bluuebunny/crossref_metadata_2025 huggingface dataset:
Two sets of conditions for expansion in a Laurent's series
Authors: Kennedy Margaret Delina
Published: November 2008

Let f(z) be a single-valued function of z, defined in the domain D given by ; suppose that f(z) is integrable in the general Denjoy sense round all circles with centre ζ lying in D, and along all portions of radial lines intercepted between two such circles.
On the Diurnal Variation of Ultra-Short Wave Wireless Transmission
Authors: Appleton E. V.
Published: November 2008

An interpretation of ultra-short wave wireless phenomena is given which indicates that the maximum number of electrons per c.c. in the atmospheric ionized layer is of the order 105 to 106.
The influence of an illuminated mercury surface on the Franck-Cario reactions
Authors: Hirst Herbert Sim
Published: November 2008

It has been found that illumination of a mercury surface by ultraviolet light strong in the line λ=2536·7 Å., exerts a marked influence on the photochemical union of hydrogen andgases such as oxygen, ethylene, and carbon monoxide. The rate of reaction is proportional to the first order to the area of surface exposed.

The formation of a mercuric oxide film has been shown to occur only in the presence of a mixture of hydrogen and oxygen, and alternative mechanisms for its formation are suggested.

In the case of nitrogen-hydrogen mixtures, hydrazine and ammonia have been identified in the reaction products, and in the case of hydrogen and oxygen, hydrogen peroxide and water have been found, and it is suggested that the reactions proceed in steps.

The catalytic efficiency of the surface depends on its cleanness and is cut down by poisons, such as the reaction products in the case of the combination of hydrogen and carbon monoxide.
A theorem on Newton's polygon
Authors: Frith R.
Published: November 2008

Castelnuovo has shown that the maximum freedom of a linear system of curves of genus p is 3p + 5, and that a system with this maximum freedom consists of hyperelliptic curves which can be transformed into a system of curves of order n with an (n − 2)-ple base point and a certain number of double base points; the only exceptions being that when p = 3 the system may be transformable into that of all quartics, and when p = 1 the system may be transformable into that of all cubics. Further, since, in the transformed system, the characteristic series is a it is non-special and hence the redundancy of the base points is zero, therefore each of the double points of this system reduces the freedom by exactly three; and hence if we remove all the double points we get a system of curves of genus p′ and freedom r′ = 3p′ + 5 with only one base point. If we take this point for origin the system of curves can be represented by a single Newton polygon containing in its interior exactly p points, collinear since the curves are hyperelliptic (p ≠ 3), and containing on its boundary 2p + 6 points. From this we can deduce immediately a theorem concerning convex polygons drawn on squared paper; I shall now give an a priori proof of this theorem.
Foucault currents in cylindrical shells and ribbons
Authors: Carter F. W.
Published: November 2008

This article deals with the currents set up in thin-walled cylinders by two-dimensional field variation in the plane of the section, and is applied particularly to the problem of the distribution of alternating currents in strip conductors, a matter of some technical importance.
Die lokale Struktur der ebenen Cantorschen Mannigfaltigkeiten
Authors: Kaufmann Boris
Published: November 2008

1. Die Fassung des lokalen Dimensionsbegriffes führt zu verschiedenen neuen, z. Teil schwierigen dimensionstheoretischen Strukturproblemen. In einer ganz natürlichen Weise ergeben sich dabei mehrere Gesichtspunkte je nach der Dimension des Raumes, in welchem die Gebilde eingebettet sind. Einer dieser Gesichtspunkte betrifft die n-dimens. Cantorschen Mannigfaltigkeiten (c.m.) in einem n-dimens. Raum; wir beginnen hier deren Untersuchung im einfachsten wesentlichen Fall—zweidimens. ebener Mannigfaltigkeiten. In dieser ersten Note werden wir beweisen, dass eine zweidimensionale c.m., welche von einer einzigen geschlossenen Kurve begrenzt wird, 1-dimensionale Mannigfaltigkeitsstellen dann und nur dann enthält, wenn die Begrenzungskurve keine einfache (Jordan-) Kurve ist. Damit wird eine Bedingung für das Auftreten 1-dimensionaler Mannigfaltigkeitsstellen gewonnen. Diese Stellen sind nämlich mit den nicht erreichbaren Randstellen des offenen Teiles der Mannigfaltigkeit identisch. Es ist bekannt, dass die hier behandelten speziellen c.m. noch schärfere Eigenschaften besitzen und deshalb sich mit viel schwächeren Begriffsbildungen erfassen lassen, nämlich unter dem Gesichtspunkt des Zusammenhangs im Kleinen. Demgegenüber sind die allgemeinen ebenen Mannigfaltigkeiten nicht erforscht und dort gelangt der lokale Dimensionsbegriff zu seiner Geltung. Die allgemeineren c.m. können ausserordentlich kompliziert sein, ja bereits im Falle, wo jeder Punkt der Mannigfaltigkeit ein 2-dimens. konzentrischer m.-Punkt ist. Die Gesichtspunkte, unter welchen sich solche c.m. behandeln lassen, sind vor allem der Zusammenhang einer ebenen c.m., die Anzahlihrer offenen Komponenten und insbesondere die Mehrfachheitihrer Punkte, je nach der (endl. oder unendlichen) Anzahl der nichtäquivalenten Folgen von c.m., welche einem 2-dimens.
An Investigation of the Upper Limit of the Thorium C + C″ β-ray Spectrum by means of the Wilson Cloud Chamber
Authors: Ho P. C.
Published: November 2008

