bluuebunny/crossref_metadata_2025_split huggingface dataset:
Improvements to the “scale of two” thyratron counter
Authors: Lewis W. B. 
Published: November 2008

An improved circuit is described for a two thyratron counting unit such as is used in the “scale of two” thyratron counter. The advantages which result are that the minimum time of resolution is reduced and a source of occasional failure (“jamming”) of the counter is removed. Any type of impulse may be counted and, with selected thyratrons, impulses separated by only 0.00015 sec. may be reliably counted as separate. This makes possible a counting speed of 8000 random impulses per minute with a probable loss of only 2%.The modification introduced into the circuit used in the original “scale of two” counter is that small condensers (0·0015 μF max) are cross connected between the grids and anodes of the pair of thyratrons, and the extinguishing condenser connected between the anodes is made small (0·02 μF max). The anode resistances are 10,000 ohms and grid leaks 100,000 ohms for the values of capacity mentioned. A valve amplifying stage is introduced, to pass the impulses from the modified circuit of the first pair to the second pair of thyratrons. For greater convenience the mechanical counting meter operated by the third pair is modified to count in multiples of eight so that no multiplication of its reading is necessary.
On Ramanujan's Arithmetical Function Σr, s(n)
Authors: Wilton J. R. 
Published: November 2008

Let σs(n) denote the sum of the sth powers of the divisors of n,and let , where ζ(s) is the Riemann ζfunction. Ramanujan, in his paper “On certain arithmetical functions*”, proves that, ifandthen , whenever r and s are odd positive integers. He conjectures that the error term on the right of (1·1) is of the formfor every ε > 0, and that it is not of the form . He further conjectures thatfor all positive values of r and s; and this conjecture has recently been proved to be correct.
Integrals of Systems of Ordinary Differential Equations
Authors: Cherry T. M. 
Published: November 2008

Letbe a system of differential equations in which X1, … Xn are analytic functions independent of t, expansible in convergent series of powers of x1, … xn. Suppose further that not all the functions X1, … Xn vanish when x1 = … = xn = 0. It will be shown in §§ 1–3 that these equations possess (n — 1) independent integrals independent of t, expansible in convergent series of powers of x1, … xn; i.e. functionssuch thatidentically. In § 4 it is shown how the series for ϕ1,…ϕn−1 may be most directly constructed, and in § 5 is briefly considered the corresponding problem when the origin is a singular point of the equations, i.e. when the expansions of X1,…Xn all begin with terms of the first or higher degrees.
On the Convergence of a certain Multiple Series
Authors: Bailey W. N. 
Published: November 2008

The theorem given by Professor Mordell in the preceding paper is capable of a slight generalisation. I first prove that the serieswhere α, β, γ, … are positive, converges only when κ > Σ (a′/α), the summation referring to those exponents a′ of a, b, c, … for which a ≥ 0; or, if there are no such exponents, the series converges only when κ > a/α, k > b/β, …
A transformation of a formula of Sommerfeld
Authors: Thomas L. H. 
Published: November 2008

Sommerfeld has shown that the electromagnetic field set up by a Hertzian oscillator placed on the infinite plane surface separating two media normal to that surface may be obtained from the Hertzian function of cylindrical coordinates r, z,where the suffixes 1, 2 refer to the two media occupying the regions z>0, z<0,andwhere ɛ1, μ1, σ1 are the dielectric constant, permeability, and conductivity (in Heaviside units) of the first medium, k2, k2 like expressions for the second, c is the velocity of light, and n the frequency in radians per second; +(λ2−k2)½ is taken to have its real part positive.
On the Formation of a Gaseous Helide of Radium Active Deposit
Authors: Morrison D. M. 
Published: November 2008

