bluuebunny/crossref_metadata_2025 huggingface dataset:
On the Mathematical Equivalence of two ways of regarding the Excitation of an Atom by a Fast Moving α-particle
Authors: Frame J. W.
Published: November 2008

1. It is possible to treat the excitation of an atom by an α-particle in two ways; we may either solve the Schrödinger equation for the system consisting of the α-particle and the atom, or we may, on account of the great mass of the α-particle, treat it as a moving centre of force, and solve the Schrödinger equation for the electrons in the field of the α-particle and nucleus, If the α-particle has velocity υ greater than the orbital velocities of the electrons, it is possible to obtain approximate formulae for the excitation probabilities by both methods; in the former case by the well known Born method, and in the latter case by a method first used in this connection by Gaunt*, and which is essentially the same as the method of variation of constants. The two methods give formally very different formulae for the excitation probabilities; it is the purpose of this paper to show that they are in fact identical if the ratio of the mass of the electron to that of the α-particle be considered vanishingly small.
The theory of the extraction of electrons from metals by metastable atoms, II
Authors: Massey H. S. W.
Published: November 2008

In a previous paper the nature of the phenomena occurring when electrons are ejected from metal surfaces by the impact of metastable atoms and positive ions was discussed. It was shown that electrons may be ejected by excited atoms at considerable distances from the surface, and by considering the case of an excited hydrogen atom at the centre of a spherical cavity in the metal the order of magnitude of the effects to be expected and the velocity distribution of the ejected electrons was determined. In this paper the theory will be extended in two directions.
A Theoretical Discussion of the Electrical Properties of the Soil
Authors: White F. W. G.
Published: November 2008

It has been shown by Ratcliffe and White that the effective specific inductive capacity (ε) and the effective conductivity (σ) of soil placed, as dielectric, in a simple condenser vary in a marked manner with frequency. The range of frequency used was from 0·1 × 106 cycles per second to 10 × 106 cycles per second. It was found that at 6·1 × 106 cycles per second the specific inductive capacity was very large (about 40 e.s.u.) and that it decreased to a value of about 12 e.s.u. at 10 × 106 cycles per second. This latter value agrees with the value found in experiments on the propagation of short radio waves over the surface of the earth. The effective conductivity of the soil had a low value at 0·1 × 106 cycles per second but increased rapidly to a constant value for frequencies above 3 × 106 cycles per second.
On the determination of the horizontal component of the earth's magnetic field by a coupled oscillations method
Authors: Vedy L. G.
Published: November 2008

An account is given of a simple experiment designed to illustrate quantitatively the phenomena of coupled oscillations. Two similar small magnets are suspended in the earth's magnetic field at a suitable distance apart so that there is appreciable magnetic interaction between the two oscillatory systems. Under the conditions employed, the equations of motion reduce to a simple form, and the experiment may be used as a method of measuring the intensity of the horizontal component of the earth's magnetic field.
Extension of a theorem of C. G. F. James
Authors: Babbage D. W.
Published: November 2008

The linear subspaces [k] of a space [n] have been represented by Grassmann by the points of a rational locus V(n−k)(k+1), of dimension (n−k)(k+1), in ; the representation is effected by taking as independent homogeneous coordinates of a [k] in [n] the determinants of order k+1 which belong to the matrix, of k+1 rows and n+1 columns, formed by the co-ordinates of any k+1 linearly independent points of the [k].
The distribution of second order moment statistics in a normal system
Authors: Wishart J. , Bartlett M. S.
Published: November 2008

Let x be a normally distributed variable, which may without loss of generality be measured from the mean of the distribution, so that E(x) = 0, E denoting mathematical expectation. Then x satisfies the differential relation

where k2 (otherwise σ2 or (2h2)−1) is the semi-invariant of order 2. Also k1 = 0, and we know that kr = 0 for r > 2.
On the theory of the recombination of ions in gases at high pressures
Authors: Harper W. R.
Published: November 2008

Previous theories of the recombination of ions in gases are shown to be inapplicable to high pressures. An approximate quantitative theory which takes into account all the relevant phenomena is developed. The effect of both thermal agitation and mutual attraction in determining encounters of ions of opposite sign is considered, and the frequency of encounters which lead to recombination calculated. The criterion for an encounter to lead to recombination is that the drift of the ions towards each other due to their mutual attraction must be greater than their tendency to separate due to their Brownian movement. Excellent agreement is obtained with such experimental results as are available. Preferential recombination between an ejected electron and its parent ion is also discussed.

