up circulating currents which possess kinetic energy and so in effect add to its inertia: consequently the frequency under water will be less than the frequency in air. The effect of the water is precisely the same as if the density of the bar had been increased: the apparent increase of mass is usually called the “added mass” of the bar and the total apparent mass the “virtual mass”, and these terms will be used subsequently.
Two Formulae for Space Curves
Authors: Archbold J. W.
Published: November 2008

We consider in space [3] a curve C of order n and genus p without multiple points. If we represent the lines of [3] by the points of a quadric Ω in [5], the chords of C will be represented by the points of a surface F of order (n−1)2−p lying on Ω. This surface has a triple curve M (with multiple points) corresponding to the ruled surface of trisecants of C (and the quadrisecants) of order ⅓(n−1)(n−2)(n−3)−p(n−2). It is the object of this note to find the genera of M and of a prime section ϑ of F; these being also the genera of the ruled surface of trisecants of C and of the ruled surface of chords of C which belong to a linear complex.
Note on Exchange Phenomena in the Thomas Atom
Authors: Dirac P. A. M.
Published: November 2008

For dealing with atoms involving many electrons the accurate quantum theory, involving a solution of the wave equation in many-dimensional space, is far too complicated to be practicable. One must therefore resort to approximate methods. The best of these is Hartree's method of the self-consistent field. Even this, however, is hardly practicable when one has to deal with very many electrons, so that one then requires a still simpler and rougher method. Such a method is provided by Thomas' atomic model, in which the electrons are regarded as forming a perfect gas satisfying the Fermi statistics and occupying the region of phase space of lowest energy. This region of phase space is assumed to be saturated, with two electrons with opposite spins in each volume (2πh)3, and the remainder is assumed to be empty. Although this model hitherto has not been justified theoretically, it seems to be a plausible approximation for the interior of a heavy atom and one may expect it to give with some accuracy the distribution of electric charge there.
Rotational Specific Heat and Rotational Entropy of Simple Gases at Moderate Temperatures
Authors: Sutherland G. B. B. M.
Published: November 2008

It is well known that the rotational specific heat of a diatomic gas is given by

where R is the gas constant, σ = h2/8π2AKT, h is Planck's constant, T is the absolute temperature, K is Boltzmann's constant, and A is the moment of inertia of the molecule.
Note on thermal instability in cylindrical condensers due to dielectric losses
Authors: Goodlet B. L.
Published: November 2008

A convenient form of cylindrical condenser consists of a stack of thin circular discs of suitable insulating material, interleaved with discs of metal foil of somewhat smaller diameter. Numbering the foils from one end, all the odd foils and all the even foils are connected together. The total capacity is then n times that between two adjacent foils, if the total number of foils is n + 1.
An experiment on dispersion
Authors: Evans R. C.
Published: November 2008

In the spectrum formed by a grating the red is more deviated than the blue, while in that formed by a prism, which does not exhibit anomalous dispersion in the visible spectrum, the reverse is the case. If, then, white light is passed first through a grating and then through a prism, the nature of the spectrum produced will depend on the relative dispersions of the grating and prism. If throughout the spectrum the dispersion of the prism is less than that of the grating, the sequence of colours will be the same as that in the spectrum due to the grating alone, red being more deviated than blue. If, on the other hand, the dispersion of the prism is throughout the spectrum greater than that of the grating, the sequence of colours will be reversed. On account of the rapid increase in the dispersion of glass near the blue end of the spectrum, it may, however, be possible that the dispersion of the prism exceeds that of the grating in the blue while in the red the reverse is the case. Under these conditions both red and blue will be more deviated than some intermediate colour for which the dispersions of the grating and prism are equal. The spectrum will thus have a sharp edge of this colour at the end of minimum deviation, while in the direction of greater deviation it will at each point consist of two superimposed colours. It is as if an ordinary white light spectrum was folded back on itself about the sharp edge.
The problem in statistics of testing several variances
Authors: Bartlett M. S.
Published: November 2008

The test of significance of the difference between two means was extended by means of Fisher's methods of analysis of variance to test jointly the differences between several means, the test being essentially the same as the test of significance of two variances. But though it is true in many cases that we are chiefly concerned with two independent estimates of variance, and the method of analysis of variance is available, the problem of jointly testing the variances obtained from several independent samples does not fit obviously into this scheme.
The inner shell ionization of atoms by electron impact
Authors: Burhop E. H. S. , Massey H. S. W.
Published: November 2008

Calculations have been made of the cross-section for ionization of the inner shells of atoms by electron impact in the cases of the K-shells of nickel, silver, mercury and of the three L-shells of silver and mercury.

