bluuebunny/crossref_metadata_2025 huggingface dataset:
The interaction of lattice vibrations and free electrons in metals
Authors: Jones H.
Published: November 2008

The electron theory of metals, established in its present form mainly by Pauli, Sommerfeld and Bloch, makes it possible to classify the phenomena connected with metals into two main divisions, which may be called first order effects and second order effects, according to whether the effect depends on the first or second approximation to the energy of a degenerate electron gas at a given temperature. Examples of the first group are the constant paramagnetism of the alkalis and Volta contact potentials. Examples of the second group are the specific heat of the electrons and all thermoelectric effects. The temperature dependence of the electrical conductivity should be included in the group of first order effects, since here the temperature is introduced through the lattice wave motion. The limits of the existing theory are now easily described. First order effects are accounted for with success, second order effects are in the main far from being adequately covered by the theory. To see clearly the reason for this, one must examine briefly the basis of the existing theory. In the first approximation the metal is regarded as composed of two independent systems; a system of lattice vibrations, and a system of electrons free to move in a given space periodic field of potential. In this way by dealing with the two systems quite independently one can reproduce many properties of metals, for example, Debye's theory of specific heats. The theory of pure lattice heat conductivity is concerned only with the former system, while the constant paramagnetism of the alkalis, and the fact that the electrons contribute only a very small amount to the total specific heat of the metal can be accounted for by considering the latter system only.
A note on Ferromagnetism
Authors: Fowler R. H. , Powell F. C.
Published: November 2008

The purpose of this note is to amplify the discussion and conclusions of a recent paper by one of us. It has recently been shown by Webster, drawing largely from the work of Weiss and Forrer, that in single crystals of both iron and nickel there is a family of natural directions of micromagnetization defined by the observed directions of easy holomagnetization. The evidence presented by Webster is completely convincing, and we therefore may refine our picture of micromagnetization by asserting that the micro-elements are all (holo)magnetized in one or other of the directions of easy holomagnetization. These directions are the family of six directions (± 1, 0, 0) (0, ± 1, 0) (0, 0, ± 1) for iron and the family of eight directions (± 1, ± 1, ± 1) for nickel. We wish to comment on Webster's theory later in connection with the properties of cobalt.
Note in regard to surfaces in space of four dimensions, in particular rational surfaces
Authors: Baker H. F.
Published: November 2008

To a non-singular algebraic surface in space of five dimensions there can generally (the Veronese surface, of order 4, and cones are exceptional) be drawn, from an arbitrary point, a finite number of chords. If such a surface be projected from a point into space of four dimensions, there will, therefore, in general, be a certain number of points upon the resulting surface, at which two sheets of this surface, with distinct tangent planes, have an isolated common point. Such points have been called improper double points. We consider an algebraic surface ψ, in space of four dimensions [4], with no other multiple points than such double points, which we shall call accidental double points. The chords of the surface ψ, drawn from an arbitrary point O of the space [4], form a surface, or conical sheet, of which a general generator meets the surface in two points. The locus of these points is a curve which we shall call the chord curve. This curve has an actual double point at each of the accidental double points of ψ There will also, generally, be a certain number of points of the surface which are points of contact of tangent planes of the surface passing through O (and therefore also points of contact of tangent lines through O, these tangent lines being generally tangent lines of the chord curve).
An attempt to detect the spontaneous transformation of helium into penetrating radiation
Authors: Tarrant G. T. P. , Gray L. H.
Published: November 2008

The spontaneous disappearance of an electron and proton in the form of radiation has often been suggested as a possible source of energy to maintain the emission of radiation from hot stars. It can be calculated that the energy of a single quantum formed in this way should be about 1000 million electron-volts. In order to account for the absorption of the ultra-penetrating radiation observed by Regener in deep lakes, and interpreted on the basis of the Klein-Nishina formula, a quantum of still higher energy is required. Sir James Jeans has recently calculated that the transformation of a helium atom into a single quantum would provide a radiation of about the required penetrating power.
On Nicod's reduction in the number of primitives of logic
Authors: Bernstein B. A.
Published: November 2008

Nicod, using Sheffer's operation “│” as a primitive (undefined) idea, obtained a drastic reduction in the number of primitive propositions (postulates) used by Whitehead and Russell in their theory of deduction for “elementary” propositions. In this reduction Nicod accepts Sheffer's replacement of the primitive ideas “∼” and “v” and the primitive propositions *1·7 and *1·71 of the Principia by a single primitive idea and a single primitive proposition, and, by adding two ingeniously devised postulates of his own, derives from the three postulates the Principia's six primitive propositions *1·1, *1·2, *1·3, *1·4, *1·5, *1·6. But the author does not consider the consistency or the independence of his three postulates; does not discuss the Principia's remaining primitives—the ideas “φx” and “⊢. φx” and the propositions *1·11 and *1·72; and fails to prove that his postulates are derivable from those of the Principia.
The numerical calculation of the scattering correction in γ ray absorption measurements
Authors: Tarrant G. T. P.
Published: November 2008