In the light of the hypothesis that the helium atom can exist in a hydrogen-like form in which one electron is relatively far removed from the nucleus with respect to the other, a series of experiments have been carried out with a view to determine whether or not helium could be obtained in the form of a compound similar to certain known hydrides. The mass of evidence points to a positive result. A gaseous compound was shown to form when excited helium was passed over a strong source of the active deposit of radium. Compound formation was proved by counting scintillations caused by the transportation of the radioactive material from the discharge tube into a bulb containing a zinc sulphide screen. That the effect observed in the bulb was due to the presence of a gaseous radioactive helide and not to radium emanation or solid particles in suspension was definitely proved; that it was due to hydrogen impurities in the helium was shown to be very unlikely. The compound in hydrogen was more efficiently condensed by a carbon dioxide-ether cooling mixture than that in helium. The catalytic effects of tap grease and mercury vapour on the reaction were noted.In conclusion the writer wishes to thank Professor Sir Ernest Rutherford, O.M., P.R.S., for his interest in the work; Dr J. Chadwick, F.R.S., for advice and helpful suggestions; Mr J. A. Ratcliffe, for the wireless set used in producing the electrodeless discharge; Mr Crowe, for the preparation of the radioactive sources; and Mr Brown, for help in counting scintillations.
On some series of functions
Authors: Paley R. E. A. C. 
Published: November 2008

We now consider some properties of “almost all” serieswhere the numbers vary independently in the interval (0, 1). What we need is a definition of measure in the space of sequences or, what is the same thing, in the spaceof infinitely many dimensions. Such a definition has been given by Steinhaus. He defines a correspondence between the cube (7.2) and the interval (0,1) in the following way:
Some calculations on the diffusion of slow neutrons in hydrogenous media
Authors: Westcott C. H. 
Published: November 2008

The paper describes calculations on the distribution in space of slow neutrons based on the application of diffusion laws to their motions. The effects of using spheres of different sizes, and of changing the composition of the hydrogenous slowing-down medium, are discussed. Curves are given which can be compared with the results of the experiments on the slowing down in different media carried out by the author in collaboration with Mr T. Bjerge, and a revision of the calculations based on these experiments leads to a value of 6 × 10−25 cm.2 for the total absorption cross-section of the water molecule, and indicates that most of this absorption must be attributed to the hydrogen atoms.The author desires to acknowledge the receipt of a Senior Research Award from the Department of Scientific and Industrial Research.
The interpretation of ionization measurements in gases at high pressures
Authors: Kara-Michailova Elizabeth , Lea D. E. 
Published: November 2008

When γ-radiation passes through a gas it ionizes by means of fast electrons which produce clusters of secondary ionization at intervals along their paths. At high pressures a considerable amount of recombination of ions takes place in these clusters; in the present paper a theory is described which enables the proportion of ions escaping recombination to be calculated as a function of the gas pressure and collecting field. A review of the available experimental data concerning the variation of ionization current with pressure and collecting field is given, and it is shown that the predictions of the cluster recombination theory are in satisfactory agreement with these experimental data.Jaffé has given a theory of initial recombination valid for a columnar distribution of ionization, and has shown that this theory is in agreement with experiments in which α-particles produce the ionization. A number of authors have applied Jaffé's equations to ionization produced by fast electrons regardless of the initial localization of the ions in clusters. It is shown in the present paper that such measure of agreement with experiment as is obtained by this procedure is only obtained at the expense of assigning incorrect values to certain known constants. Also in the case of X- and γ-rays the columnar theory predicts a variation of the proportion of recombination with the wave-length of the radiation which is much too rapid. It is further shown that the method based on Jaffé's equations which is used by Clay and his colleagues to extrapolate experimental ionization currents at finite collecting fields to saturation currents at infinite fields is liable to systematic error in a direction likely to lead to the deduction of a spurious wall effect or the exaggeration of an existing wall effect.
On the Product of Three Homogeneous Linear Forms. IV
Authors: Davenport H. 
Published: November 2008

Let L1, L2, L3 be three homogeneous linear forms in u, v, w with real coefficients and determinant 1. Let M denote the lower bound offor integral values of u, v, w, not all zero. I proved a few years ago (1) thatmore precisely, thatexcept when L1, L2, L3 are of a special type, in which case If we denote by θ, ø, ψ the roots of the cubic equation t3+t2-2t-1 = 0, the special linear forms are equivalent, by an integral unimodular linear transformation, to(in any order), where λ1,λ2,λ3 are real number whose product is In this case, L1L2L3|λ1λ2λ3 is a non-zero integer, and the minimum of its absolute value is 1, giving
The vapour pressure equation of solutions and the osmotic pressure of rubber
Authors: Miller A. R. 
Published: November 2008

