bluuebunny/crossref_metadata_2025 huggingface dataset:
Commuting flows and conservation laws for Lax equations
Authors: Wilson George
Published: November 2008

In recent years there has been great progress in the study of certain systems of non-linear partial differential equations, namely those that have a ‘Lax representation’

Here P and L are linear differential operators in one variable x, whose coefficients are l × l matrices of functions of x and t. Thus L has the form

where each ui is a matrix of functions ui,αβ(x, t), 1 ≤ α, β ≤ l. The symbol Lt means that we differentiate each coefficient of L, and as usual [P, L] = PL − LP. The coefficients of P are supposed to be polynomials in the ui, αβ and their x-derivatives, so that (1·1) is equivalent to a system of non-linear ‘evolution equations’ for the variables ui, αβ. The simplest example is the Korteweg–de Vries (KdV) equation

which has a Lax representation with

(Here l = 1, and there is only one coefficient ui, αβ. ) The connexion between the KdV equation and this ‘Schrodinger operator’ L was discovered by Gardner, Greene, Kruskal and Miura(6), but it was P. Lax (13) who first pointed out explicitly what we call the Lax representation given by (1·1) and (1·2). The notation (P, L), due to Gel'fand and Dikii(7), reflects this fact.
A frame with no admissible topology
Authors: Isbell John
Published: November 2008

Every finitary algebra is, of course, a topological algebra in the discrete topology. On the other hand, a topological Boolean σ-algebra is indiscrete; sequences (0, 0, …, x, x,…) with join x converge to (0, 0,…) with join 0, so 0 is in {x}−, and similarly x is in {0}−. For lattices with infinite joins but only finite meets, such as topologies, the matter is more interesting.
The low frequency scalar diffraction by an elliptic disc
Authors: Sleeman B. D.
Published: November 2008

Proc. Cambridge Philos. Soc.63 (1967), 1273–1280
2-convexity and 2-concavity in Schatten ideals
Authors: Jameson G. J. O.
Published: November 2008

The properties p-convexity and q-concavity are fundamental in the study of Banach sequence spaces (see [L-TzII]), and in recent years have been shown to be of great significance in the theory of the corresponding Schatten ideals ([G-TJ], [LP-P] and many other papers). In particular, the notions 2-convex and 2-concave are meaningful in Schatten ideals. It seems to have been noted only recently [LP-P] that a Schatten ideal has either of these properties if the underlying sequence space has. One way of establishing this is to use the fact that if (E, ‖ ‖E) is 2-convex, then there is another Banach sequence space (F, ‖ ‖F) such that ‖x;‖ = ‖x2‖F for all x ε E. The 2-concave case can then be deduced using duality, though this raises some difficulties, for example when E is inseparable.
Selection Rules in Nuclear Radiation
Authors: Taylor H. M.
Published: November 2008

In the first part of this paper we discuss the radiation from a single charged particle moving in an arbitrary central field of force and obeying Dirac's equation. We consider the electric quadripole and magnetic dipole radiation as well as the electric dipole. We derive the selection rules for the magnetic dipole radiation and collect together for reference the corresponding selection rules for the electric dipole and quadripole radiations. In the second part we discuss the relative intensities of the various types of radiation, treating in detail the cases where the selection rules for magnetic dipole and electric quadripole are simultaneously satisfied. Finally we show that these results have an important bearing on the theory of internal conversion of γ-rays. The internal conversion of soft γ-rays occurs with such high probability that the theory is unable to account for the experimental results unless it is assumed that the radiation is largely magnetic dipole in character. On the other hand, Fisk and Taylor (loc. cit.) were unable to account for the presence of magnetic dipole radiation in appreciable amounts. We show that this is due to the fact that, of the two possible transitions (a and e of § 2) in which both magnetic dipole and electric quadripole radiation can be emitted, Fisk and Taylor considered only the second. In the case of the second, corresponding to a transition between two distinct terms, we show that Fisk and Taylor were correct in predicting a negligible amount of magnetic dipole radiation, but in the case of the first, corresponding to a transition between two levels of one multiplet term, we find that there is indeed a high percentage of magnetic dipole radiation.
A second note on the simple group of order 6048
Authors: Edge W. L.
Published: November 2008

1. The simple group of order 6048 has a permutation representation of degree 28 and two of degree 63. These are conspicuous in SU(3,32), the representation as a group of 3-rowed matrices, unitary and of determinant + 1, whose elements all belong to GF(32), or to J as we name this field. When these matrices as linear transformations in the projective plane ϖ over J they leave invariant the unit Hermitian form

and the single capital U will henceforth signify the group so acting. There are 28 isotropic points n in ϖ satisfying H3 = 0; they furnish the representation of degree 28. The other 63 points p in ϖ are non-isotropic and furnish one of the representations of degree 63. The other is furnished by 63 triangles T; the vertices of a T are all p and every pair of them is linked in the antipolarity Π set up by H3; if their coordinate vectors are y and z then y′z = 0. Each p is a vertex of 3 T. These matters are fully explained in (7). Those points linked by II to a given point A are collinear, on the polar of A; no p lies on its polar, every n does. The polars t of points n are called tangents of H3 = 0, and the only isotropic point on t is its pole. The polars s of points p are secants of H3 = 0: each consists of 4 n with 3 pairs of linked p and, given s, it is these 3 pairs that complete those T having a vertex at the pole of s.
On the one centre expansion of scattered distorted spherical and hyperspherical waves
Authors: Edelstein L. A.
Published: November 2008