The method of applying the scattering correction in accurate γ ray absorption measurements is discussed, and it is shown that the scattering correction only diminishes, as the solid angle subtended by the chamber at the source is reduced, if the absorber diameter is reduced correspondingly. The method of calculation of the scattering correction in cases in which the absorber is placed very close to the chamber is explained, and the result of the calculation for one particular chamber is shown to be in satisfactory agreement with the experimental values. The scattering correction allows the ionization function of chambers to be determined and an (ionization function)/(wave-length) curve is shown for a steel thick-walled high-pressure ionization chamber.
The effect of a magnetic field on relative gravity determinations with invar pendulums
Authors: Bullard E. C.
Published: November 2008

The effect of changes in magnetic moment and in the earth's field on the period of invar pendulums used for relative gravity determinations is discussed, and methods of eliminating them by the use of Mumetal screens are described.
The Absolute Intensities of the γ-Rays of Radium B and Radium C
Authors: Ellis C. D. , Wooster W. A.
Published: November 2008

Considerable advances have been made in recent years in our knowledge of the γ-ray emission of radioactive bodies. It has been established quite definitely that series of definite frequencies are emitted, forming a characteristic line spectrum of the nucleus, and methods of measuring the frequencies have been devised so that at present the γ-ray spectra of most of the radioactive bodies are well known.
On the Addition of the Primary Aberrations
Authors: Steward G. C.
Published: November 2008

In the application of the Characteristic Function of Hamilton, or of any allied function, to the computation of a symmetrical optical system three steps are necessary. The performance of the system as a whole must be considered, and from this it appears that the aberrations may be derived from certain ‘aberration coefficients’ which occur in the expansion of an ‘aberration function’. In the second place, relations must be obtained giving the properties of the complete system in terms of the properties of the component systems, which, in general, will be single refracting surfaces; and finally, an evaluation of the coefficients must be made for the simple system—a single surface. For the first and third of these steps reference may be made elsewhere, and also for a general investigation of the second; a simple derivation is given, in the present note, of the necessary relations between a composite system and its components in the case where first order aberrations only are considered.
On the series of forms of Jacobi's rotating liquid ellipsoid
Authors: Sedgwick W. F.
Published: November 2008

The object of these notes is twofold. First, to connect the theory of the possible shapes of Jacobi's ellipsoidal equilibrium figure, for a given mass of homogeneous rotating liquid, with the theory of its stability and other properties given by Prof. H. F. Baker in his paper “On the stability of rotating liquid ellipsoids” in Proc. Camb. Phil. Soc. vol. 20, pp. 190–97. And secondly, to indicate two inconclusive reasonings on the subject in Thomson and Tait's Natural Philosophy (2nd edition), arts. 778 and 778′, and to complete the proofs of the results in question.
The equianharmonic and harmonic envelopes of a quadrinodal cubic surface
Authors: Timms G.
Published: November 2008

Prof. W. P. Milne has recently been investigating, in connection with the general cubic surface, the Equianharmonic Envelope S4 and the Harmonic Envelope T6, denned as the envelopes of planes cutting the surface in equianharmonic and harmonic cubic curves respectively: he suggested to me that I should consider the particular case of the quadrinodal cubic surface.
A photographic method of determining the mobility of recoil atoms
Authors: Briggs G. H.
Published: November 2008

The experiments described in this paper were carried out in the University of Sydney during November 1924 and, owing to the short time which was available, are of a purely preliminary character. The following account is given as they are not likely to be resumed in the immediate future.
Potential functions with periodicity in one coordinate
Authors: Howland R. C. J.
Published: November 2008

The general conception underlying the following analysis is that of a field of potential which is invariant under a group of geometrical transformations. The boundary is to consist of a number of parts which transform into each other and the boundary values also transform into each other. To satisfy the conditions we build up functions which are invariant under the same group of transformations and combine them to give the prescribed boundary values over one section of the boundary. The boundary conditions on the other sections are then automatically satisfied. When the groups are simply translations or rotations we get functions periodic in either a Cartesian or an angular coordinate.
On the probability of artificial nuclear transformations and its connection with the vector model of the nucleus
Authors: Goldhaber M.
Published: November 2008

1. The theorem of quantum mechanics that “the spin part of the angular momentum is approximately a constant of motion provided that the forces depending on the direction of the spins are small compared with the total interaction forces” is introduced into the discussion of artificial nuclear transformations. A definition is given for the terms “probable” and “not probable” nuclear reaction.

2. Attention is drawn to the experimental result that the reaction Li6 + H1 → He4 + He3 is more probable than the reaction Li7 + H1 → 2He4, and an explanation is suggested.