A number of errors have unfortunately been made in the last section of the above paper.(1) ‘log’ should be deleted from equation (44).(2) The simplification of equation (45) which was presented neglects some important terms, with the result that the later equations give only the limiting values of osmotic pressure and vapour pressure at infinite dilution. The section following equation (45) must therefore be ignored.
The hydrogen atom and the classical theory of radiation
Authors: Eliezer C. Jayaratnam 
Published: November 2008

The paper considers the application of Dirac's classical theory of radiating electrons to consider the straight line motion of an electron towards a fixed proton and the straight line motion of two oppositely charged electrons. The paper also gives a discussion of the probable solutions in two- and three-dimensional motions of an electron moving round a fixed proton. In all these cases there appears to be no solution which would permit a collision between the two particles.I take this opportunity to express my deepest gratitude to Prof. Dirac for his patient guidance and supervision, and also to Christ's College for a scholarship.Note added. In the problem of the rectilinear motion considered in section (2), if we take the charges on the particles to be of like signs, then we find that the equations of motion have a solution which corresponds to a collision. In place of equation (4) we haveIt can be easily verified that under suitable initial conditions there is a solution with x→0 and y→∞. Near x=0 the solution is approximately . Thus a collision is possible for like charges and not possible for unlike charges. The result, in both cases, is the opposite of what we should expect from elementary physical considerations.
The sputtering of platinum in hydrogen and deuterium
Authors: Ditchburn R. W. , Roulston K. I. 
Published: November 2008

The rates of sputtering in hydrogen and deuterium have been compared using ions drawn from a low voltage arc accelerated through a known potential. The rates are equal within a few per cent.One of us (K.I.R.) is indebted to the Industrial Research Council of Eire for a maintenance grant.
The ionization method of measuring neutron energy
Authors: Gray L. H. 
Published: November 2008

The absolute measurement of neutron flux density by ionization methods is discussed.The relation Eυ = JυW ρ between the energy Eυ absorbed per unit volume of a solid medium and the ionization Jυ produced in a gas-filled cavity in the medium, the average energy W expended by the ionizing particles in producing a pair of ions in the gas, and ρ the stopping power of the solid relative to the gas for these particles, is critically examined from the point of view of the measurement of the rate of absorption of neutron energy at any point in a hydrogenous medium. The accuracy attainable is assessed, in so far as it depends on our knowledge of the quantities ρ and W.
On lattice points in the domain |xy | ≤1, | x + y | ≤√5, and applications to asymptotic formulae in lattice point theory (II)
Authors: Mahler K. 
Published: November 2008

I. Lattice points in the domain | x |α + | y |α ≤ 1Theorem 4. Let G be the star domainwhere α > 0. Then, when α tends to zero,Proof. The linear substitutionchanges G into the similar domainand so.
Further significance tests
Authors: Jeffreys Harold 
Published: November 2008

In a previous paper (afterwards referred to as Paper I) tests have been given for the significance of some quantities found statistically. The results are given in the formP(q| θh)/P(˜q|θh); herehdenotes the previous knowledge and θ the experimental evidence used, whileqis the hypothesis that all the variations outstanding can be attributed to accidental error or random variation, and ˜qthe hypothesis that at least part of them is systematic. It has been supposed in the analysis thatqand ˜qare equally probable on the informationh; but if they are not, the only alteration is that the ratios evaluated now representIf successive batches of relevant information are available the total effect on the probability ofqcan therefore be got by multiplying the values ofgiven by the investigations separately. In each case the assumption thatqhas prior probability ½ is really a practical working rule rather than a statement of fact.
The “closest” estimates of statistical parameters
Authors: Pitman E. J. G. 
Published: November 2008