Experiments on the β-ray spectrum of thorium C + C″ by the Wilson cloud chamber method were made. From a certain number of photographs taken, the upper limit of the continuous spectrum was found at 9250 Hρ and no β-particles of higher energy were observed. This value for the upper limit agrees fairly well with the result obtained by Henderson's recent experiments by the method of Geiger-Müller counter and focusing magnet.
Further Investigations of the Amplitude Variations of Downcoming Wireless Waves
Authors: Pawsey J. L.
Published: November 2008

Experiments are described for measuring the lateral deviation of wireless waves after reflection from the E and F regions of the ionosphere. It was found that the greatest lateral deviation observed, 20° or more, was that due to the e region, and the least, about 0·5°, was due to the normal E region in the case of a distant transmitter.

The time variation of amplitude of a reflected wave was found to be consistent with a random scattering at the ionosphere.

In the theoretical discussion it is shown that changing horizontal irregularities, ion clouds, are a very important cause of fading. Values are calculated for the average fading periods which would result from the horizontal winds in the neighbourhood of the E region known to exist from other evidence. These calculated periods agree with the observed and it is inferred that horizontal winds are a very important cause of fading.
Potenzreihen mit mehrfach monotoner Koeffizientenfolge und ihre Legendrepolynome
Authors: Fejér Von Leopold
Published: November 2008

1. In den folgenden Zeilen möchte ich einige Sätze mit Beweisandeutungen mitteilen, die sich teilweise auf die Besonderheiten der Konvergenz der Partial-summen zur entwickelten Funktion teilweise auf die Eigenschaften der “Legen-dreschen Polynome” einer Potenzreihe

beziehen, in welcher die Koeffizientenfolge {αn} mehrfach monoton ist. Eine Folge {αn} heisst k-fach monoton, wenn die Zahlen der k + 1 unendlichen Folgen {αn}, {δ1αn} {δ2αn}, …,{δkαn}, n = 0, 1, 2, …, durchwegs nichtnegativ ausfallen. Hier bezeichnet δvαn die v-te Differenz von αn, d.h.
Geometrical Properties of Three Symmetric Matrices
Authors: Turnbull H. W.
Published: November 2008

In the early editions of the Geometry of Three Dimensions Salmon had stated that the equations of any three quadric surfaces could be simultaneously reduced to the sums of five squares. Such a reduction is not possible in general, but can be performed if and only if a certain combinant Λ, of the net of quadrics, vanishes. Algebraically the theory of such a net of quadrics is equivalent, as Hesse(2) showed, to that of a plane quartic curve: and the condition for the equation a quartic to be expressible to the sum of five fourth powers is equivalent to the condition Λ = 0(1). While Clebsch(3) was the first to establish this condition, Lüroth(4) gave it more explicit form by studying the quartic curve

which satisfies the condition. Frahm(5) seems to have been the first to prove the impossibility of the above reduction of three general quadric surfaces, by remarking that the plane quartic curve obtained in Hesse's way from the locus of the vertices of cones of the net of quadrics would be a Lüroth quartic. Frahm further remarked that the three quadrics, so conditioned, could be regarded as the polar quadrics belonging to a cubic surface in ∞2 ways; but that for three general quadrics no such cubic surface exists. An explicit algebraical account of these properties was given by E. Toeplitz(6), who incidentally noticed that certain linear complexes associated with three general quadrics became special linear complexes when Λ = 0. This polar property of three quadrics in [3] was generalized to n dimensions by Anderson (7).
The Electronic Structure of
Authors: Coulson C. A.
Published: November 2008

The molecular-orbital method has been applied to a study of in its ground state and excited levels, and the relative importance of the perturbation and variational methods has been considered in some detail, as well as the effect of certain integrals which, in discussions of molecular structure, have often been neglected. It appears that the ion should exist in stable equilateral form with a nuclear distance about 0·85 Å., and that all excited levels are unstable.

Reasons are given for supposing that the molecule H3 is linear and not triangular.
Jacobian surfaces of quadrics in four dimensions
Authors: Roth L.
Published: November 2008

The surfaces considered in this note are Jacobians of four quadric forms (“quadrics”) in [4], where one or more of the quadrics are space-pairs. It is believed that, with one exception, they have not been noticed previously; and so little is known of non-rational surfaces in higher space that it seems worth while putting them on record.
The β-ray Type of Disintegration
Authors: Ellis C. D. , Wooster W. A.
Published: November 2008

In three other papers in this number of the Proceedings will be found direct experimental evidence that in a β-ray disintegration the γ-rays are emitted after the electron has been ejected from the nucleus. In this paper we discuss this result in more detail and especially its bearing on the general picture of the β-ray disintegration.
On the Theorem of Pappus
Authors: Newman M. H. A.
Published: November 2008

1. The axioms necessary for the construction of a proof of Pappus' Theorem, regarded as a theorem in the geometry of projective space of three dimensions, fall into three groups:

I. Axioms of Incidence;

II. Axioms of Order, giving the properties of the relation “between”, and establishing the order type of the projective line as cyclical and dense in itself;

III. An Axiom of Continuity.

It is customary in treatises on projective geometry to adopt in Group III the Axiom of Dedekind, which states that if the points of a segment are divided into two classes, L and R, which have each at least one member, and are such that no member of L lies between two points of R, nor vice versa, then there is a point of the segment, not an end-point, which is neither between two points of L nor between two points of R. This axiom, however, assumes considerably more than is necessary for the proof of the theorem.
The Lebesgue-Stieltjes Integral
Authors: Francis E. C.
Published: November 2008

It has long been realized that a Stieltjes integral can be defined and developed on the same lines as a Lebesgue integral, using in place of the measure of a set E the variation of an increasing function ø (x) upon E; but only the more obvious—and less useful—properties of such an integral seem to have been stated. The original integral defined by Stieltjes possesses the remarkable Integration by Parts property that if either of exists, so does the other, and their sum is equal to
The Equations of Viscous Motion and the Circulation Theorem
Authors: Jeffreys Harold
Published: November 2008

The following derivation of the equations of viscous motion, and, by analogy, the equations of motion of an isotropic elastic solid, appears to have an advantage in brevity over those in general use.
An attempt to detect an Electric Moment in a Light Quantum
Authors: Clark R. J. , Watson W. H.
Published: November 2008

If a light quantum has an electric moment, we should expect its axis to coincide with the direction of the electric vector, and therefore to be perpendicular to the plane of polarisation. Now an electric doublet of moment μ on entering a uniform electric field E with its doubletaxis parallel or antiparallel to the field will have its energy changed by an amount ∓ μE. For a light quantum this is equivalent to a change in wave-length

when in the field. The method adopted to detect this change was to set up a grating with a strong electric field normal to its surface, and then have a plane polarised parallel beam of light fall on the ruled surface with the electric vector parallel to the direction of the field. The spectra formed near the normal to the grating were then examined both in the presence and absence of the field.
On the quintic surface in space of five dimensions. (Part II, Curves on the surface.)
Authors: Bath F.
Published: November 2008

Consider the pencil of [4]'s through the [3] which contains l15, l25, l34, any [4] of the pencil meets the V25 again in a conic, q.
An Involution in the Complex Variable
Authors: Grace J. H.
Published: November 2008

Suppose that ƒ1, ƒ2, ƒ3 are three quadratics in the complex variable z with complex coefficients, say

and consider the equation

the α's being real quantities. It determines two values ζ, ζ′ of z and thus two points P, P′ in the complex plane. Clearly when P is given the ratios α1: α2: α3 are in general determinate and accordingly P′ also. This is the involutory transformation I propose to discuss.
On Questions of Reality for Twisted Quartics of the First Kind
Authors: Du Val Patrick , Baker H. F.
Published: November 2008

By a twisted quartic of the first kind is intended the curve in space of three dimensions defined by two quadratic equations, i.e. the complete intersection of two quadric surfaces. If this intersection consist of the aggregate of two or more curves of lower order than the fourth, it will be called a degenerate quartic. In general, a twisted quartic of the first kind will be referred to for brevity as a quartic.
Multiple tangents to forms in four dimensions
Authors: Roth L.
Published: November 2008

Formulae for multiple tangents to the general surface in ordinary space were obtained at different times by various writers, but the discussion remained incomplete until the advent of Schubert's enumerative method, which solves the whole problem by a purely mechanical process. Schubert later extended the method to general forms in [n], in the restricted case where the multiple tangents have only a single contact. In the present paper the results for three dimensions have been used to build up the formulae for forms in [4], by means of the correspondence theorem on which much of Schubert's work is based. It would no doubt be possible to evolve a complete set of incidence formulae for four dimensions and then to proceed as in Schubert's discussion of surfaces; but the present method is preferable for two reasons. In the first place, all the results have been obtained in a very simple manner; and secondly, a large number of minor results have been found in the process.
The Deformation of Soap Bubbles in Electric Fields
Authors: Macky W. A.
Published: November 2008

Experiments on the deformation of soap bubbles in an electric field show that until the bubble begins rapid vibrations there is no difference between positive and negative bubbles.

The positive bubble always bursts very shortly after beginning to vibrate even when resting upon a large supply of soap solution.

The negative bubble, however, when resting on ample soap solution, continues vibrating until a much higher field is reached and usually does not burst until some protuberance on the positive plate allows a spark to pass, which may then strike and destroy the bubble.

The values of the field for which a particular bubble begins rapid vibrations, so that in continuous light it appears to be conical, are found to be the same within experimental error for both positive and negative bubbles. The fields for bubbles of different sizes are found to vary inversely as the square root of the radius.

Photographs are given illustrating the shape of bubbles in various fields.