3. The nuclear spin of Lie is derived from nuclear transformations and found to be 1.

4. Similar derivations for the spins of H3, He3 and B11 are summarised in a table.
The Resultant of Two Fourier Kernels
Authors: Hardy G. H.
Published: November 2008

1. A “Fourier kernel” means here a function K(x) which gives rise to a formula

of the Fourier type. Thus

are Fourier kernels. If K(x) is a Fourier kernel, λ is real, and a positive, then

are Fourier kernels.
Note on a Paper of Bell and Wolfenden on Electrolytic Separation of Diplogen
Authors: Sawyer W. W.
Published: November 2008

Bell and Wolfenden have published a theory of diplogen concentration by electrolysis, which, to a considerable extent, explained the empirical result, that the efficiency of diplogen separation is constant. This theory depends on the assumption that a quantity, γ, introduced by Gurney is constant. In a later paper Bell writes that this assumption would “probably not be justified in a strict examination of the problem”, but he gives no quantitative estimate.
An Apparatus for Recording Average Amplitudes of Wireless Echoes
Authors: Farmer F. T.
Published: November 2008

Attention has recently been directed to the measurement of the relative amplitudes of wireless echoes from the ionosphere, and some interesting methods of automatic registration have been devised (1, 2). For vertical incidence, and for wave-lengths shorter than 200 metres, visual observation shows that a single echo, even when separated from its accompanying oppositely polarized magneto-ionic component, shows considerable rapid fading (3). This fading is evident on the records published by White (1), and it is clear from these that it is not easy to determine the relative amplitudes of two echoes. This paper is an account of an apparatus which automatically integrates the amplitude of an echo over a time long compared with the period of the fading, and at the end of the period registers the integrated value on a recording galvanometer. The process is then repeated so that the final galvanometer trace represents the average value of the intensity throughout successive equal periods of time. It is further described how by using a double thread recording galvanometer, it is possible to record the averaged amplitudes of two different echoes (e.g. ordinary and extraordinary, or E and F) during alternate integrating periods.
The Dissection of Closed Sets of Arbitrary Dimension and the Generalized Brouwer-Alexandroff Theorem
Authors: Kaufmann B.
Published: November 2008

The successful development in recent years of the topology of general closed sets is largely due to the application of combinatory methods, which have led to an elaborate theory of approximation of closed sets by infinite cycles, to the generalization of duality theorems for closed sets, and to a geometrical theory of dimensions. Corresponding to the combinatory invariants involved the main results of these theories concern for the most part properties of the set as a whole, which cannot possibly express fully its internal and in particular its local structure. Still less is our knowledge of the eventual relations between the local structure of a set and its properties in the large.
Note on the Flow past a Circular Cylinder
Authors: Howarth L.
Published: November 2008

Buri's method of solution of the equations of turbulent motion in a boundary layer has been used extensively in a previous paper. Since the results of that investigation have not been compared with experiment, nor has any other application of Buri's method been checked experimentally, it seems advisable to apply this method to a problem for which the experimental results are known.
A theorem of Dr P. Zeeman
Authors: Grace J. H.
Published: November 2008

The following theorem, first enunciated by Dr Zeeman in 1899, has attracted the attention of several writers:

If five points A, B, C, D, E in three dimensions are such that the feet of the perpendiculars from E on the faces of the tetrahedron ABCD are coplanar the relation between the five points is symmetrical.
The Amount of Energy Emitted in the γ-Ray Form by Radium E
Authors: Aston G. H.
Published: November 2008

The β-ray type of disintegration is accompanied in general by two forms of radiation, the β-rays and the γ-rays. Radium E is noteworthy in that it gives a large amount of γ-radiation that it can only be detected by careful measurements. Yet, since it differs in this way from β-ray elements such as radium B and radium C, it is important for theoretical discussion that we should obtain as much knowledge as possible about this small amount of γ-radiation.
A Convenient Method of Distillation of the Alkali Metals
Authors: Clark R. J.
Published: November 2008

The alkali metals, sodium, potassium and rubidium can be distilled easily in a good vacuum and obtained reasonably free from occluded gas in the following way. As potassium is now used for the absorption of mercury vapour and many experiments are being done on the others, an account of a convenient way of preparing pure specimens may be of some service to experimenters.
Layers of Caesium, and Nitrogen on Tungsten
Authors: de Bruyne N. A.
Published: November 2008

While investigating the effect of a caesium film on the auto-electronic emission from a tungsten surface I noticed certain phenomena which seemed worth further investigation. Langmuir and Kingdon have shown that the emission from a tungsten filament in caesium vapour varies with the temperature of the filament in the manner shown in Fig. 1. The initial rise in the emission at low temperatures is due to the existence of a caesium layer on the filament. At higher temperatures the caesium evaporates and the emission drops until eventually the temperature becomes high enough to produce a thermionic emission from the clean tungsten surface.
On a Case of steady flow of a Gas in Two Dimensions
Authors: Lees S.
Published: November 2008