The agreement with experiment is reasonably good for the K-shell ionization, but only fair in the case of the rather meagre experimental data available for the L-shell. The values obtained for the relative ionization in the K- and L-shells are in good agreement with those to be expected from experiment.
On the stability of crystal lattices. I
Authors: Born Max
Published: November 2008

The stability of lattices is discussed from the standpoint of the method of small vibrations. It is shown that it is not necessary to determine the whole vibrational spectrum, but only its long wave part. The stability conditions are nothing but the positive definiteness of the macroscopic deformation energy, and can be expressed in the form of inequalities for the elastic constants. A new method is explained for calculating these as lattice sums, and this method is applied to the three monatomic lattice types assuming central forces. In this way one obtains a simple explanation of the fact that the face-centred lattice is stable, whereas the simple lattice is always unstable and the body-centred also except for small exponents of the attractive forces. It is indicated that this method might be used for an improvement of the, at present, rather unsatisfactory theory of strength.
The invariant theory of isotropic turbulence
Authors: Robertson H. P.
Published: November 2008

The statistical theory of isotropic turbulence, initiated by Taylor (3) and extended by de Kármán and Howarth (2), has proved of value in attacking problems associated with the decay of turbulence. In its application to such hydro-dynamical problems, the theory falls into two parts, a kinematical part and a dynamical part. The kinematical aspect consists in setting up correlations between velocity components, or their derivatives, at two arbitrary points in the fluid, and reducing the form of the tensor thus obtained in accordance with the severely restrictive assumption of isotropic turbulence; the success of de Kármán and Howarth's investigations is largely attributable to their improved treatment of this purely kinematical problem. The dynamical part then consists in applying the implications of the equations of continuity and motion to the functions defining the correlation tensors, in order to obtain information concerning their functional dependence on time and on the displacement between the two points for which the correlations are computed.
The scattering of neutral mesons
Authors: Booth F. , Wilson A. H.
Published: November 2008

Expressions are obtained for the scattering of neutral mesons by protons according to the quantized theory. It is found that the scattering due to the g1 (point charge) interaction is much less than for charged mesons, but that the anomaly in the scattering due to the g2 (dipole) interaction remains. The bearing of this upon recent attempts to modify the theory by introducing nuclear particles with charge 2e or − e is discussed.
On the stability of crystal lattices. IV
Authors: Born Max , Misra Rama Dhar
Published: November 2008

In order to explain the phenomena of melting, tensile strength, etc., we have to investigate the stability of crystals for finite deformations, for which deviations from Hooke's law occur. Although these deviations are in most cases of an irreversible character, it is necessary, for a systematic study, to develop mathematical methods for treating the mechanical (reversible) case of a highly strained crystal lattice, where terms of higher order than the second in the deformation energy must be taken into account.
On the correction of lattice spacings for refraction
Authors: Wilson A. J. C.
Published: November 2008

The increase in the angle of reflexion from a powdered crystal varies from (1 − n) tan θ to (1 − n) (tan θ + cot θ) as its absorption increases. In either case the true lattice spacing of a cubic crystal is obtained by increasing the extrapolated lattice spacing by a fraction 1 − n of itself. The divergence of the reflected beam by refraction is small in comparison with other causes of line width.
Note on a Paper ‘On the Theory of Successive Radioactive Transformations’
Authors: Sedgwick W. F.
Published: November 2008

Since the above-named paper was published my attention has been called, by Mr A. F. Crossley, to the fact that a solution for Rutherford's ‘Case 1’ had been given by H. Jeffreys in his Operational methods in mathematical physics (Cambridge Tracts in Mathematics), p. 35. This solution is very neat and direct. As, however, the method depends on the use of inverse operational factors, such as where it is less elementary in character than that followed in my paper.
The evaluation of the cosmical number
Authors: Eddington A. S.
Published: November 2008

