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An integral equation
Authors: Hardy G. H. , Titchmarsh E. C.
Published: November 2008

1. The integral equation

where x, f (x), and λ are real and α positive, may be regarded as a differential equation of order α. Suppose for example that α is a positive integer p, that f (x) tends to 0, when x → ∞, with sufficient rapidity, and that

Then, if we integrate repeatedly by parts, and write z for fp (x), (1·1) becomes

The only solutions are finite combinations of exponentials.
The sums of powers as simultaneous canonical expressions
Authors: Bronowski J.
Published: November 2008

1. In this paper I discuss the expression of m + 1 general forms F, F′, …, F(m), each of order n and homogeneous in r + 1 variables z0, z1, …, zl, …, zr, as the sums each of the nth powers of (the same) h + H linear forms in these variables. I take h > 0 of these linear forms to be undetermined, namely the forms

whose coefficients are undetermined; and I take the remaining H ≥ 0 linear forms to be assigned, namely the forms

whose coefficients are assigned.
A study of the intensity variations of downcoming wireless waves
Authors: Ratcliffe J. A. , Pawsey J. L.
Published: November 2008

Experiments are made on wave-lengths between 200 and 500 ms. for distances of transmission less than 200 kms. It is found that the variations of downcoming wave intensity are uncorrelated on two receivers separated by about one wave-length, and it is shown that this implies a considerable amount of lateral deviation of the waves. A special receiver is used to confirm the occurrence of lateral deviation, and an estimate of the angle of deviation is made. The possible causes of intensity variation are considered in the light of these experimental results, and it is suggested that a major cause of “fading” is the interference, at the ground, of waves “scattered” from a series of diffracting centres distributed over an area of radius at least 20 kms. in the present experiments. The possible results of such a mechanism are discussed.
Note on Losses in Cable Sheaths
Authors: Carter F. W.
Published: November 2008

In a recent communication to the Society, the author referred to cable-sheath losses, and gave formulae for computing them in certain cases. These appertained to power cables in which were comprised a group of conductors, arranged symmetrically and encased in a single conducting sheath. In some distribution systems, however, the conductors for the several phases are encased in separate lead sheaths, which are either laid in proximity as separate cables, or grouped and comprehended in an outer sheath. The analysis previously given does not include such cases directly. Moreover, it is common practice either to lay the elementary cables with sheaths in contact, or to bond the sheaths together at the ends of suitable sections, in order to prevent differences of potential between them; and, when this is done, a circulating current flows in the circuit of the sheaths and bonds, sufficient to maintain equality of potential between the several sheaths. This current, to which reference was made in the former paper, is additional to the eddy current discussed therein, the integral of which over the cross section of the sheath is zero. It is for convenience here referred to as the “circulating current,” to distinguish it from the “eddy current,” although there is no such distinction between them as the names imply.
Note on the pedal locus
Authors: Gabbatt J. P.
Published: November 2008

While the paper “On the pedal locus in non-euclidean hyperspace” was in the press, Professor H. F. Baker kindly directed the writer's attention to a reference from which it appeared that the euclidean case had first been studied by Beltrami. After publication, it was discovered that the main subject of Beltrami's paper was the non-euclidean case. He proves, by analytical methods, theorems which may be stated as follows: Let A0, A1,…, An denote the vertices of a simplex, [A], in non-euclidean space of n dimensions, and P a point such that the orthogonal projections of P on the walls of [A] lie in a flat, p; then the locus of P is an (n − 1)- fold, W, of order n + 1, which is anallagmatic for the isogonal transformation q. [A]; the isogonal conjugate of P is the absolute pole of p, and the envelope, w, of p is therefore the absolute reciprocal of W. Beltrami does not note the theorem, fundamental for the geometrical treatment of the subject, that W is the Jacobian of a certain group of n + 1 point-hyperspheres. He goes on to show that in the euclidean case the locus W breaks up into an n-ic (n − 1)-fold and the flat at infinity, while the envelope w does not, in general, break up. Finally, he notes that in two dimensions there is also a special non-euclidean case, in which W breaks up into an order-conic and a line, and w into a class-conic and a point; and that the appropriate condition is fundamentally that which is necessary for the degeneration of a certain class-conic into a pair of points. “But what are the points? And what is the corresponding condition satisfied by the absolute conic? This is a question which it would be interesting to resolve.” It does not appear that the subject has been pursued further; and in the present note an attempt is made to discuss fully the analogous case qi degeneracy in n dimensions.
On a family of constructs in higher space
Authors: James C. G. F.
Published: November 2008