The problem of finding general types of solution for the steady flow of gases in two dimensions, adiabatically and without friction, does not appear to have received a great deal of attention. An interesting and suggestive treatment of hydrodynamics applied to gases is given in Riemann-Weber's Partielle Differential-Gleichungen, but the application is to pressure propagation. Reference must also be made to a paper by the late Lord Rayleigh, who gave a general differential equation which must be satisfied by the velocity potential ø, but the problem of utilizing this equation does not appear easy.
The High Energy Groups in the Magnetic Spectrum of the Radium C β-Rays
Authors: Ellis C. D.
Published: November 2008

Many years ago Rutherford remarked that the high energy lines of the radium C β-ray spectrum showed the existence of γ-rays of very high frequency, and later he pointed out that their frequencies must be closely connected with the β-ray energies through the quantum relation. Although the truth of Rutherford's point of view has become more and more obvious, no advance in the detailed knowledge of these lines has been made in the last seven years. This spectrum is very complicated and the lines faint.
Some Methods of Estimating the Successive Ionisation Potentials of any Element
Authors: Hartree D. R.
Published: November 2008

On certain assumptions as to the quantum numbers of the normal orbits of the successive electrons captured by an atom, the successive ionisation potentials can be estimated by making use of the relations between terms of the same quantum number for different atoms of the same electronic structure. Formulae for several of the successive ionisation potentials are given, the form of the formulae being suggested by theoretical reasoning and numerical values of the constants determined from known spectra; the successive ionisation potentials of O, Fe and Ag, calculated from these formulae, are tabulated. It is pointed out how far different these ionisation potentials are from the X-ray excitation potential of the same orbit of a neutral atom.
A Note on the Hydrogen Chromosphere
Authors: McCrea W. H.
Published: November 2008

It is well known that eclipse observations show that the chief constituents of the higher levels of the sun's chromosphere are ionised calcium and hydrogen. The equilibrium of the calcium has formed the subject of Milne's important researches. The hydrogen presents a rather different problem owing to the large number of possible quantum states that have to be considered.
Note on the effect of Temperature on the Auto-electronic Discharge
Authors: de Bruyne N. A.
Published: November 2008

R. A. Millikan and C. F. Eyring have shown that the auto-electronic emission is independent of the temperature of the cathode up to 1000° K. At higher temperatures a thermionic emission takes place and this masks the auto-emission. W. S. Pforte and the author have recently shown that the thermionic emission is increased by the field in accordance with a relation derived by Schottky, so that when both thermionic and auto-electronic emissions are present it is only necessary to subtract the thermionic current, as given by the Schottky relation, from the total measured current to obtain the value of the auto-emission at the temperature considered. In this way the thermionic and auto-emissions can be separated.
Some formulae for surfaces in higher space
Authors: Roth L.
Published: November 2008

The formulae of this paper are chiefly concerned with multiple secants and tangents to surfaces in spaces of dimension four to seven; they are obtained by the functional method of Cayley and Severi, which was first applied to scrolls by C. G. F. James. The majority of the results have already been given for scrolls in James's third paper; but whereas the functional equations in this case involve only two variables, those of the present paper contain four, and the work is considerably more complicated. It is also to be noted that, though the formulae for scrolls might perhaps have been expected to follow as particular examples from the corresponding formulae for surfaces, this is not always the case. The reason for the discrepancy is not always clear, although in some instances theoretical reasons for disagreement can be assigned.
Current Distribution near edges of Discharge-Tube Cathodes
Authors: Moon P. B. , Oliphant M. L.
Published: November 2008

The experiments described below were under-taken to follow up the observation, made independently by Dr Chadwick and one of us (M. L. O.), that the disintegration by sputtering of a plane cathode in a gas discharge was not uniform but showed a maximum at a small distance from the edge. By running the discharge for many hours under steady conditions it was possible to remove a narrow ring of metal at a distance of about a millimetre from the edge of a thin platinum cathode, the middle portion and the extreme edge remaining whole. A photograph of a cylindrical cathode taken by Kaye shows the same effect very clearly.
On the stability of copper catalysts produced by thermal decomposition
Authors: Constable F. H.
Published: November 2008

The view which the author has expressed regarding catalytic activity as due to frozen groups of atoms with strong specific external fields is further tested by a study of the stability of preparations of copper at the instant of formation, and during the subsequent life of the catalyst.
Maxwell's Law, and the Absorption and Emission of Radiation
Authors: Milne E. A.
Published: November 2008

In the famous paper in which he developed his theory of radiation, Einstein investigated two problems. He investigated first the problem of what distribution of radiant energy in the steady state would be set up by an enclosed assembly of atoms in thermodynamic equilibrium radiating and absorbing according to the assumptions of his theory; he showed that the distribution so set up was given by Planck's law. He investigated secondly the velocities set up amongst the atoms in consequence of the random variations in direction of the emissions and absorptions; provided that emissions are assumed to be directed, so giving rise to recoil momentum, he showed that the mean square velocity-component in a given direction, , is equal to its equipartition value, given by .
A mathematical theory of natural and artificial selection. (Part VI, Isolation.)
Authors: Haldane J. B. S.
Published: November 2008