1. In many problems in theoretical physics one encounters situations in which it is desirable to be able to expand orbitals centred at one particle about another centre, frequently, but not always, another particle.

There have long been available methods for permitting this to be effected for bound states but no such procedures are available for the continuum.

Typical examples of physical situations in which such procedures are desirable are positron scattering from atomic and molecular systems, electron scattering from molecules, and nucleon scattering from light nuclei, to mention only a few.

The author's interest arose from a desire to establish a general theory of slow electron scattering from molecules, and an account of such a theory will appear shortly (1).
Dissociation under a Temperature Gradient
Authors: Dirac P. A. M.
Published: November 2008

If a gas, each of whose molecules is capable of dissociating into two similar molecules, be kept at a constant temperature, it would attain an equilibrium state in which the rate at which double molecules dissociate is equal to the rate at which they are formed by recombination of single molecules, there being a different equilibrium state for each different temperature. If, now, the gas be subjected to a temperature gradient, the concentrations necessary for equilibrium would be different at different points, so there would be diffusion between the two kinds of molecules, a steady state being attained when the rate of diffusion of double molecules into any region is equal to their excess rate of decomposition over their rate of formation within that region.
The kinetic theory of metallic conduction
Authors: Davies E. L. , Livens G. H.
Published: November 2008

1. The kinetic theory of metallic conduction is usually supposed to lead to formulae for the conductivity coefficients which in some respects are very much at variance with the actual phenomena. In spite of the important successes achieved by the theory, which in themselves are almost sufficient evidence that the fundamental basis is correct, these discrepancies are regarded in some quarters as so serious as to warrant the complete rejection of the theory as beine inadequate to account properly for the facts.
Convolution and the second dual of a Banach algebra
Authors: Pym John S.
Published: November 2008

We shall point out a connexion between convolution (as defined in (5); see also (7)) and the Arens multiplication on the second dual of a Banach algebra (1, 2). This enables us to demonstrate in an easy way the existence of a commutative Banach algebra whose second dual is not commutative. Examples can also be found in (3, 9) and elsewhere.
The region of convergence of simple Taylor–Dirichlet series
Authors: Read G. A.
Published: November 2008

In several recent papers G. L. Lunc (7, 8, 9, 10, 11) has examined the series

where {mn} is a sequence of positive integers or zeros, and μn > 0, μn ↑ ∞. The discussion has been aimed towards extending results which are well-known for Dirichlet series to series (1) which Lunc has very reasonably called a Taylor–Dirichlet series.
Inversion of a class of transforms with a difference kernel
Authors: Varma V. K.
Published: November 2008

1. Recently Ta li(10) Buschman(2, 3), Erdelyi(4) and Shrivastava(8, 9) obtained solutions of integral equations involving polynomial kernels in the range of integration x to 1. Widder(12) obtained an inversion of a convolution transform with a Laguerre polynomial as kernel.
Generalized canonical formalism for degenerate dynamical systems
Authors: Shanmugadhasan S.
Published: November 2008

The familiar classical dynamical theory treats only a restricted class of dynamical systems. For a Lagrangian depending on n generalized coordinates, the n generalized velocities, and (perhaps) the time the usual theory considers only the normal case in which the Hessian matrix of the Lagrangian with respect to the generalized velocities has rank n. But the rank of this Hessian matrix can be less than n in a given dynamical system. Dynamical systems with this property are called degenerate in the present work. Some important physical systems do in fact belong to this degenerate class: for instance, Einstein's theory of the gravitational field, the theory of the electromagnetic field in terms of the electromagnetic potentials, and the theory of the neutral vector meson field. This circumstance makes it necessary to have a general theory of degenerate dynamical systems (and of the associated degenerate variational problems).
Concepts of differentiability and analyticity on certain classes of topological groups
Authors: Reid G. A.
Published: November 2008

We introduce the concepts of a local seminorm on a topological group and of a locally convex group, showing that discrete groups, locally compact Abelian groups and compact groups are locally convex, and that a topological vector space is locally convex as a linear space if and only if it is locally convex as a group. We show that notions of differentiability, analyticity and derivability can be defined for locally convex groups and that these notions are suitably related and well behaved. We prove that for a locally compact Abelian group G the Fourier transforms of measures of compact support on the character group Ĝ are analytic, and for G compact the coefficients of continuous irreducible unitary representations are. Using these spaces of analytic functions we define the basic concepts of a differential geometry.
Integrals involving products of generalized Whittaker functions
Authors: Ragab F. M. , Simary M. A.
Published: November 2008