I am indebted to various members both of the Wills Physics Laboratory and of the Cavendish Laboratory for discussion of the problems dealt with in this paper.
The evaluation of Gibbs' phase-integral for imperfect gases
Authors: Ursell H. D.
Published: November 2008

Statistical mechanics is concerned primarily with what are known as “normal properties” of assemblies. The underlying idea is that of the generalised phase-space. The configuration of an assembly is determined (on classical mechanics) by a certain number of pairs of Hamiltonian canonical coordinates p, q, which are the coordinates of the phase-space referred to. Liouville's theorem leads us to take the element of volume dτ=Πdp dq as giving the correct element of a priori probability. Any isolated assembly is confined to a surface in the phase-space, for its energy at least is constant; when there are no other uniform integrals of the equations of motion, the actual probability of a given aggregate of states of the proper energy, i.e., of a given portion of the surface, varies as the volume, in the neighbourhood of points of this portion, included between two neighbouring surfaces of constant energies E, E + dE; it therefore varies as the integral of (∂E/∂n)−1 taken over the portion. If I be the measure of the total phase-space available, interpreted in this way, and i that of the portion in which some particular condition is satisfied, then i/I is the probability of that condition being satisfied.
The spectrum of the self-adjoint partial differential equation in any domain
Authors: Steen S. W. P.
Published: November 2008

The object of this paper is to obtain the general solution to the self-adjoint partial differential equation in n dimensions

where pij, q and ρ and bounded, continuous functions of (x1,…, xn) in a domain D and on its boundary, and where ∑pijXiXj≥0 for all (x1,…, xn) of D and all X1,…, Xn. The domain D is an n-dimensional domain and may be either the whole or part of a Riemann surface space of n dimensions. Its boundary is to consist of any number, zero, finite or enumerable, of continuous continua of n − 1 dimensions. These terms will be explained in paragraph II. The solution u = u(x1,…, xn; t) will be valid for (x1,…, xn) in D and t ≥ 0, and will satisfy boundary conditions of the type or similar, these conditions becoming identical at any part of the boundary of D that lies at infinity.
The testing of statistical hypotheses in relation to probabilities a priori
Authors: Neyman J. , Pearson E. S.
Published: November 2008

In a recent paper we have discussed certain general principles underlying the determination of the most efficient tests of statistical hypotheses, but the method of approach did not involve any detailed consideration of the question ofa prioriprobability. We propose now to consider more fully the bearing of the earlier results on this question and in particular to discuss what statements of value to the statistician in reaching his final judgment can be made from an analysis of observed data, which would not be modified by any change in the probabilitiesa priori. In dealing with the problem of statistical estimation, R. A. Fisher has shown how, under certain conditions, what may be described as rules of behaviour can be employed which will lead to results independent of these probabilities; in this connection he has discussed the important conception of what he terms fiducial limits. But the testing of statistical hypotheses cannot be treated as a problem in estimation, and it is necessary to discuss afresh in what sense tests can be employed which are independent ofa prioriprobability laws.
Geodetic and Dynamical Principles, a comparison and connexion
Authors: Hargreaves R.
Published: November 2008

The geodetic problem presented by a quadratic form is identical with the dynamical problem in which the same quadratic appears in the kinetic energy, and potential energy is not represented. If the quadratic in space-differentials is equated to kdt2, the problem is essentially the same whatever constant value is given to k; the corresponding feature is the arbitrary constant value of kinetic energy.
On a group of order 25920 and the projective transformations of a cubic surface
Authors: Burnside W.
Published: November 2008

The figure in three dimensions consisting of two sets of three non-intersecting lines, such that each line of one set intersects each line of the other set, will here be called a “net.” All nets are projectively equivalent to each other: and there are just 72 projective transformations that change one net into another.
On the Structure of Liquids and Vitreous Solids
Authors: Jeffreys Harold
Published: November 2008

The success of the kinetic theory in accounting for the properties of gases, and that of the ionic lattice theory in accounting for those of crystals, have been such as to warrant the belief that the main features of the constitution of gases and crystals are now understood, and that future development will proceed mainly on the lines already laid down. For liquids, on the other hand, we have hardly the beginnings of a theory; amorphous solids are in a slightly better state. The present article does not claim to have gone far towards the construction of such a theory, but it is hoped that it will at any rate indicate what are some of the chief problems to be faced when one is constructed.
On Reichinstein's Displacement Principle
Authors: Constable F. Hurn
Published: November 2008

The velocity of bimolecular reaction at a catalyst surface is often proportional to the concentration of either reactant when this is small, but at greater concentrations the rate of reaction falls off considerably, and finally the increase of concentration may actually retard the reaction. The maximum reaction velocity does not occur in general when the partial pressures of the reactants are in the ratios suggested by the chemical equation of reaction.
The quenching of resonance radiation and the breadths of absorption-lines
Authors: Ditchburn R. W.
Published: November 2008

In a paper by Slater a new theory of radiation is discussed and the following explanation of the breadths of spectral lines is given: “An atom in the ith. state has a probability Pi of suffering in unit time a transition. Thus there is a probability Pi that the vibrations of each of the oscillators will simultaneously cease. But we shall assume that, in addition to this probability Pi of ceasing its oscillation altogether, each oscillator has also an independent probability Pj of suffering an interruption in which it ceases its oscillation as if it were leaving the state, but immediately begins again as if it were entering the same state” (i.e. with an arbitrary phase difference). “This term Pj is the same as the probability that an atom in the jth state will leave that state.” The total probability of interruption of vibration is thus (Pi + Pj) which is symmetrical with regard to the two end states. This makes the breadths of absorption and emission lines equal and so satisfies Kirchhoff's law.
On points with arbitrarily assigned mutual distances
Authors: Coxeter H. S. M. , Todd J. A.
Published: November 2008