The constructs in question are represented by the vanishing of all determinants of two rows and columns drawn from a matrix of r + 1 rows and s + 1 columns, where r can be taken less than or equal to s. They are thus the multiple constructs of highest dimension on the well-known family of constructs generated by projective systems of linear spaces. In the present note the constructs are considered from a somewhat different point of view, the principal applications being to the determination of directrix loci of minimum order for the systems of spaces which they contain, and to the re-proof and extension of a known theorem.
The Behaviour of the Centres of Activity of Saturated Surfaces during the initial stages of Unimolecular Reactions
Authors: Constable F. H.
Published: November 2008

The theory of catalytic action recently advanced qualitatively by Pease, Taylor, Armstrong and Hilditch, Constable, which the author has endeavoured to treat quantitatively, is extended in this paper to the explanation of the effect of pressure and inert diluents on catalytic action on saturated surfaces. The results are shown to be in agreement with experiment, and further inferences may be made concerning the relative mean lives of the reactant and the diluent on the surface.
On quadrics poristically related to a rational norm curve
Authors: Ramamurti B.
Published: November 2008

The main object of this paper is to study the quadrics which have simultaneously certain poristic relations with a rational norm curve. We shall begin with a résumé of the work done in this direction in the ordinary space [3].
On the operational solution of linear mixed difference differential equations
Authors: Neufeld Jacob
Published: November 2008

L. M. Milne-Thomson has recently described a method of solving linear finite difference equations by using processes some-what analogous to those employed by Heaviside.
On the minimum modulus of integral functions
Authors: Cartwright M. L.
Published: November 2008

1·1. Let f (z) be an integral function of order ρ, and let

It is well known that, if ρ < 1,

This result was first conjectured by Littlewood, who proved it with cos 2πρ in place of cosπρ; it was afterwards proved independently by Wiman and Valiron about the same time. Many other authors have written on the subject, considering among other things the set of valves of r for which m (r) is comparatively large.
On isolated singularities of surfaces which do not affect the conditions of adjunction (Part III.)
Authors: Du Val Patrick
Published: November 2008

In the second part of this paper I pointed out how the presence of any singular point or points of the kind considered in the first part, on a rational surface, corresponds to a subgroup of the group of symmetry of the polytope which represents the properties of the system of base points in the plane representation of the surface. The subgroup is generated by reflexions, and may be the direct product of one or more factors (all the primes of symmetry in one factor being perpendicular to all those in any other factor). Each factor corresponds to a singular point on the surface, namely (in Coxeter's notation), a factor [ ] to a conic node C2, a factor [3n] to a binode Bn+2 (n ≥ 1), a factor [3n,1,1] to a unode Un+5, (n ≥ 1), and finally a factor [3n,2,1] to a unode The possible subgroups in the finite groups that arise have been enumerated by Coxeter; and we shall find that every subgroup generated by reflexions in the group of symmetry of the polytope in ε dimensions which represents ε base points corresponds to a possible configuration of the base points, in which just those rational curves of grade − 2 are actual which correspond to primes of symmetry belonging to the subgroup; without exception, for ε ≤ 6; with one exception—the subgroup [ ]7—for ε = 7; and with three exceptions—the subgroups [ ]7, [ ]8, and [31,1,1 × [ ]4—for ε = 8. Since moreover the system |k| of cubics passing simply through all the base points is in all these cases an actually existing system, for which all the rational curves of grade − 2 are fundamental, its projective model (or in the case ε = 8, in which |k| is only a pencil, the projective model of the system |2k|) provides a rational surface on which all the sets of curves corresponding to the subgroups in question actually appear as singular points.
Notes on the Theory of Series (XVIII): on the Convergence of Fourier Series
Authors: Hardy G. H. , Littlewood J. E.
Published: November 2008

1. In this note we give first our proof of a theorem (Theorem 1) which we stated in Note XIII. We then prove a new theorem (Theorem 2) which leads to another proof of the main theorem of Note XVII.

