bluuebunny/crossref_metadata_2025 huggingface dataset:
The Acceleration of β-particles in Strong Electric Fields such as those of Thunderclouds
Authors: Wilson C. T. R.
Published: November 2008

As is well illustrated by photographs of their tracks, the majority of β-particles of given initial velocity bave approximately the same range, measured along the path. When the track shows a visible branch, indicating that an exceptionally close encounter has resulted in the ejection of an electron with kinetic energy amounting to an appreciable fraction of that of tbe primary particle, the main track is naturally of less tban tbe average range. A diminution of range may also be expected to follow such of the nuclear deflections as result in the emission of X radiation; this decrease in range indeed affords a possible method of measuring the energy of the radiation emitted in individuai cases and so testing theories such as that of Kramers as to the nature and origin of the continuous X-ray spectrum.
The Determination of the Frequency of an Alternating Current Supply by the Vibrations of Rods
Authors: Searle G. F. C.
Published: November 2008

One method of measuring frequency depends upon resonance. If a body of unit mass, subject to elastic constraints and experiencing a damping resistance proportional to its speed, be acted on by a periodic force F sin pt, the equation of motion may be written

The amplitude of the steady vibration ultimately set up is

the maximum value occurring when p2 = q2 − 2s2. If the motion, in the absence of the dri ving force F sin pt, be only slightly damped, s2 is very small compared with q2, and little error will be made if the value of p giving maximum amplitude be taken as equal to q. When s/q is small, the resonance is “sharp,” i.e. a small discrepancy between p and q, or, more strictly, between p and (q2 − 2s2)½, causes the amphtude to fall far below its maxìmum value. Thus, when the damping is small, maximum amplitude, f or variation of p, indicates that p is very nearly equal to q.
Generalized Köthe function spaces. I
Authors: Phuong-Các Nguyen
Published: November 2008

The idea of constructing a space of functions taking values in a locally convex spaceEfrom a linear spaceof scalar valued functions is well known. We can, for example, define a spaceconsisting of allE-valued functions φ(t) such thatfor all elements e′ of the dualE′ ofE. Besides this construction there are others which arise in special cases. This idea has been used to obtain integrals of vector-valued functions (compare (2), Chapter III, § 4). Schwartz has also used it in his paper on differentiable vector-valued functions (9) whose main result is the famous kernel theorem, as well as in introducing vector-valued distributions. It is natural to expect that the space of vector-valued functions obtained will inherit some properties of the function spaceand the vector spaceE. Therefore one usually starts from some function spacewhich has interesting properties.
On the slow broadside motion of a thin disc along the axis of a fluid-filled circular duct
Authors: Shail R. , Norton D. J.
Published: November 2008

The study of the slow motion of a solid body through a viscous fluid is of importance in many fields. We may instance viscometry and chemical engineering as two subjects in which a quantitative understanding of such motions is essential. Thus, in recent years systematic attempts have been made to calculate the drag force and couple on slowly sedimenting particles in a bounded viscous fluid. Using the ‘method of reflexions’ Brenner (1) has obtained formulae which give, for a wide class of sedimenting particles, an estimate of the effect of the container walls on the drag force, the error being O(c/l)3, where c is a characteristic particle dimension and l the distance from the centre of the particle to the nearest boundary wall. The reader is referred to the recent book by Happel and Brenner (2) for a complete account of these and other pertinent investigations; also to Williams (3) for an alternative integral equation approach to these results.
On axially symmetric waves
Authors: Craggs J. W.
Published: November 2008

The methods of papers I and II (1), (2) are applied to a problem in classical elastodynamics.
On well-quasi-ordering finite trees
Authors: Nash-Williams C. St. J. A.
Published: November 2008

A new and simple proof is given of the known theorem that, if T1, T2,… is an infinite sequence of finite trees, then there exist i and j such that i < j and Ti is homeomorphic to a subtree of Tj.
On a class of renewal functions
Authors: Daley D. J.
Published: November 2008

Renewal processes in discrete time (or as they are commonly termed, recurrent events) are appropriately described byrenewal sequences{un} which aregeneratedby discrete distributions, according to the equation