In ((4)) Slater gave expansions of the generalized Whittaker functons pFp(x). (She gave this name to the generalized hypergeometric function pFp(x), since it is a generalization of the well-known Whittaker function 1F1(x).) In another paper ((2)), Ragab gave a series of products of generalized Whittaker functions in terms of such functions or in terms of generalized hypergeometric functions pFq(x). In the present paper, integrals involving products of generalized Whittaker functions will be evaluated in terms of such functions or of other generalized hypergeometric functions.
On a multiple integral representation of a kernel of Fox
Authors: Varma V. K.
Published: November 2008

A function K(x) is said to be a symmetrical Fourier kernel if the integral equation

remains valid when f(x) and g(x) are interchanged amongst themselves.
Kelvin-Helmholtz instability of a dusty gas
Authors: Michael D. H.
Published: November 2008

In a recent paper ((2)) Saffman gave a representation of the flow of a dusty gas in which the dust was described in terms of a number density of small particles with very small volume concentration but appreciable mass concentration. In the same paper the problem of the stability of plane parallel flows to small disturbances was formulated, and subsequently the author ((1)) gave some approximate results for the stability of plane Poiseuille flow.
The Theory of the Scattering of Short X-rays by Molecular Hydrogen
Authors: Massey H. S. W.
Published: November 2008

The importance of experimental investigation of the intensity of scattering of X-rays by crystals has long been realized, since it affords valuable information regarding the charge distribution in the atoms of the crystal. Recently Debye has pointed out the importance of similar investigations of the scattering by gases. Here diffraction effects may be observed due to regularities in the structure of the gas molecules as distinct from crystal diffraction, where the regularities are due to the orderly arrangement of crystal atoms. This provides a means of obtaining important information as to the structure of molecules, and Debye and his collaborators have obtained interesting experimental results in this field. Debye has also introduced an approximate theory of the results obtained, but owing to the great complexity of the molecules considered it is at present impossible to proceed any further in the theoretical discussion. However, the case of molecular hydrogen may be worked out for short X-rays, since the charge distribution in the ground state of the molecule is fairly well known. It is the purpose of this paper to calculate the intensities of short X-rays scattered from hydrogen gas. The case of other diatomic molecules will not be very dissimilar.
Note on the use of a thyratron with a Geiger counter
Authors: de Bruyne N. A. , Webster H. C.
Published: November 2008

The thyratron valve is essentially a triode which contains a small quantity of some inert gas, usually mercury vapour. If a suitable voltage be applied between anode and filament, with sufficient negative grid bias, the valve behaves like an ordinary vacuum tube; but, as the grid is made more positive, at a certain critical voltage an arc strikes between the anode and the filament, with enormous increase in the anode current. The grid then exercises no further control over the anode current, which must be limited by external resistance so as not to exceed the saturation emission current from the filament. If the voltage across the tube exceeds the “disintegration” potential, of some 20–25 volts, the cathode will be disintegrated by positive ion bombardment. Limitation of the anode current to the emission of the filament ensures that the voltage, while higher than the ionization potential of mercury, is considerably below the disintegration potential.
Characteristics of complexes of conics in space of four dimensions
Authors: James C. G. F.
Published: November 2008

In this note we obtain a system of characteristics on which depend the main enumerative properties of complexes, or systems ∞3, of conics in space of four dimensions. The method used is applicable to a large number of problems of this nature, and we select this illustration as being analogous to congruences in ordinary space. In particular a finite number of conics pass through an arbitrary point, thereby defining the order, n, of the system. Linear complexes are those for which this order is unity. The conics of a complex satisfy eight simple conditions, in general conditions of contact with a fixed form or forms, but they may include conditions of incidence with a surface, each such counting once, with a curve, each such counting twice, or with fixed points each such counting thrice. In particular only incidence conditions can occur when the system is linear, for otherwise more than one conic would pass through certain ∞3 points of space. Points through which more than a finite number of conics pass are termed singular, as well as their loci. Directrix constructs are necessarily singular, but they do not necessarily exhaust the singular system, for the complex may possess ∞2 curves lying on a surface. In the linear case through a point of a singular curve pass ∞2 conics, and through a point of a singular surface ∞1 conics. The possibilities in the nonlinear cases are too numerous to be detailed. Similarly the system of planes of the conies may possess a singular curve, through which ∞2 of the planes pass.
On Hausdorff's proof of the extended Riesz-Fischer theorem
Authors: Pollard S.
Published: November 2008

1. The extended Riesz-Fischer theorem, in the special case of the trigonometrical orthogonal system, states that, if a and β are real numbers such that

then

implies, whenever α ≤ 2, the existence of a solution of the system of integral equations,

such that

and further

When a = 2, this reduces to the original Riesz-Fischer theorem.
The number of apparent double points of certain loci
Authors: Edge W. L.
Published: November 2008