Let SsTt denote a space of s + t dimensions with the property that the square of the distance between any two points

is of the form

where s of the ε's are equal to + 1 and t of them to — 1.
On the ambiguity in the specification of a two-sheeted surface by its branch curve
Authors: Du Val P.
Published: November 2008

It is familiar that a “double plane”, i.e. an algebraic surface conceived as consisting of each point of a plane counted twice, is unambiguously determined when its “branch curve”, or locus of points in the plane which count as two coincident points of the surface (the general point of the plane counting as two distinct points), is assigned; the only condition which must be satisfied by this curve being that it is of even order. In fact, if f(x, y) = 0 is the Cartesian equation of the curve, the double plane can be regarded as the limit of the surface

where A is a large constant, or as the projection of this surface, for finite A, from the point at infinity on the z-axis. If, however, f = 0 is of odd order, it is easily seen that the line at infinity also forms part of the branch curve of the double plane. A repeated portion of the branch curve is ineffective, for if φ (x, y) = 0 is any other curve, the surface

is birationally identical with the former merely by putting
The capture of electrons by positive ions from neutral gas atoms
Authors: Smith R. A.
Published: November 2008

An apparatus for the production of beams of fast positive ions is described. Measurements of the cross-section for capture of an electron by a positive ion from a neutral gas atom are given on the case of He+ ions in helium and protons in helium and hydrogen. These are compared with the theory given by Massey and Smith, and as far as possible with previous experiments.
Note on the Formula for the Number of Quadrisecants of a Curve in Space of Three Dimensions
Authors: Maxwell Edwin A.
Published: November 2008

If pCε is a curve of order ε and genus p without singularities in space of three dimensions, the formula for the number of quadrisecants is well known, namely*,

Welchman has shown that the necessary reduction in this formula when the curve pCε has a point of multiplicity r with r distinct tangents, no three of which are coplanar, is
The Effect of Non-Normality on the t Distribution
Authors: Bartlett M. S.
Published: November 2008

Some theoretical results on the effect of non-normality on the t test of significance, though it is stressed that they are incomplete and not perhaps of much quantitative value, nevertheless tend to agree with the result of sampling investigations in showing that for moderate departures from normality this test may still be used with confidence, particularly for testing differences in means of equal numbers of observations.
A Statistical Derivation of Langmuir's Adsorption Isotherm
Authors: Fowler R. H.
Published: November 2008

The commonest form of Langmuir's adsorption isotherm is

where θ is the fraction of the surface of the solid covered by adsorbed molecules, p the gas pressure in equilibrium with the adsorbed layer and A = A (T) a function of the temperature alone. This formula is usually derived by a kinetic argument which balances the rates of deposition and re-evaporation. It is perhaps not without interest to show that formula (1) and similar formulae can be obtained directly by the usual statistical methods which evaluate all the properties of the equilibrium state of any assembly. The ordinary derivation is apt to obscure the essentially thermodynamic character of (1) and to lead one to think that its form depends on the precise mechanisms of deposition and re-evaporation, whereas in fact it depends only on the whole set of states, adsorbed and free, accessible to the molecules in question. By suitable use of the usual technique for handling assemblies obeying the Fermi-Dirac statistics the saturation effect can be naturally incorporated in the theory ab initio.
On the Real Zeros of a certain Trigonometric Function
Authors: Stein P.
Published: November 2008

Suppose that a, c and A are real, and that

Let n (X) be the number of zeros of

in ; then it is not difficult to prove that

for some K = K (A, a, c, θ). The problem in this paper is to determine the function K explicitly in terms of its variables.
Some Experiments with Neutrons slowed down at different Temperatures
Authors: Westcott C. H. , Niewodniczanski H.
Published: November 2008

Experiments are described comparing the effects of neutrons slowed down in paraffin wax at the temperatures of liquid nitrogen and liquid hydrogen with those obtained at ordinary temperatures. It is found that the absorption produced by certain substances increases as the temperature is lowered. The transformations produced in these substances also increase, but generally to a smaller extent, probably owing to more neutrons being absorbed in the paraffin: the exact figure depends on the thickness of the layer of cooled paraffin.