Any two renewal sequences {u′n}, {u″n} define another renewal sequence {un} by means of their term-by-term product {un} = {u′nu″n}, for the joint occurrence of two independent recurrent events ℰ′ and ℰ″ is also a recurrent event. Considering a renewal process in continuous time for which we shall suppose a frequency functionf(x) of the lifetime distribution exists, so that a renewal density exists, the analogous property would be that for two renewal density functionsh1(x) andh2(x), the functionh(x) =h1(x)h2(x) is a renewal density function. A little intuitive reflexion shows that whileh(x)dxhas a probability density interpretation, this is not in general true ofh1(x)h2(x)dx. It is not surprising therefore to find in example 1 a case where the product of two renewal densities is not a renewal density. Example 2, on the other hand, shows that in some cases it is true, and taken together with example 1, there is suggested the problem of characterizing the class of renewal densitiesh(x) for which αh(x) is a renewal density for all finite positive α and not merely α in 0 < α ≤A< ∞. In turn this characterization enables us to define a classof renewal densities for whichh1(x) andimply that.
The Propagation of Electromagnetic Waves in a Refracting Medium in a Magnetic Field
Authors: Hartree D. R.
Published: November 2008

The equations of propagation of electromagnetic waves, simple harmonic in time, in an optically anisotropic stratified medium are obtained from the treatment of the refracted wave as the resultant of the incident wave and wavelets scattered by the elements of volume of the medium, and are reduced to a simple form.

The primitive property of the medium, from which the other optical properties are derived, is the scattering tensor, relating the induced dipole moment per unit volume to the applied electric field.

The relation between the dielectric tensor (corresponding to the dielectric constant of an isotropic medium) and the scattering tensor is obtained.

A medium consisting of classical oscillators in an external magnetic field is then considered, the scattering tensor and dielectric tensor are evaluated for such a medium, and finally a formula for the refractive index is obtained.

For an ionised medium the formula differs from that obtained by Goldstein; the difference is due to the inclusion in the present treatment of a term omitted by Goldstein; the significance of this term is discussed, and its inclusion justified.

Taking this term into account makes an important difference to the properties of the medium for long waves; an example is given.
A Geometrical treatment of the Correspondence between Lines in Threefold Space and Points of a Quadric in Fivefold space
Authors: Turnbull H. W.
Published: November 2008

§ 1. The six Plücker coordinates of a straight line in three dimensional space satisfy an identical quadratic relation

which immediately shows that a one-one correspondence may be set up between lines in three dimensional space, λ, and points on a quadric manifold of four dimensions in five dimensional space, S5. For these six numbers pij may be considered to be six homogeneous coordinates of such a point.
On the representation of the modular group of order ½p (p2 – 1) as a group of linear substitutions on ½ (p – 1) symbols, ichen p is a prime of the form 4n + 3
Authors: Burnside W.
Published: November 2008

The actual form of the set of linear substitutions on 3 symbols, which gives a representation of the simple group of order 168, has been known for many years.

In these Proceedings (vol. 20, pp. 247–9) I have determined the form of the set of linear substitutions on 5 symbols which gives a representation of the simple group of order 660.
Some enumerative results for elliptic quartic curves
Authors: Todd J. A.
Published: November 2008

In a recent paper Mr W. G. Welchman has devised an extremely simple method for determining the number of elliptic quartic curves in space which pass through p given points and have 8–p assigned lines as chords. In the present note this method will be applied, with certain necessary extensions, to some cases in which the curves are subject to the condition of meeting certain lines in one point only, in addition to conditions of the previous type. The method in principle is perfectly straightforward, but circumstances arise which make its application somewhat limited.
A mechanical method for the solution of second-order linear differential equations
Authors: Bullard E. C. , Moon P. B.
Published: November 2008

A mechanical method of integrating a second-order differential equation, with any boundary conditions, is described and its applications are discussed.
Energy problems in the continuous spectra of β-rays
Authors: Chalmers J. A.
Published: November 2008

By their measurements on the heating effect of the β-rays of Radium E, Ellis and Wooster(1) have proved that the continuous spectra of β-rays represent the actual β-rays of disintegration. Since the β-rays in the continuous spectrum have a wide range of energies, there arise problems which can best be understood by considering a particular series of disintegrations, which we may regard as typical; the most suitable series to consider is the following:

The main problem that arises is to reconcile the facts that the β-rays from Thorium B and Thorium C are emitted with a wide range of energies, while before, in Thorium A, and after, in Thorium C and Thorium C′, we have bodies which emit α-rays which are homogeneous, or, at most, consist of a few homogeneous groups [Rosenblum(2)].
A measurement of Boltzmann's constant by means of the fluctuations of electron pressure in a conductor
Authors: Ellis H. D. , Moullin E. B.
Published: November 2008

This paper describes a repetition of J. B. Johnson's determination of Boltzmann's constant by observing the thermal agitation of electrons in a conductor. The principle of the measurement is described and the various sources of error are examined separately. The attenuator system of the calibrating circuit is discussed in detail and mutual agreement established between the three different attenuating systems. The valve amplifier had a maximum response which could be arranged to occur at any frequency between 1000 and 4000 cycles/sec. The mean value of 52 determinations of Boltzmann's constant (with the amplifier set to respond to one of four frequencies within the range 1 to 4 k.c./s.) was found to be 1·361 × 10−16 ergs per degree, with a probable error of 1·42%: this value is 0·8% less than the accepted value of 1·372 × 10−16 ergs per degree. The investigation developed out of a research initiated with a directly practical objective; we desire to express our thanks to the Hebdomadal Council of the University of Oxford, who by a grant to one of us have enabled this section of the work to be completed.
The groups of symmetries of the regular polytopes
Authors: Todd J. A.
Published: November 2008

The object of the present paper is the geometrical study of the groups of rotation and reflexion of the regular polytopes in higher space, and the extension to these configurations of known results in the cases of the ordinary regular polyhedra. It will appear from the work that the groups can be defined abstractly in terms of a certain number of operations, the relations connecting which have a particularly simple form. For a polytope in n dimensions this number is n − 1, if we consider the group composed simply of the rotations of the polytope, while if we consider the extended group, which includes the possible reflective symmetries of the polytope, n operations suffice. It appears, further, that with one exception all the groups so obtained possess the property that their operations are expressible in terms of two, so that the entire group can be generated by two operations. The relations connecting these, however, are in general complicated, and the symmetrical forms involving more operations are more convenient to use.
A note on inverse probability
Authors: Haldane J. B. S.
Published: November 2008

The problem of statistical investigation is the description of a population, or Kollektiv, of which a sample has been observed. At best we can only state the probability that certain parameters of this population lie within assigned limits, i.e. specify their probability density. It has been shown, e.g. by von Mises(1), that this is only possible if we know the probability distribution of the parameter before the sample is taken. Bayes' theorem is based on the assumption that all values of the parameter in the neighbourhood of that observed are equally probablea priori.It is the purpose of this paper to examine what more reasonable assumption may be made, and how it will affect the estimate based on the observed sample.
On the operational solution of linear difference equations whose coefficients are expressible by factorial series
Authors: Milne-Thomson L. M.
Published: November 2008

In a former paper the application of Boole's operators π and ρ to linear difference equations whose coefficients are rational functions was discussed. Reference should be made to this paper for the definitions of π and ρ and for Theorems I–V. In the present paper a more general type of equation is considered. Two new theorems are proved and used to obtain the formal solution and to establish the convergence.
On the product of two Legendre polynomials
Authors: Bailey W. N.
Published: November 2008

1. As long ago as 1878 Neumann gave a formula expressing the product of two Legendre polynomials as a sum of such polynomials. In the same year Adams gave an inductive proof, and obtained the result in the form
The calculation of atomic fields
Authors: Thomas L. H.
Published: November 2008

The theoretical calculation of observable atomic constants is often only possible if the effective electric field inside the atom is known. Some fields have been calculated to fit observed data but for many elements no such fields are available. In the following paper a method is given by which approximate fields can easily be determined for heavy atoms from theoretical considerations alone.
A mathematical theory of Natural and Artificial Selection
Authors: Haldane J. B. S.
Published: November 2008

Expressions are found for the progress of slow selection in a Mendelian population where generations overlap. The changes are very similar to those which occur when generations are separate.
The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods
Authors: Hartree D. R.
Published: November 2008

The paper is concerned with the practical determination of the characteristic values and functions of the wave equation of Schrodinger for a non-Coulomb central field, for which the potential is given as a function of the distance r from the nucleus.