These experiments were carried out at the suggestion of Professor C. T. R. Wilson, and I am greatly indebted to him for his continued advice and interest throughout their progress.
Some definite integrals involving Legendre functions
Authors: Bailey W. N.
Published: November 2008

The integral

where n is a positive integer, was obtained recently by Van der Pol by operational methods.
A Property of Continuous Arcs, II
Authors: Verblunsky S.
Published: November 2008

This note is a sequel to a former one, a knowledge of which will be assumed. We here develop the methods of that note to give a proof of Jordan's Theorem. We write ind C (P) for 1/2π times the absolute value of the change in log (z − P) as z describes the continuous arc C. If C is a Jordan curve, ind C (P) is either 0 or 1. Further, if C is a polygonal line, the index is a continuous function of P. If C is a closed continuous curve, interior to a circle to which P is exterior, then ind C (P) = 0.
The exact difference equation of the first order
Authors: Milne-Thomson L. M.
Published: November 2008

The theory of the exact difference equation in the general linear case has been fully developed, but the corresponding theory for the non-linear equation of the first order does not appear to have been considered. In this paper necessary and sufficient conditions for the difference equation of the first order to be exact and the form of the primitive are obtained. It appears that two conditions are required for a difference equation to be exact, one of which is identically satisfied in the limiting case of the exact differential equation. These conditions are applied to determining the primitive in some cases where the conditions for exactness are not satisfied.
Primary ionization in helium and hydrogen
Authors: Williams E. J.
Published: November 2008

According to recent experiments by Cosyns(1), in which the probability that a cosmic-ray electron actuates a Geiger-Müller Counter is observed, the rate of production of primary ions is practically the same in helium as in hydrogen. The value obtained for hydrogen (which agrees very closely with that previously obtained by Danforth and Ramsey (2) using the same method) is moreover little different from that found for fast β-particles in expansion-chamber experiments.
On sets of fractional dimensions
Authors: Best E.
Published: November 2008

1. Any number x can be expressed uniquely in the form

where the xr's are positive integers and where xr < r. In the present paper we consider the set of numbers Eb for which the xr's are bounded, so that 0 ≤ xr < b say, where b also is an integer. We prove that this set has dimension function

h(t) = b−u,

where t = euu−u−½(b − 1)(2π)−½, in the sense of Hausdorff.
The average number of electrons accompanying a cosmic-ray meson due to collisions of the meson with atomic electrons
Authors: Williams E. J.
Published: November 2008

The penetrating component of cosmic radiation is now believed to consist of mesons—particles with a mass about one-tenth of that of a proton—while the soft component is attributed to electrons. By collisions with atomic electrons a fast meson occasionally produces fast secondary electrons, “knock-on” electrons, thus giving rise to a certain amount of the soft component. It was first suggested by Bhabha(1) that fast electrons produced in this way may be responsible for the electron showers known to be associated with the penetrating particles. In the present note we calculate the ratio, which we denote by R, of the average number of electrons which accompany a meson due to the knock-on process, making use of the simplification which results from the fact that the rate of loss of energy by fast electrons in ionization and excitation and also the emission of soft quanta are nearly independent of their energy. In virtue of this, the total length of electron track, and therefore the magnitude of R, is only slightly affected by the cascade process of shower production, a process which causes the energy of a fast electron to be shared among a number of electrons. The difficulties associated with a treatment of the cascade process may therefore be avoided in calculating R.
Note on the shearing motion of fluid past a projection
Authors: Dean W. R.
Published: November 2008

1. A method has lately been developed by N. Muschelišvili (1) for the solution of problems of the slow two-dimensional motion of viscous liquid and of the corresponding problems of plane stress and plane strain, in cases in which the area in the x, y-plane that is concerned can be represented conformally on the interior of the circle |ζ| = 1 in the ζ-plane by a relation of the form z = x + iy = r(ζ), where r(ζ) is a rational function of ζ. In most problems in which the method has been used the function r(ζ) has been a simple one, but it is of importance to consider a rational function of as general a form as possible since, given any relation z = f(ζ), it will usually be possible to find a rational function that approximates to f(ζ) throughout the circle |ζ| = 1 and for a close approximation a complicated function r(ζ) will in general be required.
Cayley's problem of seven lines on a quartic surface
Authors: Ursell H. D.
Published: November 2008

1. The following investigations were undertaken in connection with Cayley's problem of seven lines on a quartic surface.
Coincidence formulae in geometry
Authors: Ursell H. D.
Published: November 2008

1. The principal object of this note is to give a new derivation of the coincidence formula in the general correspondence of points in a rational n-space. It is supposed that the correspondence is algebraical and that there are ∞n pairs of corresponding points P, P′ (in the geometrical sense of ∞n). Denote by the symbol (i) (n − i)′ the number of pairs P, P′ such that P lies in a general i-space and at the same time P′ in a general (n − i)-space. When (P, P′) tends to a coincidence of the correspondence, the join PP′ tends to some limiting position, which may depend on the way in which (P, P′) tends to the coincidence. In general the limiting lines will lie on a conical figure—a star—with the double point as vertex. In the simplest case of an isolated double point, the star fills the whole n-space. Denote by the symbol ε(i, n) (n − i) the number of times it happens that a double point lies on a general (n − i)-space while the limiting line PP′ meets a general i-space(and lies in the n-space).
The Gliding of a Plate on a Stream of Finite Depth. Part II
Authors: Green A. E.
Published: November 2008

7. We have discussed the gliding of a plate of finite length on a stream of finite depth. Numerical calculations have been made for the case when the angle of incidence of the plate to the stream was 30°, the results for any other angle being similar.