A few experiments were also made substituting liquid hydrogen for paraffin wax (without change of temperature). The effect of this also appears to depend on the thickness of the layer, but it is not yet possible to draw definite conclusions.
A Case of Double Reflexion
Authors: Appleyard E. T. S. , Skinner H. W. B.
Published: November 2008

We recently had occasion to have a right-angled quartz prism made with the optic axis of the quartz perpendicular to one of the short sides of the right-angled triangle (Fig. 1). The prism has the property that it gives two images when light enters perpendicular to one of the faces and, after internal reflexion at the hypotenuse, passes out at right angles to the other face. It was at first sight rather difficult to see why this doubling of the image occurs, since the incidence on both faces of the prism is normal and quartz is a uniaxial crystal, there can be no double refraction occurring.
On the Obliquity Function to be used in the Approximate Theory of Diffraction
Authors: Harper W. R.
Published: November 2008

The solution of problems in diffraction by an elementary application of Huyghens's principle is discussed. The obliquity function is investigated, using the criterion that the formula used must give the right result when integrated for the case of an undiffracted plane wave. It is shown that this is satisfied for distant points by any function which makes the integrals converge, but that to satisfy it completely a constant obliquity function is necessary. This makes a consideration of the distant boundary essential, as the integrals do not converge in this case. It is shown that a boundary distributed in a Gaussian way is completely satisfactory. The integral for the case of diffraction by a straight edge is solved exactly, leading to the usual result. Finally, the results are discussed in relation to the teaching of the subject.

I am indebted to Dr H. Jeffreys and to Mr F. P. White for their interest in this paper; also to Mr N. F. Mott, who has contributed much to the course of its development.
Note on the Condensation of Radioactive Substances on Solid Surfaces
Authors: Henderson G. H.
Published: November 2008

During my visit to Cambridge last year I was given the opportunity of carrying out in the Cavendish Laboratory some experiments with a view to studying the reflection of atoms from a solid surface. For this purpose the atoms of radioactive substances offer the advantage of extreme sensitivity and also the possibility of quantitative measurements by means of their radiations. While the main object of the experiments could not be achieved, some phenomena have been observed which seem to deserve brief mention.
A System of Linear Equations with an Infinity of Unknowns
Authors: Titchmarsh E. C.
Published: November 2008

I have collected in the present note some theorems regarding the solution of a certain system of linear equations with an infinity of unknowns. The general form of the equations is

the numbers a1, a2, … c1, c2, … being given. Equations of this type are of course well known; but in studying them it is generally assumed that the series depend for convergence on the convergence-exponent of the sequences involved, e.g. that and are convergent. No assumptions of this kind are made here, and in fact the series need not be absolutely convergent. On the other hand rather special assumptions are made with regard to the monotonic character of the sequences an and cn.
The Enumeration of the Partitions of Multipartite Numbers
Authors: Macmahon P. A.
Published: November 2008

This paper is a study of a new method of enumeration of the partitions of multipartite numbers.

Incidentally an algebraic function, which is derived from the repetitional exponents of partitions of unipartite numbers, presents itself. The generating function which enumerates the partitions of unipartite numbers is expressible in terms of these functions and finds in such expression its fullest connection with the divisors of numbers. There are also similarly derived functions connected directly with bipartite, tripartite, etc. numbers. It has not been necessary to study these for the purposes of this paper.
Motion of a Sphere in a Viscous Fluid
Authors: Bromwich T. J. I'A.
Published: November 2008

This problem was originally attacked by Stokes, and the formula obtained by Stokes for the case of steady motion (given at the end of § 1 below) is now generally used in experimental work. However, this formula throws no light on the way in which the velocity of the sphere approaches its steady value: and the only known discussion of this problem was given by Boggio (his results will be found in § 2 below). Unfortunately, when the specific gravity of the sphere exceeds ⅝ (or the constant σ, defined later, is less than 4) the solution so obtained cannot be converted into numerical results, because the constants α, β become complex numbers; and even when α, β are real, the formula is not very convenient for large values of t (as the velocity approaches the steady value).
Poisson's Summation Formula in several Variables and some Applications to the Theory of Numbers
Authors: Mordell L. J.
Published: November 2008

In a recent paper, Poisson's summation formula

was proved very simply by integration by parts, subject to the conditions:

(α) for all real values of x, f(x) and f′(x) are continuous and f (x)→0, f′ (x)→0 as |x| → ∞.

(β) f(x) and f″(x) are such that the integrals, converge.
An Analysis of Triode Valve Rectification. Part II
Authors: Landale S. E. A.
Published: November 2008

Generally, when an alternating voltage is applied to a cumulative grid rectifier no grid current flows at mean grid potential. The behaviour of a rectifier working under these conditions is examined. An expression is derived for calculating rectified current for any applied voltage. As this equation is rather cumbersome to apply, a very simply empirical formula is given which is applicable for any value of applied potential whatever.

An expression is derived for the power absorbed by the rectifier. It is shown that as the applied voltage increases, the apparent resistance of the rectifier decreases and approaches half the value of the grid leak resistance.

Further, it is shown that rectified current depends on the peak value of the applied potential and that it is almost independent of ordinary wave-form variations, even when the applied voltage is small.