The method used is to integrate a modification of the equation outwards from initial conditions corresponding to a solution finite at r = 0, and inwards from initial conditions corresponding to a solution zero at r = ∞, with a trial value of the parameter (the energy) whose characteristic values are to be determined; the values of this parameter for which the two solutions fit at some convenient intermediate radius are the characteristic values required, and the solutions which so fit are the characteristic functions (§§ 2, 10).

Modifications of the wave equation suitable for numerical work in different parts of the range of r are given (§§ 2, 3, 5), also exact equations for the variation of a solution with a variation in the potential or of the trial value of the energy (§ 4); the use of these variation equations in preference to a complete new integration of the equation for every trial change of field or of the energy parameter avoids a great deal of numerical work.

For the range of r where the deviation from a Coulomb field is inappreciable, recurrence relations between different solutions of the wave equations which are zero at r = ∞, and correspond to terms with different values of the effective and subsidiary quantum numbers, are given and can be used to avoid carrying out the integration in each particular case (§§ 6, 7).

Formulae for the calculation of first order perturbations due to the relativity variation of mass and to the spinning electron are given (§ 8).

The method used for integrating the equations numerically is outlined (§ 9).
The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part II. Some Results and Discussion
Authors: Hartree D. R.
Published: November 2008

The methods of solution of the wave equation for a central field given in the previous paper are applied to various atoms. For the core electrons, the details of the interaction of the electrons in a single nk group are neglected, but an approximate correction is made for the fact that the distributed charge of an electron does not contribute to the field acting on itself (§2).

For a given atom the object of the work is to find a field such that the solutions of the wave equation for the core electrons in this field (corrected as just mentioned for each core electron) give a distribution of charge which reproduces the field. This is called the self-consistent field, and the process of finding it is one of successive approximation (§ 3).

Approximations to the self-consistent field have been found for He (§ 4), Rb+ (§ 5), Na+, Cl− (§ 9). For He the energy parameter for the solution of the wave equation for one electron in the self-consistent field of the nucleus and the other corresponds to an ionisation potential of 24·85 volts (observed 24·6 volts); this agreement suggests that for other atoms the values of the energy parameter in the self-consistent field (corrected for each core electron) will probably give good approximations to the X-ray terms (§4).

The most extensive work has been carried out for Rb+. The distribution of charge given by the wave functions in the self-consistent field is compared with the distribution calculated by other methods (§ 6). The values of X-ray and optical terms calculated from the self-consistent field show satisfactory agreement with those observed (§ 7).

The wave mechanical analogue of the case in which on the orbital model an internal and an external orbit of the same energy are possible is discussed (§ 8).
The reciprocation of one quadric into another
Authors: Baker H. F.
Published: November 2008

It is a familiar fact that if two quadric forms, in (n + 1) homogeneous variables, be each expressible as a sum of squares, of (n + 1) independent linear functions of the variables, then they are polar reciprocals of one another, in regard to any one of 2n quadrics. The question arises whether this is true for any two quadrics. Segre states that this is an unsettled question. A solution is given, however, by Terracini for any two non-degenerate quadrics, supposed to have been reduced to the Weierstrass canonical form. The present note has the purpose of pointing out that a solution is derivable from a remark made by Frobenius; this requires a knowledge of the roots of the equation satisfied by the matrix of the two quadrics.
The stability of a strut under thrust, when buckling is resisted by a force proportional to the displacement
Authors: Goldstein S.
Published: November 2008

For a strut under thrust when buckling is resisted by a force proportional to the displacement, it is shown that for both a clamped and a pinned strut there is stability for any length when the thrust is less than a certain critical value; and the relation is found between the length and the first value of the thrust above this that will give buckling. A notion of the wave-length and number of nodes after buckling is obtained.

The results are applied to the theory of the formation of mountains by horizontal compression, and it is shown that the crust of the earth would be able to transmit, without buckling, stresses right up to the breaking stress across regions of continental extent, unless the depth down to the level of no strain were less than 16 metres.