It was found that, for a given depth of stream and a given height of the trailing edge above the bed of the stream, the value of the lift increases with the length of the plate, until finally, when the plate is infinitely long, the lift assumes a maximum value. Further, for a given depth of stream, the total normal lift on the plate is independent of its height above the bed of the stream, when the length of the plate is small, except when the trailing edge of the plate is above the surface of the stream. Finally, when the depth of the stream is very large and the plate is near the middle of the stream, then our solution approximates to the classical Rayleigh flow past a plate in an infinite fluid.
On the Radiation Emitted by a Multipole and its Angular Momentum
Authors: Heitler W.
Published: November 2008

1. By expanding the radiation field in a series of spherical waves the complete expressions for the field emitted by an electric and magnetic 2l-pole are obtained (§§ 1, 2).

2. It is shown that the wave emitted by a (electric or magnetic) 2l-pole has an angular momentum about the z-axis Mz = mU/ν (U = total energy), where m, according to the orientation of the 2l-pole in space, can assume the values −l, −l + 1, …, + l (§ 3). The angular momentum is contained in that region of the field in which the product EH decreases as r−3.

3. By quantizing the waves it is shown that the angular momentum of a light quantum emitted by a 2l-pole behaves like the angular momentum of an electron in a central field of force without spin (commutation relations, etc.) (§ 4). The angular momentum of a single light quantum is an integral multiple of ħ.
The Dissociation Equilibrium of Hydrogen and its Adsorption on Tungsten
Authors: Roberts J. K.
Published: November 2008

The various processes occurring at the surface when hydrogen is adsorbed on tungsten are considered together with the dissociation equilibrium of hydrogen in the gas phase. The form of the adsorption isotherm is deduced from the principle of detailed balancing and is in agreement with that obtained by Fowler using a statistical method. A detailed interpretation of the experimental results now available shows that either (a) measurements of the rate of removal of the adsorbed film of oxygen on tungsten do not measure the rate of evaporation of oxygen atoms or (b) it is not possible to obtain a general first approximation formula giving the rate of evaporation of adsorbed atoms in terms of the heat of desorption. The desorption of hydrogen from tungsten is discussed and it is shown that the agreement between the temperature at which the film evaporates at an appreciable rate and that deduced from a desorption formula of the type mentioned in (b) assuming that the hydrogen evaporates as atoms must at present be regarded as a coincidence.
Experiments on the Efficiencies of Production and the Half-Lives of Radio-Carbon and Radio-Nitrogen
Authors: Allison Samuel K.
Published: November 2008

These experiments were performed at the Cavendish Laboratory with the high voltage apparatus of Cockcroft and Walton and may be considered as a continuation of the experiments of Cockcroft, Gilbert and Walton on the production of induced radioactivity by high velocity protons and deuterons.
Some canonical triple surfaces
Authors: Bronowski J.
Published: November 2008

1. In 1891, Castelnuovo suggested that there may exist canonical surfaces which consist of three sheets covering the same surface. In particular, he remarked that, if the canonical curves of a surface contain a then the canonical model of the surface is certainly a three-sheeted surface. An example is the quintic surface with a tacnode, in space of three dimensions; its canonical model is a triple plane, branching along a curve of order 12 with twenty-four cusps which lie on a quartic.
On multiple curves and surfaces as limits
Authors: Semple J. G.
Published: November 2008

As particular cases of the general problem of finding the most general limiting tangential form of a degenerate manifold, we discuss here the degeneration of a curve on a surface into one or more multiple components, and the degeneration of a surface on a threefold into a single surface counting multiply, the method used consisting essentially in finding the approximate form of the curve or surface as the limit is approached. The approach system is supposed in all cases to be contained in a linear system of curves or surfaces, on the surface or threefold considered.
Note on the velocity distribution in the wake behind a flat plate placed along the stream
Authors: Rosenhead L. , Simpson J. H.
Published: November 2008

1. The velocity distribution in fluid of small viscosity, or at large Reynolds number, in the laminar boundary layer associated with a thin flat plate along the main direction of flow, has been worked out by Blasius on the basis of the Prandtl theory. These original investigations covered the region from the forward edge of the plate to the downstream edge. In a recent paper Goldstein investigated the velocity distribution in the wake immediately behind the downstream edge over a region within about 0·5lfrom it,lbeing the length of the plate. Later

Tollmien obtained a first approximation to the flow at a great distance down-stream. Goldstein § extended Tollmien's solution by finding a second approximation and by joining it up with his previous work on the flow very near the plate.
On certain surfaces which have unpostulated singularities
Authors: Maxwell Edwin A.
Published: November 2008

1. While the general theory of surfaces has received a great deal of attention in recent years, there remain a number of difficulties in the application of the results of that theory to the study of particular surfaces. I propose to discuss here certain details which, from their nature, do not seem to be amenable to very general treatment. I shall therefore consider a number of examples with a view to illustrating the way in which certain classes of surfaces may be expected to behave.
On the canonical systems of certain algebraic surfaces
Authors: Pedoe D.
Published: November 2008