By slightly modifying the expression for rectified current we find that, in an amplifier, the grid current is a measure of V and, in an oscillator, the grid current is a measure of the output. The only condition is that V > 2b in both cases.

In conclusion I wish to express my indebtedness to Professor C. E. Inglis for placing at my disposal the facilities of the Cambridge University Engineering Laboratory, and to Mr E. B. Moullin for the interest he has taken in the work.
On Mellin's inversion formula
Authors: Burkill J. C.
Published: November 2008

The extension of Mellin's inversion formula expressed by the equations has been considered by Fowler who shows that some form of Stieltjes integral is essential to Poincaré's proof of the necessity of the quantum hypothesis. Fowler confines his discussion to a restricted type of function φ (y) which is sufficient for the physical problem. It will be proved here that the formulae hold with a general Stieltjes integral in the first equation.
A mathematical theory of natural and artificial selection
Authors: Haldane J. B. S.
Published: November 2008

Expressions are found for the changes caused by slow selection in populations whose characters are determined by incompletely dominant, multiple, or polyploid factors, and for the equilibria attained in certain of these cases.
Note on the velocity of gas-reactions
Authors: Christiansen J. A.
Published: November 2008

In a recent paper, G. N. Lewis and D. F. Smith have discussed the problem of the velocity of chemical reactions, especially that of the only gas-reaction, the decomposition of N2O5, which had then been proved with sufficient accuracy to be kinetic unimolecular, i.e. unimolecular in the purely empiric sense of the word.
The Draining of a Vertical Plate
Authors: Jeffreys Harold
Published: November 2008

When a vessel of liquid has been emptied and put aside, a thin film of liquid clings to the inside and gradually drains down to the bottom under the action of gravity. The layer being thin, the motion is very nearly laminar flow, and the curvature of the surface in a horizontal direction may be ignored. Thus the problem for a cylindrical vessel is reducible to that of a wet plate standing vertically.
Some Measurements on the Absorption of X-Rays of Long Wave-length
Authors: Martin L. H.
Published: November 2008

An account is given here of the measurements of the mass absorption coefficients of iron, nickel, copper and aluminium over a wave-length range 0·705 Å.U. to 1.932 Å.U.
The average value of an exponential function over the lattice points of a circle
Authors: Wilton J. R.
Published: November 2008

1. The subject-matter of this communication I believe to be new, but after Lemma 1 the method is classical; Lemma 1 is itself a particular case of a theorem which I have given elsewhere, and is a straightforward extension of a well-known result.
The Wave Mechanics of an Atom with a non-Coulomb Central Field. Part III. Term Values and Intensities in Series in Optical Spectra
Authors: Hartree D. R.
Published: November 2008

In two recent papers the writer has given an account of a practical method of finding the characteristic values and functions of Schrödinger's wave equations for a given non-Coulomb central field. For terms of optical spectra the method is effectively the following. We take the wave equation in the form

and require the values of ɛ for the solutions which are zero at the origin and at r = ∞. We consider the result of integrating this equation outwards from P = 0 at r = 0 to a radius r0 at which the deviation from a Coulomb field is negligible, and inwards from P = 0 at r = ∞ to the same radius, with a given value of ɛ; the characteristic values are those values for which these two solutions join smoothly on to one another, i.e. for which they have the same value of η = −P′/P at this radius. For a given ɛ, the solution zero at the origin depends on the particular atom; the solution zero at infinity can be expressed in a form independent of any particular atom.
A suggested theory of electric conduction
Authors: McCrea W. H.
Published: November 2008

Sommerfeld has recently made a very important advance in the theory of electric conduction by employing the conception of electron waves and the new mechanics. His theory has been most successful in explaining a large number of phenomena. It treats them, however, from the macroscopic point of view and does not examine the atomic processes involved. The present note, intended to be merely tentative, seeks to attack the latter aspect of the problem.
Note on some linear Diophantine Inequalities
Authors: Mordell L. J.
Published: November 2008

Let A, B, C, D, P, Q be any real constants such that

.

A well-known theorem by Minkowski states that integer values of x, y exist such that

.
Inverse Probability
Authors: Fisher R. A.
Published: November 2008

I know only one case in mathematics of a doctrine which has been accepted and developed by the most eminent men of their time, and is now perhaps accepted by men now living, which at the same time has appeared to a succession of sound writers to be fundamentally false and devoid of foundation. Yet that is quite exactly the position in respect of inverse probability. Bayes, who seems to have first attempted to apply the notion of probability, not only to effects in relation to their causes but also to causes in relation to their effects, invented a theory, and evidently doubted its soundness, for he did not publish it during his life. It was posthumously published by Price, who seems to have felt no doubt of its soundness. It and its applications must have made great headway during the next 20 years, for Laplace takes for granted in a highly generalised form what Bayes tentatively wished to postulate in a special case.
The heavy isotope of hydrogen. The Liversidge Lecture, 1934
Authors: Fowler R. H.
Published: November 2008