An attempt is made to employ the results also to consider the stability of solid surface films under compression, and it is seen that other factors, making for stability, must enter besides rigidity and gravity. This is supplied by surface tension, and it then appears that collapse due to a weakness in the film must precede buckling.

A rough calculation is given of the frequency of vibration of an atom in a solid film.
Oxidations on Charcoal
Authors: Wright W. M.
Published: November 2008

It is well known that charcoal possesses catalytic properties. It accelerates the decomposition of hydrogen peroxide, the removal of bromine from ββ-dibromo propionic acid and enables certain oxidations to take place in air at ordinary temperatures.
On the annihilation radiation of the positron
Authors: Klemperer Otto
Published: November 2008

Boron and carbon activated by diplon and proton bombardment have been used as positron emitting sources in experiments on the annihilation radiation.

By coincidence counting with two Geiger-Müller tubes it has been established that the radiation is emitted in pairs.

By the coincidence method of Becker and Bothe it has been shown that the annihilation radiation consists only of “soft” quanta.

By ordinary absorption measurements it has been shown that the annihilation radiation is homogeneous with a hardness corresponding to 0·5 million e-volt.
The influence of screening on the creation and stopping of electrons
Authors: Bethe H. A.
Published: November 2008

The cross-sections for the emission of radiation by very fast electrons (energy great compared with mc2) and for the production of pairs of positive and negative electrons by very hard γ-rays is calculated, considering the screening of the atomic field in which the processes happen. The main part of the paper deals with the integration of the differential cross-section over the angles. It is found that the integral cross-section increases steadily with increasing energy of the primary particle, reaching an asymptotic value for extremely high energies.
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.
The Two-Dimensional Electrostatic Solutions of Born's new Field Equations
Authors: Pryce M. H. L.
Published: November 2008

The general solution of Born's new field equations is found for the two-dimensional electrostatic case, by which the coordinates are expressed as functions of the field vectors, Conditions for inversion are discussed. Special cases are worked out, namely: singnle charge, two charges, charge in a constant field. Expressions are given for forces acting on the charges. A singular solution is also discussed, with reference to the neutron. The implication of the solutions on the general theory and the equations of motion is discussed in the conclusion.
Some Tests of Significance, Treated by the Theory of Probability
Authors: Jeffreys Harold
Published: November 2008

It often happens that when two sets of data obtained by observation give slightly different estimates of the true value we wish to know whether the difference is significant. The usual procedure is to say that it is significant if it exceeds a certain rather arbitrary multiple of the standard error; but this is not very satisfactory, and it seems worth while to see whether any precise criterion can be obtained by a thorough application of the theory of probability.
Limitations of Crystal Symmetry on Physical Phenomena with Particular Reference to Diamagnetic Magnetostriction
Authors: Shoenberg D.
Published: November 2008

A direct method of obtaining the limitations of crystal symmetry on physical phenomena is described and discussed in greater detail for the case of diamagnetic magnetostriction. Formulae are obtained for the variation of the longitudinal and transverse effects with crystal orientation in the case of bismuth.
On the Asymptotic Periods of Integral Functions
Authors: Scott Sheila
Published: November 2008

A period of a function f(z) is defined to be a number ω (≠ 0) such that

is identically zero; and it can be shown that an integral function may either have no periods or else a single sequence kλ (k = ± 1, ± 2, …).
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, if

and

then , whenever r and s are odd positive integers. He conjectures that the error term on the right of (1·1) is of the form

for every ε > 0, and that it is not of the form . He further conjectures that

for all positive values of r and s; and this conjecture has recently been proved to be correct.
On the Transport of Sediments by Streams
Authors: Jeffreys Harold
Published: November 2008

1. The power possessed by water and air in motion of lifting and carrying solids considerably denser than themselves is familiar to all; but it does not seem to have been pointed out that classical hydrodynamics provides a simple explanation. If a solid rests on the bottom of a stream, the points of contact are points of zero velocity; and the velocity just above the solid, by the equation of continuity, must be greater than the general velocity. Hence the velocities produce high pressures under the solid and low ones above it, and the difference tends to lift the solid up. If the resulting thrust exceeds the weight of the solid in the liquid, the solid will be raised, and will be unable to rest in equilibrium on the floor of the stream.
Integrals of Systems of Ordinary Differential Equations
Authors: Cherry T. M.
Published: November 2008