A complete linear system of curves on an algebraic surface may have assigned base points. The canonical system, from its definition, has no assigned base points at simple points of the surface. But we may construct surfaces on which, all the same, the canonical system has “accidental base points” at simple points of the surface. The classical example, due to Castelnuovo, is a quintic surface with two tacnodes. On this surface the canonical system is cut out by the planes passing through the two tacnodes. These planes also pass through the simple point in which the join of the two tacnodes meets the surface again. This point is the accidental base point of the canonical system on the quintic surface.
On temperature in relation to quantal phenomena
Authors: Larmor Joseph
Published: November 2008

In an excursus developing a modern view of the Carnot-Kelvin aspect of thermodynamics, appended recently to the letters from Clerk Maxwell to his friend W. Thomson reporting the early tentative evolution of the theory of the electro-dynamic field, the account presented by the writer stopped short at putting emphasis on the mystery of temperature, as a unique and supremely significant property of matter in bulk, with its trend to uniformity as presenting the basic thermal problem. The side of the subject involving dynamical analogy was there absent from the development, which was purely formal. It may be well, however, now to offer some general notions such as may be put on record, which arise naturally from that mode of approach to the subject.
Wave functions for negative ions of sodium and potassium
Authors: Hartree D. R. , Hartree W.
Published: November 2008

Solutions of Fock's equation for the self-consistent field with exchange have been carried out for the negative ions Na−, K−. The results indicate that these negative ions can certainly exist; and calculations of the energy values, and estimates of the errors of these values, indicate that these ions are probably stable and not liable to autoionization.

The wave functions for these negative ions are required in calculating atomic cross-sections for the capture of electrons by the neutral atoms.
Current effects in cylindrical superconductors
Authors: Misener A. D.
Published: November 2008

Experiments are described which show the effect of current on the transition of cylindrical superconductors to the normal state produced by increasing an external magnetic field. The transition curves were determined with an accuracy of 0·1%. Both longitudinal and transverse fields were used.

The following results were found.

(a) Transverse field

(i) The field strength at which the first traces of resistance appeared was independent of current strength for reasonably small currents (up to about 100 mA. for 0·010 cm. wire).

(ii) With increase of current the transition curve at first changed shape, becoming more and more nearly linear, and then was displaced parallel to itself toward smaller values of the applied field by an amount in accordance with Silsbee's hypothesis.

(iii) The change in shape of the transition curve was accurately reproducible and was identical for specimens of the same material and diameter.

(iv) The results for specimens of different diameter could be accurately correlated if the measured values of the resistance were plotted against the current divided by the specimen diameter raised to the power (l + R/Rn), i.e. I/(diam.)(1+R/Rn), where R/Rn is the ratio of the measured resistance to the resistance just above the transition point.

(b) Longitudinal field

(i) With increase of current, however slight, the transition curve was displaced parallel to itself, toward smaller values of the applied field, without showing any change in shape.

(ii) The displacement of critical field for a given change of current was completely independent of the diameter of the specimen.

(iii) The observed shift to smaller fields was much greater than could be accounted for by the additional field produced by the current at the surface of the wire.

(iv) If we denote the (extrapolated) value of the critical field for zero current by (Hc) and the difference between this and the observed value for a current (I) by (ΔHc) then I and Hc are connected by the empirical relation

where A and n are constants depending on the nature of the superconducting metal.

when Hc is expressed in gauss and I in milliamperes.
On Frullani Integrals
Authors: Iyengar K. S. K.
Published: November 2008

The object of this note is to obtain necessary and sufficient conditions for the existence of

on the assumption that ø is integrable in any finite positive interval not including zero (0 < α ≤ x ≤ β < ∞). On reference to the literature on the topic, I find no exhaustive analysis of this problem anywhere.
Sur les familles indenombrables de suites de nombres naturels et les problemes concernant la proprieteC
Authors: Rothberger Fritz
Published: November 2008

Nous ne considérerons, en général, que les sous-ensembles de l'intervalle [0, 1] et nous désignerons paretrespectivement l'ensemble des nombres rationnels et l'ensemble des nombres irrationnels de cet intervalle.

Nous allons employer pour les fractions continues la notation

Étant donnée une suite de nombres naturels {zn}, nous dirons que le nombrecorrespondà la suite {zn} et vice versa, et nous écrironnsz=v({zn}). Pareillement, nous dirons qu'un ensemblecorrespond à une famille Φ de suites de nombres naturels si l'on aE=v(Φ).
Deuteron bombardment of gold
Authors: Krishnan R. S.
Published: November 2008

The paper describes the results obtained from a study of the bombardment of gold with deuterons of energy up to 9·1 M.e.V. The reactions found to take place in gold are the formation of 2·7 day Au198 by a (d-p) process and of 32 hr. Hg198* in a metastable state by a (d-n) process. The β-ray and γ-ray energies associated with each radioactivity have been measured by absorption methods. Excitation functions for the two reactions have been determined and the results have been compared with those to be expected on theoretical grounds.