A definition of “closer” and “closest” as applied to estimates of statistical parameters is given, and it is shown that we can sometimes prove that estimates properly derived from sufficient statistics are the closest possible.The scaling of a gamma distribution, and the location and scaling of an exponential distribution and of a rectangular distribution are discussed in detail, and the closest estimates of the parameters obtained.
Further aspects of the theory of multiple regression
Authors: Bartlett M. S. 
Published: November 2008

This paper may be regarded as a sequel to a previous papers(1) in these Proceedings. The vector and matrix notation of that paper used for a statistical sample is systematized somewhat further, so that while a sample S refers as before to the matrix of nm values (a sample of m observations in one variate only being a row vector), we writefor the linear regression formula between the dependent and independent variates into which a sample is supposed partitioned (in place of equation (12) of (1)). More generally, a third submatrix S0 is partitioned off, and its effect eliminated (corresponding to equation (13) of (1)), but without loss of generality we assume that S2 in equation (1) above can always stand for S2.0 if necessary.
The use of minimum χ2 as an approximation to the method of maximum likelihood
Authors: Jeffreys Harold 
Published: November 2008

Suppose that we have a set of observed frequencies classified in groups, the number in a typical group being nr, and that we wish to estimate the expectations in the groups according to some law with adjustable parameters. If these expectations are mr we takeΣ denoting summation over the groups, and according to the law in question the probability that one observation should come in the rth group is mr/N. The joint probability, for given mr, that there should be n1 observations in the first group, n2 in the second and so on, is therefore
The mean value of the fluctuations in pressure and pressure gradient in a turbulent fluid
Authors: Green A. E. 
Published: November 2008

1. Taylor has shown (1) that two characteristic lengthsλ and λη may be defined for turbulent fluid motion. The length λ, which is connected with the dissipation of energy, is, for isotropic turbulence, given bywhere is the mean rate of dissipation of energy per unit volume and represents the mean square value of any component of velocity. The length λη can be defined in terms of thuswhere For isotropic turbulence Taylor assumed thatwhere B is a constant. Since the turbulence is isotropic,and so, from (1), (2), (3) and (4) we have
The rates of unimolecular reactions in gases
Authors: Slater N. B. 
Published: November 2008

The molecule of a dissociating gas is treated as a dynamical system with n normal modes of vibration with incommensurable frequencies. The assumption that the molecule dissociates when one coordinate attains a critical high value leads to a unimolecular velocity constant expressed as an integral; analytical expressions are derived for bounds to its value. Calculations in a simple case indicate that the velocity constant is approximately of the formThis is in general agreement with experiment if the constant W is identified with the corresponding empirical energy constant.
Base conditions and covariant systems in an algebraic threefold
Authors: Semple J. G. 
Published: November 2008

If F is a free linear system of surfaces in an algebraic threefold V which is either non-singular or possesses only normal singularities, then F has Jacobian and adjoint surfaces, J2(F) and A2(F), and Jacobian and adjoint curve systems, J1(F) and A1(F), such thatwhere X2, X2 are the canonical systems of surfaces and curves on V, and X1(F) is the canonical system of curves of F. The imposition of base elements (points or curves) Ei, of assigned multiplicities λi, on F defines a system F1 which we may represent formally by the equationand it is natural to enquire how the Jacobian systems of F1 differ from those of F, and how we may define adjoint systems A2(F1) and A1(F1) which cut on F1 its canonical curves and sets respectively.
Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions
Authors: Rankin R. A. 
Published: November 2008

In this and the succeeding paper I solve two problems suggested by Prof. Hardy, namely (1) that of proving that Ramanujan's functionhas no zeros on the line and (2) that of finding an asymptotic formulawhere A is a constant. I also prove similar results concerning the coefficients of general modular forms. I am indebted to Prof. Hardy and Mr Ingham for various suggestions, and in particular to Mr Ingham's paper, “A note on Riemann's ζ-function and Dirichlet's L-functions”.
An apparent inconsistency arising from tests of significance based on fiducial distributions of unknown parameters
Authors: Yates F. 
Published: November 2008