The first of these theorems requires some preliminary explanations. We are concerned with an integrable function f (θ) with the period 2π. We write

being the complex Fourier series of f (θ).
On the Bicanonical Sets of a Certain Class of Curves
Authors: Frith Ronald
Published: November 2008

W. G. Welchman in his work on fundamental scrolls* obtains as directrix curves to such scrolls a canonical curve pK2p−2 in [p − 1] and a non-special curve pCn in [n − p]. These latter curves may not, however, be general curves regarded projectively, and it is an interesting question to find out the geometrical interpretation of their particularity. The curves C and K are, of course, in birational correspondence, and the prime sections of C correspond to the sections of K by quadrics through a contact set*, i.e. a set of points such that there is a quadric which touches K at every point of the set. For k small enough it is clear that every set of k points is a contact set and in this case the curves C are quite general. For larger k the fact that the set is a contact set simply means that, on C, the points of the bicanonical sets residual to a prime section lie themselves in primes (when k is sufficiently small the number of points in this residual set is such that they always lie in a prime).
On the Power Series of an Integral Function having an Exceptional Value
Authors: Pfluger A. , Pólya G.
Published: November 2008

1. It is known that the distribution of the vanishing coefficients in the power series of an integral function is closely connected with the analytic properties of the function. But the following theorem has apparently not yet been noticed, although simple and elegant both in statement and in proof.
Some Types of Irregular Surfaces
Authors: Roth L.
Published: November 2008

It is a problem of considerable interest in the theory of surfaces to determine the irregular non-singular surface of minimum order, not referable to a scroll; in previous investigations the author has discussed the regularity or referability of surfaces in higher space, reaching the conclusion that all non-singular surfaces of order n ≤ 10 in S4 are regular or referable, with the possible exception of the surface of order n = 10 and sectional genus π = 6, which may be elliptic (pg = 0, pa = −1) or hyperelliptic (pg = 1, pa = −1). In their memoir on hyperelliptic surfaces, Enriques and Severi have obtained for the irregular hyperelliptic surface of general moduli a model 6F10 of minimum order, situated in S4, with the characters n = 10, π = 6. Using transcendental methods, Comessatti has constructed a class of irregular hyperelliptic surfaces the properties of which he has examined in detail; this class includes a member 6Π10 which is a special case of 6F10; and since Comessatti has shown that Π10 is without singularities, so also is F10, whence it follows that F10 is a solution of the proposed problem.
On Positive Harmonic Functions in a Half-Plane
Authors: Verblunsky S.
Published: November 2008

1. Let ξ, η denote the rectangular Cartesian coordinates of a point in a plane. Let J (ξ, η) denote a harmonic function which is positive in the half-plane η > 0. In this paper, we first show (Theorem I) that every such function J determines a non-negative number d, and a bounded non-diminishing function G(x), such that
The Double Excitation of Helium by Electron Impact
Authors: Massey H. S. W. , Mohr C. B. O.
Published: November 2008

The probability of excitation of helium by electron impact to the more important doubly excited levels is calculated, and comparison made with the experimental results of Priestley and Whiddington who have observed two lines due to double excitation. The calculated relative intensities and angular distributions are found to be in qualitative agreement with experiment, and reasons are given for the lack of quantitative agreement.
On the Hesse-Cayley Algorithm for a Plane Quartic whose Bitangents are all Real
Authors: Val Patrick Du
Published: November 2008