It is generally believed that isolation has played an important part in evolution. If an organism is to evolve so as to adapt itself to a special type of environment, e.g. a cave or a desert, it must not be swamped in each generation by migrants from the original habitat.
On the Fifth Book of Euclid's Elements
Authors: Hill M. J. M.
Published: November 2008

There are in the Fifth Book of Euclid's Elements twelve propositions, the object of which is to prove the equality of two ratios.
The total energy of binding of a heavy atom
Authors: Milne E. A.
Published: November 2008

Thomas's differential equation for the average field inside a heavy atom is analogous to Emden's equation for the polytropic equilibrium of a star. Emden's result that the total gravitational potential energy of a star is calculable once the differential equation has been solved is adapted to give the total electrostatic energy, and hence the total energy of binding, of an atom built on Thomas's model. This should be equal to the sum of the successive ionisation potentials. The total energy is found to be proportional to N, where N is the atomic number. The values found agree with Hartree's calculations of the successive ionisation potentials of certain atoms.
The Hall effect in Single Crystals of Iron
Authors: Webster W. L.
Published: November 2008

In this paper the results of some measurements of the Hall effect in single crystals of iron are described. This work was undertaken to determine whether the Hall coefficient varied with the direction in the crystal for which it was measured. Such a variation had been observed in other magneto-electric phenomena—the change of resistance in longitudinal* and transverse† magnetic fields—and had thrown some light on the nature of these effects. It was thought possible, therefore, that the study of the Hall effect in single crystals might yield some fresh light on this unexplained phenomenon.
The triode oscillation generator and amplifier: limitations on sinoidal performance
Authors: Turner L. B. , Meacham L. A.
Published: November 2008

The characteristic relationship subsisting between anode current ia, anode potential ea, and grid potential eg in a well-evacuated thermionic triode, viz.

leads to very simple analysis of the circuit conditions when μ, is a constant and φ is a linear function. In all the usual applications of triodes, μ is very nearly a constant; in some applications, e.g. in good acoustic amplifiers, over the working range φ is very nearly linear; in others, e.g. in rectifiers, non-linearity is essential to the performance. In oscillators, and in power amplifiers used with them, both conditions are met. Where a sinoidal oscillation is desired, i.e. an alternating current as devoid of harmonics of the fundamental frequency as can be contrived, the working range must be sensibly linear. A feature of such a régime is that the major portion of the power supplied to the triode must be dissipated in heating the anode. Where higher efficiency is required, the cycle must be made to extend beyond the linear range, and harmonics are necessarily introduced.
The Petersen-Morley theorem, I
Authors: Lyons R. J. , Frith R.
Published: November 2008

Two perpendicular intersecting straight lines will be said to be normal to each other. Then the Petersen-Morley theorem is: Given arbitrarily three lines a, b, c, let a′ be the unique normal to b and c, b′ that to c and a, and c′ that to a and b; then the lines x, y, z, normal respectively to the pairs of lines a and a′, b and b′, c and c′, have a common normal.
On the calculation of the energy in unsaturated hydrocarbon molecules
Authors: Coulson C. A.
Published: November 2008

It frequently happens that we wish to evaluate the total energy of the mobile, or π, electrons in an unsaturated hydrocarbon molecule. According to the method of molecular orbitals(1) this energy is the sum of individual electronic energies ∊r, and the summation is over all the occupied orbitals. The energies ∊r are the roots of the secular determinant, and, if we write

and suppose that there are n unsaturated carbon atoms, then the equation that gives the energies zr is
A relativistic formula for the scattering of mesons under the influence of radiation damping
Authors: Ma S. T.
Published: November 2008

It has recently been pointed out by Heitler(1) that the well-known discrepancy between the theoretical expression and the experimental results for the cross-section of scattering of charged mesons by nuclear particles can be removed by a proper consideration of the effect of radiation damping in the quantum theory. The radiation damping in quantum theory was first considered in complete detail for free electrons by Waller(2). A rigorous deduction of the integral equation set up by Waller was given by Heitler on the basis of a method developed by Góra. An alternative rigorous derivation of the integral equation has also been given by Wilson(3). Exact solutions of the integral equation for the simple scattering of mesons by nuclear particles have been found by Heitler in the non-relativistic approximation. An exact solution has not so far been given for physical problems in which the integral equations are complicated, but Wilson has given a general approximate formula for the scattering cross-section, which should be valid for all problems.
The Mobility Distribution and Rate of Formation of Negative Ions in Air
Authors: Hamshere J. L.
Published: November 2008

A modification of the alternating field method of measuring ionic mobility in a gas gives an experimental curve showing upper and lower limits to k. From this a distribution curve is derived, which has a calculable resolving power.

The mobility of negative ions in dry air shows a continuous distribution between the limits 2·15 and 1·45, with a peak value about 1·8.