1. The cosmical number N = . 136 . 2256 is most picturesquely described as ‘the number of protons and electrons in the universe’. This in itself would be a matter of idle curiosity. But N has a more general significance as a fundamental constant which enters into many physical formulae; it determines the ratio of the electrical to the gravitational forces between particles, the range and magnitude of the non-Coulombian forces in atomic nuclei, and the cosmical repulsion manifested in the recession of the nebulae. Its special interpretation as the number of particles in the universe arises in the following way. If we consider a distribution of hydrogen in equilibrium at zero temperature, the presence of the matter produces a curvature of space, and the curvature causes the space to close when the number of particles contained in it reaches a certain total; this total is N. We cannot say with the same confidence that the number of particles in the actual universe is precisely N, because the admission of radiation, complex nuclei, and unsteady conditions takes the problem outside the range of rigorously developed theory; but to the best of our belief these complications do not affect the total number of protons and electrons composing the matter of the universe.
Absorption and coincidence experiments with 198Au
Authors: Feather N. , Dainty J.
Published: November 2008

The radiation from 198Au (half-value period 2·76 ± 0·02 day) have been studied, by absorption and coincidence methods, in a wide variety of experimental conditions. For the most part the results confirm those obtained by other investigators, but, in addition, a fairly intense γ-radiation of about 65 KeV. energy has been detected. This radiation has been distinguished from the weaker fluorescent K X-radiation of mercury, emitted as the result of internal conversion of one of the γ-rays of 198Au, and it has been shown that it is not the K X-radiation of platinum. Disintegration of 198Au by K-electron capture, if it occurs at all, cannot take place in more than about 15% of all disintegrations.
The Passage of β-rays through matter
Authors: Eddy C. E.
Published: November 2008

1. The form of the absorption curves of homogeneous β-rays in aluminium has been investigated by means of a Geiger counter. It is found, in agreement with Schonland, that the main portion of the curve is linear, with pronounced initial and final flattenings. Similar initial flattenings were found for copper and silver, but not for gold.

2. The range of β-rays emergent at different angles has been determined, and it appears that the loss of range is due to the increased path consequent upon scattering.

3. Some evidence has been obtained of the angular distribution of plurally scattered particles, and of the existence of a most probable angle of scattering, similar to that found for α-rays by Geiger.
The dispersion electrons of lithium
Authors: Hargreaves J.
Published: November 2008

An attempt has been made in the work described in the present paper to use the method initiated by Hartree, for the numerical solution of the Schrödinger wave equation for an atom with a non-Coulomb field of force, to estimate the number of dispersion electrons (hereinafter denoted by “ƒ” for brevity), corresponding to the lines of the principal series of the optical spectrum of lithium, and also to the continuous spectrum at the head of the series. Various attempts have been made to do this for hydrogen and other atoms by an application of the Correspondence Principle, but the first successful attempt at a complete description was made by Sugiura†, who has calculated ƒ for the Lyman, Balmer, and Paschen series and the corresponding continuous spectra, by using the known analytical solutions of the wave equation for an electron in a Coulomb field. The same author has also calculated ƒ for the first two lines of the principal series of sodium, by the utilisation of an empirical field of force in the atom calculated from the observed term-values by a method based on the old quantum theory. He has estimated the contribution to Σƒ (summed for the whole series) due to the continuous spectrum by the theorem that Σƒ = 1 in the one-electron problem§. This property provides a useful check on the work when ƒ for the continuous spectrum is also calculated. In the present paper ƒ for the continuous spectrum is actually calculated and it is found that Σƒ = 1 to a good approximation.
The propagation of electromagnetic waves in a stratified medium
Authors: Hartree D. R.
Published: November 2008

The equations of propagation of electromagnetic waves in a stratified medium (i.e. a medium in which the refractive index is a function of one Cartesian coordinate only—in practice the height) are obtained first from Maxwell's equations for a material medium, and secondly from the treatment of the refracted wave as the sum of the incident wave and the wavelets scattered by the particles of the medium. The equations for the propagation in the presence of an external magnetic field are also derived by a simple extension of the second method.

The significance of a reflection coefficient for a layer of stratified medium is discussed and a general formula for the reflection coefficient is found in terms of any two independent solutions of the equations of propagation in a given stratified medium.

Three special cases are worked out, for waves with the electric field in the plane of incidence, viz.

(1) A finite, sharply bounded, medium which is “totally reflecting” at the given angle of incidence.

(2) Two media of different refractive index with a transition layer in which μ2 varies linearly from the value in one to the value in the other.

(3) A layer in which μ2 is a minimum at a certain height and increases linearly to 1 above and below, at the same rate.