1. A sketch of the first part of this paper, together with some of the second part, was first written out in October last, shortly after the publication of Mr Babbage's paper, A series of rational loci with one apparent double point. It is well known that the Del Pezzo quintic surface the only non-ruled quintic surface in [5], has one apparent double point; Babbage shows that this surface is a member of a series of loci in [2n + 1], each of which has one apparent double point; he establishes the existence of these loci from their representations on flat spaces ∑n, these representations being analogous to the plane representation of by means of cubic curves through four fixed points.
Some factors in the design of hot cathode X-ray tubes for steady running
Authors: Harper W. R.
Published: November 2008

Some general considerations regarding the design of hot cathode X-ray tubes for producing an accurately constant and repeatable beam of X-rays are first given. The construction of a tube in accordance with these considerations is then described. This tube has been used to show that except with small tube-currents the quality of the vacuum in such a tube has little influence on the steadiness of the beam if the electrical input to the tube can be satisfactorily controlled.

It is shown that the X-radiation associated with the fluorescence often observed in X-ray tubes has usually but not always a negligible effect on the steadiness of the beam.

Some factors limiting tube current when the filament is enclosed in a box containing a focussing slit are mentioned, and the mechanism of formation of the focus is discussed with reference to its size. The use of an auxiliary electrode to increase the tube current and improve the focussing is also discussed.
Some surfaces containing triple curves, II
Authors: Roth L.
Published: November 2008

1. A general threefold in [5] has an apparent double surface whose projection on to a [4] is a surface which contains a triple curve and which is characterised by the fact that it has no improper nodes; such surfaces have been considered in a previous paper. In the present note we consider a class of surfaces in [5] which possess triple curves and project into surfaces in [4] having improper nodes: these are the surfaces which represent the chords of a general curve of ordinary space upon a quadric primal Ω of [5]. For the representation of a line congruence of [3] by the points of a surface on Ω, reference may be made to a previous note, the results of which are used in the present work.
Related quadrics and systems of a rational quartic curve
Authors: Telling H. G.
Published: November 2008

1·1. The points, tangents, osculating planes, …, osculating primes of a curve may be said to form a system which is characterised by the number of these elements which are incident with a prime, …, line, point, respectively. For the normal rational quartic curve the system is (4, 6, 6, 4); projection of this system from a general point gives the system (4, 6, 6) in [3]; section by a general prime gives the system (6, 6, 4). These two systems in [3], which are the systems with which we are concerned in this paper, are duals of one another, and will be called systems of the first and second kinds respectively.
The Pedal (3, 2) Correspondence
Authors: Vaidyanathaswamy R.
Published: November 2008

This is a sequel to recent communications to the Society, on “The General (m, n) Correspondence” and “The (2, 1) Correspondence.” For convenience the reference numbers here adopted will be consecutive with what has preceded, so that the present work opens with § 6 as a direct consequence of previous results in earlier sections.
Configurations defined by six lines in space of three dimensions
Authors: Todd J. A.
Published: November 2008

The investigations which follow were originally suggested by the now classical problem of Cayley, the determination of the condition that seven lines in space, of which no two intersect, should lie on a quartic surface. This problem suggests the consideration of the linear system of quartic surfaces which pass through six given lines, and this, essentially, is the basis of all that follows.
The number of α-particles emitted by thorium C + C′
Authors: Watson S. W. , Henderson M. C.
Published: November 2008

The number of α-particles emitted per second by thorium (C + C′) has been determined by an ionisation method. The number obtained is 4·26 ± ·08 × 1010 α-particles per second per curie equivalent γ-ray activity when in equilibrium with radio-thorium and when measured by the γ-rays of thorium C″ through 18 mm of lead. The result agrees within 1 per cent, with that extrapolated from Shenstone and Schlundt's values.

The same apparatus was used to determine the slope of the Bragg curve over the first three centimetres of the range. The data fit the curve as given by I. Curie and Behounek within the accuracy of experiment. The curve given by G. H. Henderson falls too rapidly as it approaches the axis of ordinates.
A note on the theory of the photo-electric current across a metal semi-conductor contact
Authors: Fowler R. H.
Published: November 2008

When any metal film is deposited on the surface of a massive copper oxide slab and illuminated with visible light, a current flows in such a direction that the electrons pass from the Cu2O to the metal in greater numbers than from the metal to the Cu2O. This is the reverse direction to the easier flow of electrons under an applied voltage when the contact is allowed to act as a rectifier. This effect is sometimes called the “Vorderwandeffekt” to distinguish it from the similar group of phenomena which occur at the natural interface between massive copper and Cu2O when the surface of the copper has been oxidized in situ (“Hinterwandeffekt”).
The theory of ionisation measurements in gases at high pressures
Authors: Lea D. E.
Published: November 2008

The columnar theory developed by Jaffé to account for the recombination of ions in alpha particle tracks is extended to beta rays by taking account of the clusters of secondary ionisation. Reasonable agreement is obtained with experiment. Recombination in proton tracks produced in hydrogen by neutrons is shown to be in agreement with the columnar theory, but in the case of nitrogen nuclear tracks in nitrogen the recombination is only a hundredth of that predicted by the theory. An explanation of this effect is advanced, and it is suggested that recombination is likely to be abnormally small for all heavy nuclei of velocities not exceeding 5 × 108 cm. per sec.