At low pressures the current is resolved into ions and free electrons. From the relative numbers reaching the electrometer it is found that the electron makes an average number 9·4. 104 collisions before capture, independent of field strength and pressure, and therefore independent of the electron speed over a range W = 2×105 to W = 7×105.
A Generalization of Picard's Method of Successive Approximation
Authors: Hirschfeld H. O.
Published: November 2008

It is well known that the boundary value problem for the non-linear differential equation

can be reduced with help of a Green's function K (x, ξ) to a non-linear integral equation of the type
A Maximum-Minimum Method of Determining the Cardinal Points of a Lens System
Authors: Searle G. F. C.
Published: November 2008

The method of determining the focal length f of a thin converging lens by finding the minimum distance Δ0 between image and object is well known. When the thickness of the lens is negligible, f = ¼Δ0.

In the general coaxial system, the media at the two ends have refractive indices μ1, μ2, and there is a finite distance t between the principal or unite plances. The two focal lengths are unequal, since, f1/f2=μ1/μ2, and the nodal points do not coincide with the principal points.
Atomic disintegration by particles of low energy
Authors: Burhop E. H. S.
Published: November 2008

A source of ions consisting of a low voltage arc, yielding homogeneous ion currents of the order of ½ milliampere at low potentials is described. With such a source, atomic disintegrations have been observed with ion beam energies of less than 8kV. Approximate values of the absolute yield for protons from the deuteron on deuterium disintegration, and α-particles from the protons on lithium disintegration are obtained.
The theory of the shot effect. II
Authors: Rowland E. N.
Published: November 2008

Summary of the previous paper

The previous paper dealt first with the shot effect without secondary emission. We considered a stream of electrons arriving at random, with a fixed probable density N, at the anode of the resistance-capacity coupled first valve of a linear amplifier. This does not allow for the effect whereby the arrival of electrons alters the anode potential, and consequently the probability of arrival of succeeding electrons. The valve must be working with what is conventionally called an infinite internal resistance, and without space-charge effects. Each electron arriving at time tr is supposed to produce at time t an output effect f(t − tr) from the amplifier. We considered the hypotheses, first, that electrons may, with specific probabilities, have lives of any length on the anode system, during which they add − ε/C to the anode potential, and alternatively that each electron adds to the anode potential of the valve whose anode to earth capacity is C and whose feed resistance is R. These hypotheses determine the form of f(t − tr) when the behaviour of the main part of the amplifier is known. Since the amplifier is linear, the effects of various electrons are additive, so that the total output is ∂(t) = ∑rf(t − tr).
On the velocity and temperature distributions in the turbulent wake behind a heated body of revolution
Authors: Goldstein S.
Published: November 2008

Thevelocity distributionin the wake behind a body of revolution calculated by Miss Swain on the momentum transfer theory withlconstant over a cross-section of the wake is not in accordance with experiment.

On the generalized vorticity transfer theory the equation for the first approximation to the velocity defectufar downstream is

There are two methods of simplifying this equation on the analogy of the successful simplification of the corresponding equation for the two-dimensional case. One method gives the same equation as if the turbulence were symmetrical; the other gives the equation of the modified vorticity transfer theory.

For the former there is no real solution iflis constant over a cross-section; for a real solutionlmust be infinite on the axis of symmetry. The equation was integrated withl∝r−1over a cross-section; this variation oflwas suggested by putting the change in the area enclosed by a circular vortex-line proportional to the area of the section of the wake. The agreement with experiment is fairly good over the middle portion of the wake, but not near the edge, where the theoretical value of ∂u/∂ris not zero. Ifl2∝rp, ∂u/∂rvanishes at the edge of the wake only ifp= − 3. The solution is carried through for this case; it shows only fair agreement with experiment over the middle of the wake, and none at the edge.

The calculation on the modified vorticity transfer theory is carried out withlconstant over a section; in this case there is no definite edge to the wake, nor doesufall off sufficiently rapidly to make the momentum integral for the drag converge. (u∝r−2for larger.) Even over the middle of the wake there is a discrepancy with experimental results, rather like that shown by the momentum transfer theory.

Thetemperature distributionbehind a heated body was calculated according to each of the foregoing theories.