The author takes this opportunity of expressing his gratitude to the past and present members of the Cavendish Cyclotron Laboratory for assistance in running the cyclotron. He wishes to thank the Royal Commission for the Exhibition of 1851 for the award of a Science Scholarship.
Expansions for a particular class of exponential-logarithmic integrals
Authors: Orr William J. C.
Published: November 2008

In the course of the development of a special method of interpreting thermo-dynamic data in terms of a generalized force field for the mutual interactions of the molecules concerned, expansions were required for integrals of the following general type, which apparently have not hitherto been discussed. We require explicit expansions of F(α, s, n), in terms of the parameter α, for the integral values 0, 1, 2, 3, … of s and n, where
On multiple curves. III.
Authors: Hodge W. V. D.
Published: November 2008

A curve Γ has been defined as a normal multiple of a curveCif it contains an involutionIn, of ordern, which has the properties: (1) the sets ofInare in (1–1) correspondence with the points ofC; (2) each setInconsists ofnpoints which are distinct in the birational sense; (3)Inis generated by an Abelian groupGof ordernof birational transformations of Γ into itself. We shall denote the field of complex numbers byk, the function field ofCbyk(C), and the function field of Γ byk(Γ). Conditions (1) and (3) imply thatk(Γ) is a commutative normal (i.e. Galois) extension ofk(C). The object of this note is to show how, given the curveCand the Abelian groupGof ordern, we can construct a curve Γ with the properties (1), (2), (3).
Two moment problems for bounded functions
Authors: Verblunsky S.
Published: November 2008

1. Let (a, b) denote an unbounded interval. Given the numbers s0, …, sn−1, consider the problem of finding what conditions they must satisfy in order that there shall be a function f(x), 0 ≤ f ≤ 1, such that
The Parseval formulae for monotonic functions. I
Authors: Edmonds Sheila M.
Published: November 2008

The Parseval formulae for Fourier cosine and sine transforms,

are of course most widely known in connexion with the classical theorems of Plancherel on functions of the class L2 (whose transforms are defined by mean convergence), and with their generalizations. We cannot expect to obtain anything as elegant as the ‘L2’ results when we consider (1) for functions of other kinds. Nevertheless, since the most obvious way of defining Fourier transforms is by means of Lebesgue or Cauchy integrals, we naturally wish to know how far the formulae (1) hold good for transforms obtained in this way. The two most familiar classes of functions having such transforms are:

(i) functions f(t) integrable in the Lebesgue sense in (0, ∞), whose transforms Fe(x) and Fs(x) are defined by the Lebesgue integrals respectively; and

(ii) functions f(t) which decrease in (0, ∞), tend to zero as t → ∞, and are integrable over any finite interval (0, T); in this case the transforms are defined by the Cauchy integrals .
Configurations of closed ring molecules
Authors: Miller A. R.
Published: November 2008

The physical significance of the parameters which appear in the combinatory factors g(Ni) and g(Ni, Xij) in the statistical mechanics of mixtures is examined. Hitherto, closed ring molecules have been excluded from the jurisdiction of the formulae for these combinatory factors. The means by which this restriction on the validity of the formulae can be removed is indicated. It is concluded that the equations given by Guggenheim are formally applicable in all cases provided that appropriate values, which differ according as the molecules are simple and branched chain or closed ring molecules, are substituted for certain parameters which occur in the equations.
On a new analytical representation of curves in space
Authors: Pedoe D.
Published: November 2008

It was in a paper bearing this title that Cayley(1) first considered the problem of representing a curve in projective space of three dimensions by means of the complex of lines which meet the curve. He took the conic given by the equations

and found that the line

with dual Grassmann coordinates (…,pij,…), where

intersects the conic if, and only if,

where F(u0, u1) is homogeneous and of degree 2 in both sets of indeterminates u0 and u1 and G(…,pij,…) is a form of degree 2 in the pij. Both F(u0, u1) and G(…,pij,…) are easily determined in this case.
Combinants of a pencil of quadric surfaces. I
Authors: Todd J. A.
Published: November 2008

The classical account of the invariants and covariants of a pair of quadric surfaces is due to Salmon. The actual determination, in explicit form, of the complete system of concomitants of two quaternary quadratic forms is much later in date, and is due to Turnbull. This system, which includes mixed concomitants of various kinds, is complicated. As originally determined, it comprised 125 forms, three of which were later shown to be reducible. In the accounts of both these authors, however, the forms are considered as belonging to a particular pair of quadric surfaces, and the problem of determining these covariant forms which are invariant, not merely under change of coordinate system but also under change of base in the pencil denned by the two quadrics, is not alluded to. Such forms, which are of obvious geometrical interest, are called combinants. It is rather surprising to find that very little seems to be known about the combinants attaching to a pencil of quadric surfaces; the Encyklopädie scarcely mentions them, and I have been unable to trace any references in the literature except to the two invariants whose vanishing expresses the condition that the parameters of the four cones in the pencil form an equianharmonic or a harmonic set.
The arithmetical character of the Wishart distribution
Authors: Lévy Paul
Published: November 2008

Let ξ and η be two independent and normal random variables, with zero means and with standard deviations each equal to ½ Put

The joint distribution of X, Y, Z is a particular case of the Wishart distribution (1). It may be defined by the generating function of its cumulants (c.g.f.)
The half-life of thorium C′
Authors: Hill J. M.
Published: November 2008

1. The half-life of Th C′ was determined by a new method of coincidence technique.

2. Previous investigators have recognized difficulties in regard to this investigation. These were: (a) random delays in the counters; (b) coincidences due to other disintegrations; (c) possibility of difference in delays in recording α- and β-particles. In this investigation the difficulties arising from these effects have been overcome.