When I was invited by Christ's College to deliver this year the Liversidge Lecture, it was indicated to me kindly but firmly that I was to talk about heavy hydrogen. I appreciate most deeply the honour you have done me by this invitation, especially by coupling my name (as it were) with this distinguished American newcomer to physical and chemical science. I cannot claim myself to be much more than an interested spectator of the rapid progress which has already been made in the study of this new substance, and the facts that I shall comment on (and the many others which time will force me to neglect) have been established by the collaboration of many workers—some of them our colleagues here, who would be much better qualified to speak at first hand about heavy hydrogen than I. Professor Liversidge laid it down in his deed of gift that his lecturer shall not deal with generalities or give a mere review of his subject, or give an instructional lecture suitable for undergraduates, but shall primarily try to encourage research and to stimulate himself and his audience to think, and to acquire new knowledge. In trying to fulfil his wish I shall not therefore catalogue all the new interesting properties of heavy hydrogen; this will mean, I fear, that I can make little or no further reference to its extremely exciting properties as an atomic projectile in disintegrating lithium and other nuclei, or to the equally exciting prospect that heavy water may have entirely distinct biological properties from ordinary water, being even lethal to many organisms, or to the wonderful possibilities of the new complex organic chemistry.
The lateral force on a keel and rudder
Authors: Daymond S. D. , Rosenhead L.
Published: November 2008

The following theoretical investigation of the two-dimensional flow of an inviscid fluid past a keel and rudder, and of the consequent lateral force, follows experiments performed by Prof. T. B. Abell in the Department of Naval Architecture of the University of Liverpool, and we wish to acknowledge our indebtedness to him for the information given in many discussions.
The approximate local monotony of measurable functions
Authors: Good I. J.
Published: November 2008

1. The description “approximate”, as applied to properties of a measurable function at a point, has come to mean, roughly speaking, “with the neglect of sets of measure zero”. If a function has, at a point x0, a certain property, such as differentiability or continuity, then it has the same property in the approximate sense, but not conversely.
Note on the method of molecular orbitals
Authors: Coulson C. A. , Rushbrooke G. S.
Published: November 2008

It is shown that the usual molecular orbital treatment for the mobile electrons in unsaturated hydrocarbon molecules is not always satisfactory, since it does not yield a self-consistent field. The conditions under which the field is self-consistent are analysed, and are shown to be satisfied in many cases.
The influence of magnetization on the benedicks homogeneous thermoelectric effect
Authors: Mo W. W. , Hsü Y. K. , Band W.
Published: November 2008

In a previous piece of work(1) numerous data were obtained on the dependence upon magnetizing field of the Benedicks e.m.f. in nickel wire. The work was restricted to a study of the effect of the initial increase of magnetizing field, no hysteresis loops being examined. The curves were most remarkable in that the change of e.m.f. depended upon the direction of the magnetizing field with respect to the temperature asymmetry. In the paper quoted, it was suggested that the curves should be resolved into two components, symmetrical and anti-symmetrical with respect to the field. In the present paper the former component is termed the normal component, because, as is shown later, a simple explanation of it can be given at once. We call the antisymmetrical component the anomalous effect.
The thermodynamics of crystal lattices
Authors: Bradburn Mary
Published: November 2008

The purpose of this paper is the determination as accurately as possible of the equation of state for the simplest stable crystal, the cubic face-centred lattice, and the comparison of the result with those obtained by rough approximations(1) in order to get an estimate of the reliability of the latter if applied to more involved problems where exact calculations are impossible.
Note on the theory of series (XXIII): On the partial sums of Fourier series
Authors: Hardy G. H. , Littlewood J. E.
Published: November 2008

1. In what follows f(θ) is a periodic function of L2,

is the Fourier series of f(θ), and

is the nth partial sum of T(θ). We denote by n(θ) any function of θ which is measurable, is finite p.p. †, and assumes non-negative integral values only, and by n(θ, H) an n(θ) none of whose values exceeds H. We shall sometimes write N for n(θ), and N(H) for n(θ, H), to simplify the set-up of formulae.
Scattering of charged mesons under the influence of radiation damping
Authors: Ma S. T. , Hsüeh C. F.
Published: November 2008

An alternative derivation of the integral equation of Waller, Heitler and Wilson is given on the basis of the stationary method of the perturbation theory. The previous theoretical treatment of the scattering of charged vector mesons by nuclear particles through the nuclear interaction is extended to pseudoscalar mesons. Numerical results calculated from the theory are given for the total and differential scattering cross-sections.
On the number of planes in hyperspace which satisfy a certain set of conditions
Authors: Todd J. A.
Published: November 2008

The problem here considered is a particular case of the more general problem of finding the number of [k]s, lying in a space of dimension n(n > k), which satisfy a certain number of conditions of the following type: viz. to meet a given [a0] in a point, a given, [a1] containing the [a0], in a line, etc., finally to lie in a given [ak] containing [ak−1]. The number of such conditions is such that there are just a finite number of [k]s which satisfy them all. The method which we employ is the one introduced by Schubert and commonly known as the “degeneration method” it is explained very briefly below.
A note on Bateman's expansion in Bessel functions
Authors: Bailey W. N.
Published: November 2008