The temperature distribution was also calculated from the experimental velocity distribution, without any assumption being made concerning the kinematic coefficient of turbulent diffusion, which was eliminated between the equations for the velocity and temperature. The vorticity transfer theory with symmetrical turbulence gives an impossible result. The modified vorticity transfer theory gives θ/θmax<u/umaxwhen the experimental velocity distribution is used.
Birational transformations possessing fundamental curves
Authors: Todd J. A.
Published: November 2008

On any algebraic variety of d dimensions there exist certain systems of equivalence which are relative invariants under birational transformation. These systems include the canonical systems (Todd(3,4)) which can be defined for each dimension from zero to d − 1, and the systems defined by the intersections of these canonical systems among themselves. It is natural to enquire what is the precise behaviour of these invariant systems under birational transformations. Relatively few results of this kind are known. For threefolds, Segre has recently (2) investigated the transformations undergone by the invariant systems in any algebraic (not necessarily birational) transformation whose fundamental points and curves are of general character. More recently, A. Bassi (1) has obtained by topological methods the relations between the Zeuthen-Segre invariants of two Vd in an (α, α′) correspondence. I have recently (6) discussed the transformation of the invariant systems on a Vd for birational transformations on the assumption that the fundamental points are isolated and of general character.
A further note on Ramanujan's arithmetical function τ(n)
Authors: Hardy G. H.
Published: November 2008

This note is a sequel to two in earlier volumes of the Proceedings, the first by myself and the second by Wilton†.

Suppose that

for |z| <1; that x > 0; that

forr ≥ 0; and that

where τ(x) is to mean 0 if x is not an integer. Thus

where the dash shows that the last term is to be halved when x is an integer;

forr > 1; and
Concerning secondary flow in straight pipes
Authors: Howarth L.
Published: November 2008

In this paper the condition for the existence of a secondary flow in a straight non-circular pipe is determined according to the modified vorticity transfer theory, with Goldstein's assumed form for the tensorIt is shown that a secondary motion arises if the mixture length is not constant on the curves along which | gradu| is constant,ubeing the velocity parallel to the pipe axis.

In problems of turbulent flow treated by means of the modified vorticity transfer theory, the quantitywhereis the mean value of the square of the velocity fluctuation andpthe mean pressure, plays a part analogous to the pressure in laminar flow. In two-dimensional flow through a channel the theory shows the existence of a gradient ofacross the channel from the central plane to each wall. Qualitative arguments such as are used to explain the existence of a secondary flow for laminar motion in a curved pipe are applied here to show that a secondary flow is to be expected near the short sides in the turbulent flow through a straight rectangular pipe of large length/breadth ratio.

The equations to determine the secondary flow through an almost circular elliptic pipe are discussed, the mixture length being assumed constant on ellipses similar to and concentric with the pipe section. For a first approximation the problem is reduced to the numerical solution of three simultaneous ordinary linear differential equations.
Note on the velocity and temperature distributions behind a heated body of revolution
Authors: Goldstein S.
Published: November 2008

The measurements of Hall and Hislop (see the preceding paper) were made in a wind tunnel considerably less turbulent than has usually been used in such measurements.
Corrigendum
Authors: Gillis J.
Published: November 2008

Note on a Theorem of Myrberg*

On p. 421, line 30, of that note I remark “δ(x) is a lower semi-continuous function of x”. That statement is, in general, untrue. The proof should rather run as follows:

The subset of E1 where δ(x) ≥ a > 0 is clearly closed and, as α tends to 0, tends to h−mE1. Hence, by taking α sufficiently small, we find a closed subset of E1, of positive h-measure, every point x of which has the property that

whenever 0 < d ≤ α. We may call this subset again E1 and the proof, from the top of p. 422, goes as before.
The γ-rays of ionium
Authors: Ward A. G.
Published: November 2008

The absorption of the harder γ-rays from ionium has been studied, the first, of energy 68 ± 1 k.e.V., by the method of bracketing, making use of the K absorption discontinuities of tantalum and tungsten; the second, of energy 190 ± 20k.e.V., by analysis of the absorption curves. The intensity of these γ-rays is estimated to be 1 quantum in 10−3 disintegration for each γ-ray.
An automatic curve follower for use with the differential analyser
Authors: Blackett P. M. S. , Williams F. C.
Published: November 2008

When the differential analyser (1) is being used it is often necessary to supply to the machine information in the form of a functional relation between variables occurring in the equation. This is usually done by keeping a pointer on a graph on an input table. The x-coordinate lead screw of the input table is driven by the machine, and the y-coordinate lead screw is rotated by hand so as to keep the pointer on the given curve. The disadvantages of this method, namely the necessity for the continuous attention of an operator and the possibility of personal errors, would be obviated by the use of an automatic follower instead of a hand control.
Generalized Helices in an Ordinary Vn
Authors: Wong Yung-Chow
Published: November 2008

A linear vector m-space Rm defined along a curve (C) in a Vn and lying in the complete osculating space of (C) will be called a characteristic Rm of (C) if it is auto-parallel along the curve and makes constant angles with its tangent and the principal normals. Curves admitting a characteristic R1 have been studied by Hayden under the name of generalized helices and generalized by me§. In this paper we give a complete determination of the curves with a characteristic R2. Curves whose curvatures are proportional to a set of constants, which have been considered by Syptak for the particular case when Vn is an Rn, form one of the classes of curves of this type. As a consequence, the existence of the curves admitting a characteristic Rm (m > 2) is partly established, but the problem has not been completely solved. At the end we prove two theorems in connexion with two particular types of characteristic Rm's.
On the Parseval formulae for Fourier transforms
Authors: Edmonds Sheila M.
Published: November 2008