3. From this type of experiment it is possible to determine the order in which the particles are emitted.

4. It would appear that the time of breakdown of a counter depends slightly upon the magnitude of the initialionization.

5. The value for the half-life of Th C′ was found to be 3.0 ± 0.15 × 10−7 sec.
On Boussinesq's problem and penny-shaped cracks
Authors: Green A. E.
Published: November 2008

Harding and Sneddon(3) and Sneddon(6) have shown that the systematic application of Hankel transforms to the problem of the indentation of the plane surface of a semi-infinite elastic solid by a rigid punch, reduces the problem essentially to one of solving a pair of integral equations belonging to a class which has been studied by Titchmarsh(8) and Busbridge(1). The same method has also been applied by Sneddon(7) to the problem of the distribution of stress in the neighbourhood of a circular crack in an elastic solid. In both types of problem the results are expressed in quite simple forms independent of Bessel functions and this suggests that there should be an alternative method of solution. It is now possible to supply such a method with the help of a recent paper by Copson(2) on the problem of the electrified disk. In addition, some generalization of the stress systems considered by Harding and Sneddon can be made since they confined their analysis to stresses which were symmetrical about an axis.
The Bose-Einstein statistics of particles, with special reference to the case of low temperatures
Authors: Dingle R. B.
Published: November 2008

In this paper, it has been shown that the usual demonstrations of the phenomenon of Bose-Einstein condensation are invalid. Exact formal solutions have been derived for the partition functions and the mean occupation numbers in classical and Bose statistics, and the theory of fluctuations in the case of Bose statistics has been given.

In a second paper, convenient approximations will be derived for the mean occupation numbers in Bose statistics, and it will be shown that for temperatures well above the critical temperature of the supposed Bose-Einstein condensation the well-known resultfor the mean occupation numbers is valid, whilst for temperatures rather lower than this critical value it will be shown that particles will condense into the lowest available state. The critical temperatureT0is given bywhere the summation is over all energy levels except the lowest.
Point-curve correspondences
Authors: Scott D. B.
Published: November 2008

In this paper we consider the theory of point-curve correspondences on a single surface F, i.e. correspondences in which to each point of F corresponds an algebraic curve of F. The results previously obtained for induced and extended correspondences between two surfaces require some modification here, as we can consider the self-correspondences induced on a curve C of F, and the related theory of extended correspondences, which is now complicated by the existence of the identical correspondence on C. We also develop a theory of correspondences with (non-zero) valency, and show that for a surface whose Riemann matrix is pure and without complex multiplication (and hence for surfaces of ‘general moduli’), all correspondences are valency correspondences, this result being exactly analogous to the well-known theorem for curves. We also consider generalized valency correspondences which are an extension to surfaces of the concept of correspondences of multiple valency on a curve. The analogy between our theory and the known results for curves does break down in one important respect. It is not true that every surface possesses valency correspondences even of the generalized kind, the existence of such correspondences involving restrictions on the intersection group of the surface.
Collisions through liquid layers
Authors: Eirich F. R. , Tabor D.
Published: November 2008

The initial collision (see pp. 571–2). The instantaneous velocity change of the hammer (from V0 to V) when it first strikes the liquid film should be calculated from energy and not momentum considerations, since the hammer has momentum in the vertical direction whilst the liquid is expressed in a horizontal direction and its total momentum is, by symmetry, zero at every instant. The kinetic energy dE of an annulus of liquid of radius r will be ½mc2, where m = 2πrhρdr, and c, the radial velocity of flow, is ½rV/h, since the liquid starts moving in plug flow. Integrating from r = 0 to r = R, we find that the total kinetic energy E imparted to the liquid film is . Assuming that extraneous energy losses, including any energy imparted to the anvil, are negligible, we may equate this to the energy loss of the hammer . Hence

To a first approximation this yields

The term at the right-hand side of the denominator is one-half that given in the original derivation (equation (20)), so that the instantaneous decrease in the velocity of the hammer is even less marked. For the given case where M = 400g., R = 1 cm., ρ = 1.6 and h = 5 × 10−2 cm., the velocity decrease is less than 2%.

Vol. 44 (1948), pp. 566–80.
Elastic deformations of infinite strips
Authors: Hopkins H. G.
Published: November 2008

In this paper, Fourier integrals are used to solve some elastic problems of generalized plane stress and small transverse displacements in infinitely long, rectangular, isotropic plates stressed only at their edges. The Airy stress function and the transverse displacement satisfy the two-dimensional bi-harmonic equation, and the basic mathematical problem is to solve this equation subject to different sets of boundary conditions. Little attention has been given hitherto to problems in which some of the boundary conditions depend directly upon displacements. Here the general problem is solved when one long edge is fixed, and stresses or displacements are arbitrarily prescribed at the other, with no stresses and displacements at infinity. The problem of a concentrated edge force is discussed in detail and numerical values of the stresses at the fixed edge are given.