It is well known, having been pointed out in the first instance by Plücker, that a plane quartic curve consisting of four unifolia, in Zeuthen's sense, i.e. four even circuits each containing one concave portion or bay, has all its twenty-eight bitangents, and all their fifty-six points of contact, real; since each bay has a bitangent whose points of contact bound it, and each pair of circuits have four common tangents. It has not, so far as I know, been remarked that the algorithm of Hesse and Cayley, in which each of the bitangents is indicated by a pair out of the symbols 1 2 3 4 5 6 7 8, can be applied in a very simple way to such a curve.
Some series of squares of Bessel functions
Authors: Bailey W. N.
Published: November 2008

The expansion of a general function in a series of squares of Bessel functions has been considered by Neumann. As, however, there appear to be only a fairly small number of known expansions of this type, it is perhaps worth while adding to their number. I give here some new expansions of this type, and deduce some simple inequalities satisfied by Bessel coefficients of low order.
The Thermodynamics of an Elastic Solid
Authors: Jeffreys Harold
Published: November 2008

1. We consider a small cube of the solid, of edge l, in the actual strained position. If the stresses were removed and the cube cooled to the absolute zero, the displacements of its particles would be (−u, −v, −w). Then the particles may be considered to have displacements (u, v, w) from a standard position; and the cube has strain components of the usual forms
The correspondence between lines in threefold space and points of a quadric fourfold in fivefold space, established by a geometrical construction
Authors: Wren T. L.
Published: November 2008

This correspondence has been established by Mr H. W. Turnbull, Proc. Camb. Phil. Soc. vol. XXII. (1925), pp. 694–9, a construction which leaves uncorrelated the lines of a special linear complex and the points of the quadric lying in one tangent fourfold; these have to be correlated by means of an independent construction.
On compressional waves in two superposed layers
Authors: Jeffreys Harold
Published: November 2008

The assumption that the compressional waves of earthquakes follow the ordinary laws of refraction, the energy within any pencil of rays remaining permanently within that pencil, has been found to lead to too small amplitudes for the indirect waves from near earthquakes. In this paper a system consisting of two superposed compressible, but non-rigid, media is considered; the lower is supposed to transmit compressional waves with the greater velocity. It is found that ah explosion within the upper medium produces a disturbance at the upper surface involving the direct wave and all the reflected waves that might be expected; but in addition a wave is found that appears to have travelled along the interface with the velocity of sound in the lower medium. This indirect wave would have zero amplitude on the simple laws of refraction with plane boundaries. The variation of its amplitude with distance from the focus is in reasonable accordance with seismological observation, and its time of arrival agrees with that inferred from the laws of refraction for boundaries with slight curvature. But if the direct wave begins with a finite velocity, the indirect one will begin with a finite acceleration; in seismological language an iPg will be associated with an ePn. The indirect wave will also take longer than the direct one to give its maximum displacement.
A New Method of Examination of Surface Films
Authors: Schulman J. H. , Rideal E. K.
Published: November 2008

For the quantitative examination of the behaviour of two-dimensional insoluble films on water surfaces the trough apparatus devised originally by Langmuir has been hitherto employed. With the aid of this apparatus it has been possible to demonstrate the existence of two-dimensional films in the solid, smectic, liquid and vaporous states, to observe the phenomenon of two-dimensional dimorphism, to study the conditions of phase equilibria and to measure the entropy changes associated with the process of two-dimensional fusion, vaporisation and sublimation. No very definite concept of molecular orientation has been obtained from the force area diagrams. Apart from the fact that the polar groups adhere to the surface and that the hydrocarbon chains in long chain compounds adhere to one another, our picture of all forms of films except possibly the solid and the highly attenuated or vaporous states is by no means clear and the various suggestions made as to the molecular architecture are somewhat problematical. Two other methods suggest themselves for gaining a further insight into molecular structure: optical methods on the lines outlined by Rayleigh and by Raman and measurements of the interphasic potential differences. While it may be possible in the future to develop the former into a method of precision, we have succeeded in rendering the latter as accurate as the method of the Langmuir trough. That interphasic potentials between air and liquid exist and are of not inconsiderable magnitude has long been known, although the first systematic investigations on changes in these potentials on the addition of capillary active materials to the liquid phase were made by Kenrick (Z.P.C. xix, 625,1896). The technique was considerably improved by Guyot (C.R. clix, 307, 1914, Ann. d. Phys. x, 2501, 1924) and by Frumkin (Z.P.C. cix, 34, 1924, cxi, 196, 1924, cxv, 485, 1925).
A case of distinction between Fourier integrals and Fourier series
Authors: Grimshaw Margaret Eleanor
Published: November 2008