1. The following proof of Bateman's expansion may be of interest since it avoids the use of Appell's hypergeometric functions of two variables, though it suffers from the defect that it presupposes a knowledge of the expansion.
A Geometrical Study of the Alternating and Symmetric Groups
Authors: de B. Robinson G.
Published: November 2008

Let a finite group Τ be represented as an irreducible group of order N of linear substitutions on n variables,

The variables may be chosen so that the substitutions of the group leave invariant the Hermitian form
The Relations between the Invariants of Two Surfaces in (1, n) Cyclic Correspondence
Authors: Frith Ronald
Published: November 2008

On a surface F′ an algebraic self-correspondence T of period n defines a cyclic involution In of sets of n points. Then if there exists a surface F whose points are in (1, 1) correspondence with the sets of In, the surfaces F, F′ will be said to be in (1, n) cyclic correspondence. The purpose of the present paper is to show that, when n is a prime number and with certain restrictions upon the united curve of the self-correspondence T, the irregularities of the surfaces F and F′ are equal.
The Stability of Viscous Fluid Flow under Pressure between Parallel Planes
Authors: Goldstein S.
Published: November 2008

If we consider, by the method of small oscillations, the stability of a viscous fluid flow in which the undisturbed velocity is parallel to the axis ofxand its magnitudeUis a function ofyonly (x, y, zbeing rectangular Cartesian co-ordinates), and if we assume that any possible disturbance may be analysed into a number (usually infinite) of principal disturbances, each of which involves the time only through a single exponential factor, then it has been proved by Squire, by supposing the disturbance analysed also into constituents which are simple harmonic functions ofxandz, and considering only a single constituent, that if instability occurs at all, it will occur for the lowest Reynolds number for a disturbance which is two-dimensional, in thex, yplane. Hence only two-dimensional disturbances need be considered. The velocity components in the disturbed motion will be denoted by (U+u, v). Since only infinitesimal disturbances are considered, all terms in the equations of motion which are quadratic inuandvare neglected. Whenuandvare taken to be functions ofymultiplied byei(αx−βi), the equation of continuity becomes

and the result of eliminating the pressure in the equations of motion then gives the following equation forv, where ν is the kinematic viscosity of the fluid:
On the Stability Problem in Hydrodynamics
Authors: Pekeris Chaim L.
Published: November 2008

The paper deals with the problem of the effect of two-dimensional first order disturbances on the linear and parabolic flows of a viscous fluid. If the laminar flow is in the direction x and is bounded by the planes y = ±h, and if the stream function of the disturbance is assumed to be of the form , it is found that, for small Reynolds numbers R (= Uh/v), C can be developed in the power series

The first term in (1) corresponds to a disturbance in still water and is known from Rayleigh's investigation of that problem; the succeeding C's are real and can be obtained by an application of Schrödinger's perturbation theory of wave mechanics. In the case of the parabolic flow, expressions for C0 and C1 were thus derived for both the symmetrical and anti-symmetrical types of disturbance.

The imaginary part of C, as represented by in this approximation, is found to be positive. When α is of the order of unity, the value of R which makes the second term of the first is about 50 for the symmetrical disturbance and 100 for the antisymmetrical disturbance. C0, the first term in the expression for the phase velocity, is less than 1 and increases with α, which implies a group velocity greater than the phase velocity.

In the case of the linear flow it is found that the C's with even subscripts vanish, implying a vanishing phase velocity. This verifies the assumption to that effect made by Southwell and Chitty, and it also suggests that the region of convergence of (A) is limited to the range of values of R covered by these authors. Expressions are derived for C1 and C3 and are evaluated for α=1. The formulae thus obtained for the damping constant give an initial decrease with R, which is not shown in the corresponding curve of Southwell and Chitty.

An alternative method of determining C is outlined in the last section. Use is made of the orthogonal system of the characteristic functions for the vorticity of a disturbance in still water. The equation determining C is reduced to an infinite determinant in which C occurs only in the diagonal elements. When R→0, the determinant can be evaluated and the roots C are found to coincide with the first two terms in (A).