Let

be 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. functions

such that

identically. 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 zeros of the Riemann zeta-function
Authors: Littlewood J. E.
Published: November 2008

Let N (T) denote, as usual, the number of zeros of ζ (s) whose imaginary part γ satisfies 0 < γ < T, and N (σ, T) the number of these for which, in addition, the real part is greater than σ. In this definition we suppose, in the first place, that no zero actually lies on the line t = T: if the line contains zeros we define
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 series

where 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, …
Note on the extension to higher space of a theorem of Wallace
Authors: Gabbatt J. P.
Published: November 2008

The theorem is attributed to Wallace that, in a euclidean plane, the circumcircles of the triangles determined by four lines, of general position, meet at a point. It is further known that, in euclidean space of n dimensions, the circumhyperspheres of the simplices determined by n + 2 flats, of general position, meet at a point, if and only if n be even.
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,

and

where 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.
A Mathematical Theory of Natural and Artificial Selection, Part V: Selection and Mutation
Authors: Haldane J. B. S.
Published: November 2008

New factors arise in a species by the process of mutation. The frequency of mutation is generally small, but it seems probable that it can sometimes be increased by changes in the environment (1,2). On the whole mutants recessive to the normal type occur more commonly than dominants. The frequency of a given type of mutation varies, but for some factors in Drosophila it must be less than 10−6, and is much less in some human cases. We shall first consider initial conditions, when only a few of the new type exist as the result of a single mutation; and then the course of events in a population where the new factor is present in such numbers as to be in no danger of extinction by mere bad luck. In the first section the treatment of Fisher (3) is followed.
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.
Note on Kühne's Theorem
Authors: Lob H.
Published: November 2008

Given four lines in a plane, we know that the circumcircles of the four triangles formed by them meet in a point. Kühne's theorem states the analogous property for Euclidean hyperspace, viz., that if from n + 2 primes in space of n dimensions we obtain n + 2 simplexes by omitting each prime in turn, the hyperspheres circumscribing the simplexes meet in a point when n is even, but not when n is odd. Mr Grace, in a recent reference to the property, gives a short and interesting proof by help of the rational curves which osculate the given primes and the prime at infinity. It is the purpose of the following note to obtain the result by considering the points of intersection of certain cubic loci.
On some series of functions
Authors: Paley R. E. A. C.
Published: November 2008

We now consider some properties of “almost all” series

where 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 space

of 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:
The polarizability of the helium atom and the lithium ion
Authors: Hassé H. R.
Published: November 2008

This investigation owes its origin to a remark by Professor Lennard-Jones that it should be possible to calculate the approximate force between two atomic systems at large distances apart by a fairly direct method. The inter-atomic force is known to be of the form kR−7 at a large distance R, and an attempt to obtain the value of the constant k for two hydrogen atoms was made by Wang. The method suggested to the writer of this paper by Lennard-Jones was used by London and Eisenschitz for this case, and they obtained a value ofk differing from that given by Wang. The value given by London and Eisenschitz has been confirmed by following the method given in this paper, but further applications to the calculation of the inter-atomic force at large distances were postponed as the problem of calculating the polarizability of helium provided, on Lennard-Jones' suggestion, a simpler case on which the method could be tested.
A new group of action functions in the unitary field theory. II
Authors: Infeld L.
Published: November 2008

There exists a group of action functions, depending on a parameter γ giving for every γ ≥ 0 a static solution with central symmetry and finite energy, both for the electric and magnetic field, and going over into Maxwell's action function for weak fields. For γ = 1 we obtain Born's case distinguished by a perfect symmetry between electric and magnetic field. For γ = 0 the action function takes a particularly simple form and the comparison with the results of Heisenberg, Euler and Kockel allows of an approximate determination of the fine structure constant.
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.
A class of fundamental [n−1] Scrolls
Authors: Woodcock E. R.
Published: November 2008