For cases (2) and (3) curves are drawn showing the variation of reflection coefficient with thickness of the stratified layer.

Case (3) may be of some importance as a first approximation to the conditions in the Heaviside layer.
The Distribution of Charge and Current in an Atom consisting of many Electrons obeying Dirac's equations
Authors: Hartree D. R.
Published: November 2008

The approximation is made that in a many-electron atom each electron is in a stationary state in the field of the nucleus and the remaining electrons, and further that this field is central.

To this approximation it is known that, if the electrons obey Schrödinger's equation, then a complete n, l group of electrons [i. e. 2 (2l + 1) electrons with the same n, l] can be divided into two half-groups for each of which the distribution of charge is spherically symmetrical. To the same approximation it is shown that, if the electrons obey Dirac's equation, a complete Stoner sub-group [i. e. 2j + 1 electrons with the same n, l, j (j = l ± £)] can be divided into two halves, for each of which the distribution of charge is spherically symmetrical.

The distribution of current and the resultant magnetic moment for a solution of Dirac's equation in a central field are obtained, and it is shown that for a complete Stoner sub-group the net current and magnetic moment are zero. The Landé g-formula for one electron is derived from the formula for the magnetic moment by neglecting ‘relativity’ terms.

The formula for the magnetic moment suggests the question whether it is ever possible to specify the direction of the spin axis of the electron (as distinct from the magnetic moment of the whole atom). This is investigated, and it seems justifiable to specify the direction of the spin axis for states for which the magnetic quantum number m has its extreme values ±j, but not otherwise.
On the Contacts of Circles
Authors: Baker H. F.
Published: November 2008

There is a theorem for three circles in a plane, that if three tangents, each of two of these circles, either all transverse or one transverse and two direct, can be drawn to meet in a point, then the three tangents, each of two of the circles, respectively conjugate to those first taken, likewise meet in a point. The theorem was stated by Quidde, with a proof for the necessity of the condition as to the tangents to be taken, in a paper designed to establish Steiner's solution of Malfatti's problem. Casey gives the theorem with omission of the condition for the character of the tangents, as does Salmon, who, however, gives a proof depending on the right choice of certain square roots which enter. Quidde's theorem is stated, accurately, in the Nouvelles Annales, and a simple metrical proof, from the diagram drawn (essentially Quidde's, see 6 below) is given later in the same Journal by Mannheim; this is practically repeated by Hart. Recently, Prof. Neville, emphasizing the necessity of the condition for the character of the tangents, has called attention to Quidde's paper.
On the Annihilation of Positrons
Authors: Jaeger J. C. , Hulme H. R.
Published: November 2008

We shall discuss the annihilation of positrons by a process in which only one quantum of light is emitted. The positive electron is regarded as a hole or unoccupied state in the distribution of electrons in negative energy states, and is destroyed by an atomic electron jumping down into it with the emission of a single quantum. This process is only possible in the presence of a nucleus, which can take up the momentum liberated, and is therefore most probable when the atomic electron is a K-electron.
Potential functions related to groups of circular cylinders
Authors: Howland R. C. J. , McMullen B. W.
Published: November 2008

In a previous paper (2) by one of the authors some types of periodic potential functions were obtained and were applied to problems in which the boundaries were cylinders or spheres arranged at regular intervals in a row or ring. It is now our object to extend the methods of that paper to a number of other cases.
Probability relations between separated systems
Authors: Schrödinger E.
Published: November 2008

The paper first scrutinizes thoroughly the variety of compositions which lead to the same quantum-mechanical mixture (as opposed to state or pure state). With respect to a given mixture every state has a definite probability (or mixing fraction) between 0 and 1 (including the limits), which is calculated from the mixtures Statistical Operator and the wave function of the state in question.

A well-known example of mixtures occurs when a system consists of two separated parts. If the wave function of the whole system is known, either part is in the situation of a mixture, which is decomposed into definite constituents by a definite measuring programme to be carried out on the other part. All the conceivable decompositions (into linearly independent constituents) of the first system are just realized by all the possible measuring programmes that can be carried out on the second one. In general every state of the first system can be given a finite chance by a suitable choice of the programme.