An experimental determination of the coefficient of recombination of ions in nitrogen and hydrogen at pressures of 20, 40 and 90 atmospheres is reported.

My thanks are due to Dr Chadwick for interest in this work, and to Dr Gray and Dr Tarrant for advice on the experimental technique of high pressure ionisation measurements. I am indebted also to the Department of Scientific and Industrial Research for a maintenance grant.
On isolated singularities of surfaces which do not affect the conditions of adjunction (Part II.)
Authors: Du Val Patrick
Published: November 2008

In the first part of this paper I investigated the nature of the isolated singular points which can appear on an algebraic surface without affecting the conditions of adjunction and the arithmetic genus of the surface. It appeared that such points have neighbourhoods analysable into connected chains or trees of curves, each rational and of virtual grade − 2, and arranged either in a single chain (giving a binode, or a conic node if there is only one curve) or in three chains of, say, n, p, q curves respectively, one end curve of each chain meeting one which we may call the central curve of the tree. The values of n, p, q are not arbitrary, but satisfy either p = q = 1, or n ≤ 4, p = 2, q = 1; the unodes given by the former case we call U, those given by the latter case U, with a suffix indicating the reduction in class in each case. I pointed out the similarity between these results and Coxeter's enumeration of groups generated by reflexions, there being a one-one correspondence between these singularities and Coxeter's groups, with the restriction that the only groups with which we are concerned are those in which any two primes of symmetry are inclined at either π/2 or π/3. In fact, the curves which make up a complete neighbourhood correspond to the bounding primes of a fundamental region, two curves which do not meet corresponding to mutually perpendicular, primes, and two which do to primes inclined at π/3.
The Genus of a Developable Surface
Authors: Baker H. F.
Published: November 2008

Cayley's remark that the formula by which the genus of a surface, according to Clebsch's definition, may presumably be computed leads to a negative number in the case of a cone, or a developable surface, or a ruled surface in general, has great importance in the history of the theory. But it would appear, from various indications, that, for a developable surface at least, it is more often quoted than read. I have thought therefore that the following simplifying remarks may have a use. Cayley uses formulae, due to Salmon and Cremona, without reference to the memoir where these are given in detail. Of two of these, for the number of tangents of a curve which meet it again, and for the number of triple points of the nodal curve, proofs by the theory of correspondence are extant; for the present purpose it is only necessary to have the sum of these two numbers. I do not know whether it has been remarked that there exists a remarkable formula for this sum, very similar to, and including the ordinary formula for the number of triple points of a general ruled surface (and like this probably capable of a direct proof by the theory of correspondence). For the genus of the nodal curve, deduced by Cayley from the Salmon-Cremona formulae, a proof by the theory of correspondence (in the general case, sufficient for the purpose in hand, in which i = τ = δ = δ′ = 0) is added here, which seems to have a certain interest.
The Cantor Manifolds Lying on a Closed Surface. Part II
Authors: Ursell H. D.
Published: November 2008

Let F be a regular surface in ordinary space and let Z be a cut on F. Then there is on Z a point z which is the limit of a decreasing sequence of 2-dimensional Cantor manifolds lying on F.
The Mass of the Neutrino
Authors: Henderson W. J.
Published: November 2008

From experimental results on the shape of the upper portion of the continuous β ray spectra of thorium C and thorium C″ the distribution of energy is deduced. This distribution is compared with Fermi's theory wherein the shape of the curve depends on the mass of the neutrino. The comparison supports Fermi's conclusion that the mass of the neutrino is zero or certainly not more than a very small fraction of the electronic mass.
Note on the development of the Circulation around a thin Elliptic Cylinder
Authors: Howarth L.
Published: November 2008

In a previous paper the present writer gave a solution of the problem of determining the circulation around a thin elliptic cylinder in a steady stream of slightly viscous fluid when the major axis of the cylinder is inclined at a small angle to the direction of the flow at infinity. The present note gives a largely qualitative analysis of the manner in which this circulation is built up when the motion of the fluid or cylinder is started instantaneously from rest with a given velocity.
The Dispersion Electrons in the One-Electron Problem
Authors: Hargreaves J.
Published: November 2008