In a paper published in the Proceedings of the London Mathematical Society, (2), 40 (1936), 143, Welchman introduced the idea of fundamental scrolls from which all special scrolls may be obtained by projection, and he developed the theory of fundamental line scrolls. These fundamental line scrolls are generated by joins of pairs of corresponding points in a cyclic (1, 1) correspondence of period 2 on a canonical curve. In this paper I generalize some of Welchman's results, and then go on to consider certain fundamental scrolls which have a particular interest.
On semi-groups
Authors: Rees D.
Published: November 2008

This paper is an attempt to apply the methods of the theory of algebras to the more general problem of the structure of semi-groups, i.e. systems with one composition, satisfying the associative law.
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 of

for integral values of u, v, w, not all zero. I proved a few years ago (1) that

more precisely, that

except 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
Some Enumerative Results in the Theory of Forms
Authors: Hodge W. V. D.
Published: November 2008

In a recent note I attempted to obtain the postulation formula for the Grassmannian of k-spaces in [n] by the consideration of forms of a certain type in k + 1 sets of r + 1 homogeneous variables, which I called k-connexes. My attempt was not entirely successful; I obtained a formula for k-connexes which suggested what the required postulation formula should be, but was unable to prove it. D. E. Littlewood has now written a paper to show that my problem is intimately connected with the theory of invariant matrices, and has thereby established the truth of the postulation formula which I had conjectured. Littlewood's proof requires a considerable knowledge of the theory of invariant matrices, and this paper results from an attempt to re-write his proof in a form which is intelligible to a student not having this specialized knowledge. Prof. H. W. Turnbull has pointed out to me the importance of the so-called k-connexes in the theory of forms, particularly in connexion with the Gordan-Capelli series, and for this reason I am taking the k-connexes as the principal topic of this paper, leaving the deduction of certain postulation formulae which are the more immediate concern of a geometer to the end.
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.
La théorie des noyaux reproduisants et ses applications Première Partie
Authors: Aronszajn Par N.
Published: November 2008

La théorie que nous nous proposons de présenter ici provient d'une synthèse de plusieurs théories particuliéres.
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 have

It 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 algebra of meson matrices
Authors: Kemmer N.
Published: November 2008

In an interesting recent paper Schrödinger(5) contributes much new information on the properties of the hypercomplex algebra used in meson theory(1, 2, 4, 6, 7) which is defined by the relations(3)
On the fractional parts of the powers of a number (II)
Authors: Vijayaraghavan T.
Published: November 2008

Let G(θ) denote the limit points of the fractional parts of θn (n = 1, 2, 3, …), and T the set of numbers θ for which G(θ) contains an infinity of points. In an earlier note it was proved that if θ is a rational fraction greater than unity then θ belongs to T. The results of this note concern the limit points of the powers of algebraic irrationals.
Deuteron bombardment of the heavy elements
Authors: Krishnan R. S. , Nahum E. A.
Published: November 2008

Experiments on the bombardment of platinum with 9 M.e.V. deuterons are reported. Radioactive isotopes of platinum of periods 32 min., 18 hr. and 2·8 days, and isotopes of gold of periods 2·6 and 5·6 days are reported. The 2·6 day gold isotope is formed directly by deuteron bombardment as well as by decay of the 32 min. platinum isotope. The absorption of the β-rays in aluminium and of the γ-rays in lead has been measured, and assignments have been made for most of the isotopes reported. Excitation function measurements have been made for the 32 min. and 18 hr. platinum isotopes, and the form of the energy-yield curve is the same for both these elements. The form of the curve seems to favour the Oppenheimer-Phillips process for the formation of these radioelements by the reaction.
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 ≤ 1

Theorem 4. Let G be the star domain

where α > 0. Then, when α tends to zero,

Proof. The linear substitution

changes G into the similar domain

and so

.
Double Fourier series and boundary value problems
Authors: Green A. E.
Published: November 2008

1. Problems in elasticity which are concerned with isotropic rectangular plates have attracted the attention of many writers both from the theoretical and practical points of view. When the boundary conditions are of the simply supported type the solution of the problems is usually simple, although when double Fourier series are used the validity of such solutions is not very clearly shown in most cases. Satisfactory exact solutions for many classical problems in which the edges of the rectangular plate are clamped have only been obtained in recent years, but approximate strain energy methods often gave results which were useful for practical purposes.