It is suggested that these conclusions, unavoidable within the present theory but repugnant to some physicists including the author, are caused by applying non-relativistic quantum mechanics beyond its legitimate range. An alternative possibility is indicated.
Statistical theory of adsorption with interaction between the adsorbed atoms
Authors: Peierls R.
Published: November 2008

In a recent paper, Fowler discusses the adsorption isotherm for a monatomic layer, assuming an interaction between neighbouring atoms in this layer. In this paper we do not intend to give a contribution to the physical problem, but merely to show how the statistical problem involved can be solved by a method due to Bethe.
Note on the slow motion of fluid
Authors: Dean W. R.
Published: November 2008

In this paper we consider the slow two-dimensional motion of viscous liquid past a sharp edge projecting into and normal to the undisturbed direction of the stream. The liquid is supposed bounded by rigid planes represented by ABCDE in Fig. 1, and, apart from the disturbance caused by the projection, is assumed to be in uniform shearing motion. The stream function is then a bi-harmonic function that must vanish together with its normal derivative at all points of the boundary, and must be proportional to y2 at a great distance from the projection.
Some applications of conformal transformation to airscrew theory
Authors: Westwater F. L.
Published: November 2008

Recent reports contain tables of a parameter K required in calculating the performance of an airscrew by a new method.

The method of calculating K (due to Goldstein) is unsuitable for large values of the pitch especially near the tip of the airscrew.

In the case of infinite pitch we fall back, for a two-bladed airscrew, on the problem of a rotating lamina in two dimensions.

The solution for a cross lamina (corresponding to a four-bladed propeller) is given below and the tables of K for four blades are completed.

A formula for the limit of K/Kp at the airscrew tip is given for a propeller with any number of blades, where Kp is an approximate value of K due to Prandtl.

K for any number of blades is given in the form of an infinite series. The case of three blades is discussed in detail.
A comparison of wave functions for the normal helium atom
Authors: Baber T. D. H. , Hassé H. R.
Published: November 2008

The object of this paper is to compare some of the wave functions which have been suggested for the normal helium atom, including one calculated in the first section of the paper, as regards energy, magnetic susceptibility and electric polarizability. Since this atom provides the simplest two-electron problem, such a comparison is of special interest and may indicate the direction in which improvements might be made in the calculation of wave functions for more complicated atoms.
Fiducial distribution of several parameters with application to a normal system
Authors: Segal Irving E.
Published: November 2008

The notion of fiducial probability was introduced by R. A. Fisher and is now widely used in statistical work involving a single unknown parameter. Fisher has also considered the joint fiducial distribution of several parameters for which there exists a sufficient set of statistics, and has derived the fiducial distribution of the two parameters of the one-variate normal law*. Since Fisher's account is brief, we give a more extended description of the fiducial distribution of several independent parameters possessing a sufficient set of statistics.
The diffraction of light by supersonic waves in solids
Authors: Nath N. S. Nagendra
Published: November 2008

An elementary theory of the diffraction of light by supersonic waves was put forward by Sir C. V. Raman and myself (4) to explain many of the important features of the phenomenon observed by Bär and others. Recently Mueller (2) pointed out the essential idea that the optical diffraction effects in solids may be interpreted on the basis of this theory if due account is taken of the photoelastic effects arising from the periodic strains caused by the sound waves. He was thus able to give an explanation of the results of Schaefer and Bergmann (5, 6). Mueller's idea, however, required a still further refinement, since it did not consider the phase relationships between the differently polarized components of the diffraction orders. This refinement has, however, been recently incorporated by Mueller and myself (3). This led us to offer an explanation of the diffraction effects observed by Hiedemann and Hoesch (1) in glass blocks.
A discussion of the thermal expansion and conductivity in liquid helium II
Authors: Jones H.
Published: November 2008

The negative coefficient of thermal expansion of He ii is shown to be a consequence of the disordering process which takes place as the temperature rises. The conclusion is reached that the thermal energy of He ii is mainly energy of disorder; the vibrational energy of the sound waves amounts only to a few per cent of the whole energy. A theory of heat conduction considered as a flow of disorder is put forward and is shown to be not incompatible with the experimental data at present available.
Electronic states at the surfaces of crystals
Authors: Goodwin E. T.
Published: November 2008

It is shown that in a crystal there exist states in which the electron is bound to a surface of the crystal and has an energy lying within a forbidden band. The wave functions and energies of these states are calculated, on the nearly free electron approximation, in terms of the constants of the crystalline potential field, which is represented by a triple Fourier series having the periodicity of the lattice. The method is shown to be applicable to a general crystal having a surface parallel to any one of the crystal planes.