A Fourier integral is said to be of finite type if its generating function vanishes for all sufficiently large values of ¦x¦. Because the coefficient functions are defined by integrals over a finite range, the behaviour of such a Fourier integral usually resembles closely that of the corresponding series.
The Photo-Electric Threshold and the Heat of Dissociation of the Potassium Molecule
Authors: Ditchburn R. W.
Published: November 2008

The absorption curves and the photo-ionisation experiments agree to give a value for the molecular photo-electric threshold of λ 2555 ± 20 a.u. If our view of the ionisation process is correct, the energy of dissociation is 0·505 ∓ ·01 volts. If the picture of the ionisation process is wrong, the energy of dissociation may be a little lower (a few hundredths of a volt). The corresponding heat of dissociation is 11400 calories. In the range of temperature (200° C. to 500° C.) generally used in experiments on potassium vapour, the fraction associated varies from 10−4 to 10−2. An equation for calculating the fraction associated at any temperature is given by the writer in an earlier paper.
Thunderstorms and the penetrating radiation
Authors: Halliday E. C.
Published: November 2008

Observations are described in which a cloud chamber was used to detect a penetrating radiation apparently produced in thunderclouds. Comparisons are made between the results obtained by Schonland and Viljoen in South Africa, there being a similarity in the magnitude of the effect observed. An analysis of the directions of arrival of the particles shows that the most favoured direction is from the north at a large angle to the direction of the magnetic field of the earth. The results obtained, though not conclusive, encourage a continuation of the experiments.
On the theorem of Riemann-Roch
Authors: Maxwell Edwin A.
Published: November 2008

The theory of surfaces and of manifolds of higher dimension is greatly handicapped by the lack of exact information concerning the extension of the theorem of Riemann-Roch for curves. Allied to this is the difficulty of finding an expression for the arithmetic genus Pa of primals in terms of their defining elements. This paper seeks to examine certain particular applications of the theorem.
The evaluation of certain integrals occurring in studies of molecular structure
Authors: Coulson C. A.
Published: November 2008

A new method is developed for evaluating certain integrals occurring in molecular problems, where distances measured from three different centres of force are involved. The convergence of the results is discussed numerically and theoretically.
Photomagnetic disintegration and magnetic moment of the deuteron in the meson theory
Authors: Ma S. T.
Published: November 2008

The interaction between an external electromagnetic field and a nuclear system can be expressed in terms of the multipole moments. The electric quadripole and the magnetic dipole moments of the deuteron have been calculated, taking into account the exchange forces as given by the meson theory. The cross-section of the photomagnetic effect of the deuteron has been calculated.

This work was carried out under the guidance of Dr Heitler and Dr Fröhlich. The writer wishes to express his sincerest thanks to them for suggesting the problem and many valuable comments. The writer is also indebted to Dr Kahn for discussions during the early stages of this work.
On the physical aspects of accretion by stars
Authors: Hoyle F. , Lyttleton R. A.
Published: November 2008

The presence of H2 molecules reduces the temperature of the cosmical cloud to a value that is small compared with the estimate given by Eddington. The necessary conditions for the molecules to persist at the capture radius of hot stars are investigated in the present paper, and it is shown that provided that the density of the cosmical cloud is sufficiently high the molecules will not suffer appreciable dissociation, and that radiation pressure will have only a negligible effect on the hydrogen. The critical density for a typical B star appears to be about 5 × 10−21 g. per c.c.
Bases for the Prime Ideals Associated with Certain Classes of Algebraic Varieties
Authors: Goddard L. S.
Published: November 2008