It has been observed that the Behrens and Fisher test of the difference of the means of two samples gives a smaller percentage of significant results than might be expected on the analogy of the ordinary t test with a pooled estimate of variance. The cause of this apparent anomaly is explained, and it is shown that the criticisms of the test to which the anomaly has given rise have their origin in (a) neglect of the relevant information provided by the estimated values of the variances, and (b) an insufficient appreciation of the fiducial basis of all tests of significance (including the ordinary t test) on small samples.It is pointed out that Sukhatme's table (constructed for the Behrens and Fisher test) also provides a test for the weighted mean of the means of two sets of observations, concerning whose relative accuracy no prior knowledge is available.
The influence of radiation damping on the scattering of mesons
Authors: Heitler W. , Peng H. W. 
Published: November 2008

I. It is shown that a general argument can be given for omitting the diverging parts of any quantized field interacting with a particle. The field equations thus freed from all singularities still contain the reaction of the field on the particle to the same extent as the reaction can be derived classically from the conservation of energy. The new field equations can be solved generally in terms of a simultaneous set of integral equations and do not give rise to any fundamental difficulties whatsoever.II. The new field equations are applied to the multiple processes of the meson theory which occur, for instance, when a meson collides with a nuclear particle, the meson splitting up into a number n of secondary mesons. The cross-sections γn are worked out for n = 1, 2, 3, 4. γn has a maximum at some energy εn which increases with n. For ε ≫ εn, γn decreases like ε−2n−4 except for n = 1, when γ1 ∞ ε−2. γ1 (ordinary scattering) is larger than all the other γ's at all energies, but the probability of high multiplicities is comparable with that for low multiplicities or even greater in some energy regions.
Relativistic dynamics of a spinning magnetic particle
Authors: Mathisson Myron 
Published: November 2008

The author's general variational method is applied to the case of a particle for which second moments are important but third and higher moments are negligible. Equations of motion are obtained for the angular momentum and for the centre of mass, equations (12·35) and (12·41), with arbitrary external forces X.The angular forces are then calculated for a charged particle with electric and magnetic moments moving in a general electromagnetic field, on the assumption that the effect of a certain part of the energy tensor, Tiii of (15·17), is negligible. This leads to the equations of angular motion, (17·13), from which it is inferred that, in order that the magnitude of the angular momentum may be integrable, the angular momentum, electric and magnetic moments must all be parallel in a frame of reference in which the particle is instantaneously at rest.The linear forces are then calculated for the case of no electric moment, leading to the equations for linear motion (18·10). From these it is inferred that, in order that the mass may be integrable, the ratio of the magnetic moment to the angular momentum must be constant.
On multiple curves. I
Authors: Hodge W. V. D. 
Published: November 2008

Let γ be an irreducible variety of dimension d, and let In denote an involution of order n on Γ. The image of In is a variety C, also irreducible and of dimension d, and Γ is said to be a multiple of C. Among the possible multiples of a given variety C there is a class of particular interest; this arises when the involution In possesses the following properties:(1) it is unramified, that is, in every set of the involution the n points are all distinct;(2) it is generated by an Abelian group G of order n of birational transformations of Γ into itself. Given any point P of Γ the n transformations of G transform P into n points P1 = P, P2, …, Pn which constitute a set of In. The two conditions together imply that the function field κ of Γ is an unramified Abelian Galois extension of the function field k of C, and this shows that the study of the particular class of multiple varieties—which we may call normal multiple varieties when it is necessary to give a name to them—is analogous to the study of a well-known branch of the theory of numbers.
A new type of boundary-value problem in hyperbolic equations
Authors: Chandrasekhar S. 
Published: November 2008

In developing the theory of radiative transfer in expanding atmospheres (Chandrasekhar, 1945 a, b) the author has recently encountered certain novel types of boundary-value problems in hyperbolic equations which appear to merit consideration for their own sake. As related to the equationthe boundary-value problems which occur are of the following general typei.e. along AB in Fig. 1. Along AD (x = 0 and 0 ≤ y ≤ l2) and BC (x = l1, 0 ≤ y ≤ l2) we are further given thatandwhere φ(y) and ψ(y) are two known functions. The problem is to solve equation (1) in the rectangular strip ABCD satisfying the stated boundary conditions.