Finally it is shown that the two-dimensional laminar flow is stable for values of R less than the smaller of the quantities 2α and α3.
The New Action Function and the Unitary Field Theory
Authors: Infeld L.
Published: November 2008

A new form of the variation principle is given using the sum T of the Lagrangian L and the Hamiltonian as an action function. This new form of the variational principle enables us to find a new special action function, which conserves the chief features of Born's theory while changing some of its former results. To a given charge correspond two static solutions with central symmetry, one giving a finite, the other an infinite energy. The potential of the one (light) particle is analogous to that in Born's theory while the potential of the other resembles a potential barrier. Also, by using the new action function, the symmetry between electric and magnetic fields ceases to exist.
Note on hyperelliptic surfaces and certain kummer surfaces
Authors: Baker H. F.
Published: November 2008

In 1907 Enriques and Severi published an extensive and fascinating account of hyperelliptic surfaces. In general a hyperelliptic surface is that expressed by the necessary relation connecting three meromorphic functions of two variables which have four columns of periods. Such functions arise naturally by associating the two variables, in accordance with Jacobi's inversion problem for hyperelliptic integrals of genus 2, with a pair of points of a hyperelliptic curve. When the primitive periods of the functions are those arising for the curve, and the set of three functions chosen is representative, in the sense that only one pair of (incongruent) values of the variables arises for given values of the functions, the surface is called by Enriques and Severi a Jacobian surface; but, if several sets of (incongruent) values of the variables arise for given values of the functions, say r sets, the surface is said to be of rank r. For example, when the three functions are all even, to each set of values of these there belong not only the values u, v of the variables, but also the values −u, − v, and r is thus even, being 2 at least, as in the case of the Kummer surface. In the paper referred to, many cases in which r > 1, corresponding to particular hyperelliptic curves possessing involutions of order r, are worked out. In general the method followed consists in arguing, from the character of the associated group of order r, to the character and equation of the hyperelliptic surface Φ of rank r; and from this the Jacobian surface F is inferred upon which there exists an involution of sets of r points, the surface Φ being the representation of this involution. The argumentation is always beautiful, but often not very brief. The hyperelliptic surfaces for which the primitive periods of the functions are not those of a hyperelliptic curve are also shown in the paper to arise from involutions on the Jacobian surface; with these I am not here concerned.
Note on a theorem enunciated by Todd
Authors: Du Val P.
Published: November 2008

J. A. Todd has enunciated the following theorem:

“If the canonical adjoints ∑ of an irreducible curve Γ contain fixed parts, then, if Γ is not elliptic, all the fixed curves are simple, rational, determined uniquely by the base points, which present independent conditions for them, and have no free intersections with each other or with the variable part of the system ∑.”
The disintegration of nitrogen by slow neutrons
Authors: Burcham W. E. , Goldhaber M.
Published: November 2008

The disintegration of nitrogen by slow neutrons has been studied in photographic emulsions of different sensitivity, which enable an unambiguous distinction to be made between the emission of α-particles and protons. Evidence has been obtained that the disintegration takes place according to the reaction

with a cross-section of about 10−24 cm.2
Self-oscillation in a retroacting thermal conductor
Authors: Turner L. B.
Published: November 2008

In automatic temperature-controlling apparatus a temperature-sensitive element, commonly called a thermostat, serves to increase or decrease a supply of heat to the body or oven whose temperature is to be prevented from changing. The thermostat and heating elements are commonly both electrical, the thermostat comprising a “master coil” whose resistance changes with temperature, the heating element consisting of a “slave coil” carrying the heating current. The temperature of the master coil is made to control the current in the slave coil, usually by the interposition of some form of relay with the appropriate translating and amplifying apparatus. Since every form of relay exhibits “backlash”—the critical values of the operative signal at make and at break are not exactly equal—such a system necessarily oscillates or “hunts” through a range at least as great as the backlash of the relay; and since further there is thermal separation between the master coil and the slave coil, the hunting has a range exceeding the backlash of the relay. It is a common experience with such thermostatic apparatus that, owing to this action, continued improvement towards constancy of temperature is not attainable by increasing the delicacy with which the temperature of the master coil controls the current in the slave coil.
The postulation of a multiple surface
Authors: Roth L.
Published: November 2008

1. In the present note we obtain the postulation of a multiple surface of S4 for primals of sufficiently high order; the method is that previously used by Campedelli to determine the postulation of a multiple curve of S3.
A critical examination of Benedicks' homogeneous electro-thermal effect
Authors: Band William
Published: November 2008

In his work on mercury, Benedicks (1) observed that the small temperature peak in the neighbourhood of the middle of the long tube containing the conductor was displaced in one direction or the other along the tube according to the direction of flow of the electric current by which the conductor was heated. Normally this effect increased with the cube of the current intensity, and was explained as due to Thomson heat. If the temperature was made as uniform as possible by means of controlled heaters at the ends of the tube, the effect became of opposite sign and increased linearly with the current. This was called the “electro-thermal effect” and was explained as due to an actual change of thermal equilibrium by the electric current, or the production of a temperature gradient by a flow of electricity.
Note on the derivation of fluctuation formulae for statistical assemblies
Authors: Bartlett M. S.
Published: November 2008

1. A familiar device in the study of statistical distributions is to form the moment-generating function

where the bar denotes averaging over all values of the statistical variate x. The moments μr of x are the coefficients of αr/r!, and the derived coefficients in the expansion of K ≡ log M are termed the semi-invariants kr. In particular,

and, for the normal (Gaussian) law,

we have the simple formula