1. The object of this paper is to discuss conditions of validity of the Parseval formulae for Fourier integrals:

where the transforms are defined by ordinary convergence; we shall not be concerned with the more elegant theory in which they are given by convergence in mean.
A theorem on s-dimensional measure of sets of points
Authors: Besicovitch A. S.
Published: November 2008

We shall consider plane Borel sets of points, though the result and the proof hold in space of any number of dimensions and for more general sets.
Point-curve correspondences
Authors: Scott D. B.
Published: November 2008

In this paper we consider the properties of a point-curve correspondence between two (generally distinct) surfaces F and G, paying special attention to the idea of ‘induced’ and ‘extended’ correspondences. We also outline the theory of point-curve corre spondences between a curve and a surface, which is not only of interest in itself, but is of importance because such correspondences arise, as do correspondences between curves, as induced correspondences of point-curve correspondences between two surfaces. The results depend chiefly on Lefschetz's theory of correspondences between algebraic curves(1) and the properties of intersections of cycles on an algebraic surface given by the author (2), as well as on the preceding paper of this series (3). Use is also made of a paper by Severi (4) on the self-correspondences on a variable curve of an algebraic surface. The theory of correspondences on a single surface will be further developed in the next paper on this subject.
Note on the velocity of propagation of gravitation
Authors: Clark G. L.
Published: November 2008

The approximate value of the world invariant at a point in a plane containing a rotating cohesive system such as a rod is calculated for a point at great distances from the system. It is found that the invariant has a stationary value when the rod is seen to be pointing towards the point in question. It is concluded that the speed of propagation of gravitation is the same as that of light.
A note on relativistic second quantization
Authors: Chang T. S.
Published: November 2008

The relativistic second quantization of free bosons is extended to fermions. To know how relativistic creation and annihilation operators operate on bra and ket vectors, it is necessary to have a relativistic scalar product which in the case of free particles can be constructed. For particles interacting with each other or capable of emitting and absorbing particles of the same kind, it is pointed out that adequate wave equations for an indefinite number of particles each taking a separate time coordinate have not yet been found and so no scalar product can be found. Thus creation and annihilation operators can be defined only by their operation on a bra vector or a ket vector but not by both. With the help of these, the relativistic wave equations referred to above are proposed and their consistency conditions studied. For the particular case of scalar particles, an illustration is given that such a theory may admit certain expressions as probabilities.
A theory of the performance of radar on ship targets
Authors: Wilkes M. V. , Ramsay J. A.
Published: November 2008

The paper contains a theoretical discussion of the scattering by a ship of the wave sent out from a radar transmitter. An expression is derived for the power returned to the receiver of the radar installation, on the assumption that the ship is composed of a large number of scattering elements which scatter energy non-coherently. Calculations based on this expression are compared with the results of experiment and shown to be in good agreement as regards the variation of signal strength with range, provided that a suitable value for the effective radius of the earth (a quantity which changes with meteorological conditions) is taken.

An experiment is described in which the absolute level of the received signal from a ship was measured by comparing it with that from a metalized sphere flown from a balloon. The result was found to be in satisfactory agreement with theoretical expectations.
X-ray excitation of lutecium
Authors: Dunworth J. V. , Pontecorvo B.
Published: November 2008

The radioactivity of the naturally and artificially produced lutecium isotope has been investigated on a number of occasions (l, 2). A 4 hr. activity was observed when lutecium was activated by slow neutrons and later a 6-day period was observed by Hevesi and Levi (3). Heyden and Wefelmeier (4) discovered the natural radioactivity of lutecium, while Libby (5) made measurements indicating that this natural activity consisted of a negative electron spectrum of maximum energy 0·215 MeV. Mattauch and others (6) showed from mass spectrograph evidence that in addition to the main isotope of lutecium of mass 175 there was an isotope of mass 176 with an abundance of 2·5%. The second isotope had been reported previously by Gollnow (7), but not identified. Since Hf176 and Yb176 are known to be stable, whereas Hf175 and Yb175 do not exist in nature, the radioactivity of natural lutecium must be assigned to Lu176. Flammersfeld and Mattauch (8) measured the energy of the natural spectrum by the absorption method and found it to be 0·4 MeV. They observed also a γ-radiation (1–2 quanta per β-particle) of 0·26 MeV. In addition, they investigated the radiations resulting from slow neutron bombardment of lutecium. They estimate the shorter period, previously reported by Hevesi and Levi, to be 3·4 hr. and to consist of a β-spectrum of maximum energy 1·150 MeV. They estimate the longer period to be 6·6 days and to consist of β-spectrum of maximum energy 0·440 MeV. In view of the closeness of this latter energy to 0·4 MeV., Flammersfeld and Mattauch allocate the 6·6 days' activity to an isomeric state of Lu176. The short period is allocated by these authors to Lu177.