In this paper we use Dirac's relativity quantum mechanics to derive the well-known Kramers-Heisenberg dispersion formula for an atom with one electron. The treatment is not limited to the case of a central field, but is quite general. An expression is also obtained for the dipole moment to which is due the incoherent scattering. The formulae obtained are similar to those obtained by O. Klein for the case of a central field. We find in this way an explicit expression for f, the number of dispersion electrons for any line of the optical spectrum (being a measure of the intensity of the line), in terms of the solutions of the four wave equations of Dirac's theory. It is further shown that the sum of the number of dispersion electrons for any state of the atom is not exactly unity, but differs from it by an amount of the order of 10−4. The result Σf = 1 has been shown by London to hold exactly for the simple wave equation as originally given by Schrödinger. It is here shown that the exact relativity treatment has a very small effect.
Some Experiments with an α-Particle Counter
Authors: Appleton E. V. , Emeléus K. G. , Barnett M. A. F.
Published: November 2008

Experiments carried out with the aim of elucidating the action of the α-particle counter are described. A normal counting circuit including an ionization chamber with point and plane electrodes was used. The “wave-form” and intensity of the transient impulses resulting from discharges stimulated by single electrified particles have been determined using a cathode-ray oscillograph. It is found that the nature of the impulse is determined largely by the capacity and resistance of the system, and not by the nature of the electrical stimulus. The potential impulse normally detected by string electrometer methods is shown to be proportional to the current responsible for the recharging of the electrostatic capacity of the counter after the rapid discharge of the capacity initiated by the electrified particle.
A Helical Method of focussing β-Rays
Authors: Tricker R. A. R.
Published: November 2008

Experiments with a helical method of focussing β-rays are described. Although the method does not appear to be so generally useful for obtaining ordinary β-ray spectra, it possesses certain advantages, of which may be mentioned:

(1) Large dispersion;

(2) Greater intensity;

(3) Good screening.

The method produces some evidence on the question of the continuous background, and experiments with it in this connection are described.
Notes on Optical Constants
Authors: Darwin C. G.
Published: November 2008

In a recent paper the present writer made an attempt to reduce the ordinary optical constants of matter to a more primitive form by introducing “scattering indices.” Their use solves half the problem of optics, for general formulae connect these indices with the behaviour of matter in bulk. There remains the deduction of the scattering indices from whatever we are assuming to be the reaction to light of the actual atoms, and even in the classical theory this encounters rather formidable difficulties of convergence. As long as we are content with no great rigour it is a fairly simple matter to deduce the scattering indices for any assumed model without any of those rather difficult arguments about the polarisation of the medium which play a part in the ordinary presentation.
The mechanism of replacement as illustrated by metasomatism of the Whin Sill
Authors: Wager L. R.
Published: November 2008

Where the solutions which carried the lead vein minerals pass along joint planes through the Whin Sill, the dolerite is altered to a white, compact rock for a distance of several feet. The original texture is well preserved in the altered dolerite; the ferromagnesium minerals are replaced by calcium, magnesium and iron carbonate, the feldspars by a minutely crystalline mixture of kaolin and muscovite and the ilmenite by small rutile crystals. The former areas of interstitial quartz are increased and apatite remains completely unchanged. The microscopic evidence that there is neither gain nor loss of alumina during the metasomatism, taken in conjunction with analyses of altered and unaltered rock, may be used to prove that there was a contraction of about 14% during metasomatism. This result is corroborated by the occurrence of cracks which are formed by shrinkage during metasomatism and which are now filled with carbonates.
Some Applications of Fourier series in the Analytic Theory of Numbers
Authors: Mordell L. J.
Published: November 2008

It is a familiar fact that an important part is played in the Analytic Theory of Numbers by Fourier series. There are, for example, applications to Gauss' sums, to the zeta functions, to lattice point problems, and to formulae for the class number of quadratic fields.
Note on the Gibbs Phenomenon
Authors: Verblunsky S.
Published: November 2008

The Fourier series of the function defined by

is

and if

then

while

The expression (1) exceeds (2); this is the Gibbs phenomenon.
Note on Pauli's Exclusion Principle
Authors: Ursell H. D.
Published: November 2008

A simple explanation of Pauli's principle was first given with the wave mechanics. Its interpretation in the new theory was that the wave functions of Schrödinger were antisymmetrical in all the electrons concerned. Thus when the interactions of the electrons may be neglected, the wave function (for a system of n electrons) can never be of the form

in nature, but only of the form

Hence σ, τ,…ω must be all distinct.
The Real Representation of the Commutator S−1 T−1 ST in Four Dimensions
Authors: Robinson G. de B.
Published: November 2008

It has been remarked in a former paper that perhaps a fuller understanding of the theory of linear groups may be arrived at by considering the real representations. It is sufficient that these be irreducible in the real field. In the present paper we continue this investigation and deal with the angles of rotation; in particular we find the form of the commutator

where S and T are substitutions of order p1 and p2 and have angles of rotation θ, θ′, θ″, … and φ, φ′, φ″, …. If certain conclusions regarding the orders of S, T, and C may be drawn we shall then be able to attack a very important problem—;that of Primitivity. This is in essence very similar to the method of Blichfeldt, since the commutator is the product of

which are both of order p2.
Experiments with a simple type of cloud chamber
Authors: Ho P. C. , Halliday E. C.
Published: November 2008

Tests of the simple type of cloud-chamber previously described by C. T. R. Wilson(1), using various ionising sources, indicate that satisfactory photographs can be obtained over a range of final pressures in the chamber extending from atmospheric down to 7 cm. of mercury.
On a theorem of Paley and Wiener
Authors: Hardy G. H.
Published: November 2008

The conjugate g of a periodic and integrable function f is not necessarily integrable, even when f is monotone inside its fundamental interval. Paley and Wiener, however, proved that g is integrable if f is monotone and odd. A simpler proof was given later by Zygmund‡.
Some properties of the vibrational spectrum of a lattice
Authors: Blackman M.
Published: November 2008

The vibrations of the particles which constitute a crystal can be represented in terms of normal vibrations subject to the condition that the energy of the crystal can be expressed as a homogeneous function of the second order in the displacements. This has been shown by Born and by Waller. It has been proved by Debye and, more generally by Born, that the density of the normal vibrations varies as ν2 when the frequency is very small. The spectrum has recently been worked out for a Born-v. Kármán lattice in the two- and three-dimensional cases. Otherwise practically nothing is known about the spectrum in individual cases, except perhaps in isolated cases where the frequency branches split up and the frequency spectrum shows certain gaps.
The structure of a β-ray line by semicircular magnetic focusing
Authors: Li K. T.
Published: November 2008

The actual distribution of a homogeneous group of electrons by semicircular magnetic focusing is calculated for different types of source.

The dependence of maximum intensity on the radius of curvature of the path of electrons is investigated and the approximate power laws assigned to different cases. The results can be used in the measurement of the intensity of β-ray line spectra.

Tables of numerical values of effective solid angle subtended by the photographic plate at the source are given for some special cases.

The results of the above calculations have been verified experimentally.
The azimuthal dependence of processes involving mesons
Authors: Chang T. S.
Published: November 2008

The elastic scattering of Yukawa's particles, or mesons, by a static electric field is recalculated by considering the scattering as a process in which the incident meson is absorbed and another meson, i.e. the scattered meson, is created, and Laporte's result is confirmed. This method promises an easier handling of the second approximation.

A similar dependence of the angular distribution of the emitted quanta upon the spin of the incident mesons in a process like Y− + P → N + hν is found. Thus, the angular distribution contains a dependence upon the azimuthal angle in general. For the reverse process, the angular distribution of the emitted mesons, both longitudinal and transverse, contains also a dependence upon the azimuthal angle, which disappears only when the incident light is circularly polarized.

The writer wishes to thank Prof. W. Pauli for calling his attention to the azimuthal dependence in the problem of elastic scattering and Dr C. Møller for help given during the preparation of the paper.
The adsorption of dipoles
Authors: Miller A. R.
Published: November 2008

In the present paper Bethe's approximation is applied to the statistical treatment of the adsorption of dipoles considering (a) the electrostatic forces alone, and (b) the electrostatic and van der Waals forces jointly. In each case formulae are obtained for the adsorption isotherm and for the variation of the heat of adsorption with the fraction of surface covered. In case (a) the heat curves are compared with those obtained by Wang who used a different approximation to take into account the electrostatic interactions due to particles adsorbed on outer sites. The curves are of the same general shape; sources of differences in them are discussed. In case (b) the results are compared with those obtained by Roberts using a different method. This treatment confirms his result that the electrostatic and van der Waals forces give contributions to the variation of the heat of adsorption which are of opposite sign and almost counterbalance one another, so that the resultant variation in the heat of adsorption is very much less than would be expected from a consideration of forces of one type only. This comparison shows further that the distribution of particles on the surface, which is taken into account in the statistical method used in this paper, has a considerable effect on the variation of the heat of adsorption.
Invariant groups associated with an algebraic surface
Authors: Scott D. B.
Published: November 2008

Alexander (1, 2) has introduced certain topological invariants of a manifold which arise from the intersections of cycles of non-complementary dimensions, and he points out that they are not derivable from the Betti and torsion numbers, nor from the fundamental group. In the present paper we consider some topological invariants of this type on an algebraic surface, and, although we cannot define them completely, we show that they are intimately connected with the multiplications of the period matrix of the simple integrals of the first kind. We are then led to a concept which we call the “intersection group” of the surface, which is, by its definition, topologically invariant, and we show that it is also invariant under birational transformations. The proofs are based on Lefschetz's theory of cycles for an algebraic surface (4) and some simple properties of the period matrix of an algebraic curve. The results obtained here have a number of applications to the theory of ∞3 correspondences between algebraic surfaces, as we propose to show in a later paper.
On the subadditivity of the transfinite diameter
Authors: Schiffer Menahem
Published: November 2008

Let M be a bounded and closed set of points in the complex z plane; d(M), a set-function which is of great importance in potential and function theory, may then be defined as follows. n points z1, z2, …, zn in M are so chosen that the product of the mutual distances

has the greatest possible value Then it can be proved that

exists. Thus the set-function d(M), named by Fekete the transfinite diameter of M, is defined.
The geometry of the binary quintic form
Authors: Todd J. A.
Published: November 2008

1. It is familiar that the properties of the invariants and covariants of binary forms of the first four orders admit of elegant geometrical interpretations on rational normal curves and their projections. For forms of order higher than four the number of irreducible concomitants which appear in the complete system increases rapidly. It is the purpose of this note to exhibit geometrically the 23 irreducible concomitants of the binary quintic, using the rational normal curve R5 in space of five dimensions and its projection R′5 on to a prime as a foundation.
On the transverse circulation in streams
Authors: Jeffreys Harold
Published: November 2008

It is well established by observation that in the ordinary flow of a stream there is a slow transverse flow outwards from the middle near the bottom, and inwards near the top; and such a circulation seems to be required to explain the fact that the filament of maximum onward velocity is depressed some distance below the centre of the free surface. Qualitative explanations of the phenomenon have been offered, depending on the extra friction near the sides, but it seems that the true explanation must depend on something more fundamental. For any explanation to be satisfactory it must show uniform linear flow to be impossible; and actually it can at least be rendered highly probable that such flow can exist over a very wide range of conditions.
Secondary Electron Emission from Solid Metal Surfaces
Authors: Sharman C. F.
Published: November 2008

A considerable amount of experimental work has been done in recent years on the properties of the electrons which leave a metal surface when it is bombarded by an incident beam of electrons. Farnsworth, who first studied the energy distribution of the secondary beam systematically, showed that it is made up of two groups. The first has the same energy as the primary beam and may be regarded as consisting of reflected primary electrons; and the second has for the most part a very small energy, of the order of a few volts, and contains the true secondaries derived from the metal. The present writer in a preliminary note has described some measurements of the differential energy distribution by means of the magnetic spectrum method. Owing to an inadequate technique those experiments are considered to be valid only in the upper half of the energy range, where the form of the distribution has several features of particular interest when the mechanism of secondary emission and of electron reflection are considered. In this note, the author called attention to the peculiar depression in the curve immediately preceding the “reflected” peak. It was impossible then to insist on the reality of such an effect, owing to the probable presence of an instrumental cause (which was given there as the explanation). That it is real, however, has been demonstrated in a striking manner by Whiddington and Brown, who have succeeded in recording electrons as slow as 100 volts photographically on a specially sensitised film.
On surfaces of sectional genus six
Authors: Roth Leonard
Published: November 2008

In two previous notes I have given the various types of non-singular surfaces with sectional genera 4 and 5. In this paper the methods previously adopted are applied to the non-singular surfaces of sectional genus 6. For purposes of classification these may be divided into the four classes:

(1) Regular, non-rational.

(2) Rational.

(3) Irregular, non-referable.

(4) Referable.
The wave equation in conformal space
Authors: Bhabha H. J.
Published: November 2008

In a recent paper Dirac has shown that by passing from the ordinary Euclidean space to a four-dimensional conformal space, some of the equations of physics can be written in a tensor form, the indices of which take on six values. Those equations which can be written in this form are then invariant under conformal transformations of the Euclidean space. Among the equations of physics which have this more general invariance are the Maxwell equations, as was proved by a direct transformation a long time ago by Cunningham, and Bateman, so that Dirac's paper provides an alternative and more general proof of this result. Certain errors ∥ in Dirac's paper, however, necessitate a reformulation of the proof. Before we do this in § 2, we briefly recapitulate in § 1 some of the general results derived there. In § 3 we investigate further the conformal invariance of the wave equation for an electron in the presence of a general electromagnetic field.
The dissociation of hydrogen by tungsten
Authors: Bryce G.
Published: November 2008

The dissociation of hydrogen by a hot tungsten filament has been studied under conditions such that all the atomic hydrogen produced is effectively removed by reaction with molybdenum or tungsten oxide. The rate of production of atomic hydrogen is many times greater than was inferred from earlier work. With the tungsten at constant temperature the rate of dissociation is proportional to the square root of the pressure. A formula is given for the rate of production of hydrogen atoms per sq. cm. of the tungsten per second.
A note on molecular orbitals
Authors: Devonshire A. F.
Published: November 2008

A method of obtaining an approximate solution of Schrödinger's equation with a non-central field of force is given for the case when a solution of a similar equation involving the angular coordinates only is known.

In conclusion I should like to thank Prof. J. E. Lennard-Jones, F.R.S., for suggesting the problem and for his continued interest in it, and also the Department of Scientific and Industrial Research for a grant.