The fact that the prime ideal associated with a given irreducible algebraic variety has a finite basis is a pure existence theorem. Only in a few isolated particular cases has the base for the ideal been found, and there appears to be no general method for determining the base which can be carried out in practice. Hilbert, who initiated the theory, proved that the prime ideal defining the ordinary twisted cubic curve has a base consisting of three quadrics, and contributions to the ideal theory of algebraic varieties have been made by König, Lasker, Macaulay and, more recently, by Zariski. A good summary, from the viewpoint of a geometer, is given by Bertini [(1), Chapter XII]. However, the tendency has been towards the development of the pure theory. In the following paper we actually find the bases for the prime ideals associated with certain classes of algebraic varieties. The paper falls into two parts. In Part I there is proved a theorem (the Principal Theorem) of wide generality, and then examples are given of some classes of varieties satisfying the conditions of the theorem. In Part II we find the base for the prime ideals associated with Veronesean varieties and varieties of Segre. The latter are particularly interesting since they represent (1, 1),without exception,the points of a multiply-projective space.
The energy loss by radiation of fast electrons in a Coulomb field
Authors: Jaeger J. C.
Published: November 2008

The problem of the energy loss by radiation of an electron in a Coulomb field has been solved, using relativistic equations and the Born approximation, by Bethe and Heitler, and, using exact non-relativistic equations, by Sommerfeld; the first of these solutions is valid only for relatively high and the second for relatively low energies. In this paper an exact solution of the problem is given using the relativistic Coulomb wave functions, but depending on a final stage of numerical computation. The method is an extension of that used to determine cross-sections for pair production.
On lattice points in the domain |xy| ≤1, | x+y| ≤ √5, and applications to asymptotic formulae in lattice point theory (I)
Authors: Mahler K.
Published: November 2008

Preface. In order to make this paper intelligible to the reader, I repeat the following definitions and results from my paper ‘On lattice points in star domains’, which is to appear in the Proceedings of the London Mathematical Society.
On the quartic surface
Authors: Segre B.
Published: November 2008

1. The projective transformations of F into itself … 121

2. The flecnodal curve, and the lines of F … 122

3. A geometric characterization of F, and the six different types of F in the real domain … 123

4. The τ-points and τ-planes, and a notation for the lines of F … 124

5. The incidence conditions for the lines of F … 125

6. The tetrads of the first kind … 126

7. The tetrads of the second kind … 126

8. The pairs of lines of F … 127

9. The tetrads of the third kind … 129

10. The 16-tangent quadrics of F … 130

11. The conics of the first kind … 131

12. The conics of the second kind … 132

13. No other irreducible conics lie on F … 133

14. The tangent planes of F of multiplicity greater than 3 … 136

15. On twisted curves, especially cubics and quartics, lying on F … 138

16. The T-transformations … 139

17. Construction of an infinite discontinuous group of birational transformations of F into itself … 142

18. Deduction of an infinity of unicursal curves lying on F … 143
The classification of orbits
Authors: Pars L. A.
Published: November 2008

The present paper is concerned with a simple type of dynamical system, a system having only two freedoms and of “separable” type. The essential simplification introduced by this restriction is that the motions in the two coordinates can be discussed to some extent independently of each other. In Part I a method of classification of the possible orbits is explained. Four constants are needed to define the motion completely, but the general nature of the orbit depends on two constants suitably chosen, and the classification is effected by reference to a plane in which the two constants are taken as Cartesian coordinates. It is shewn that the plane of reference is divided into regions by critical curves, and that the orbits represented by points in the same region (each point representing a set of orbits, not a single orbit) are of the same general type. The various possible types of orbits, and their stability, in a special sense, are discussed. In Part II the theory is applied by way of illustration to a somewhat trivial example, the orbits for a particle under a central attraction proportional to the inverse (n + 1)th power of the distance. The most striking thing here is that the possible types of orbits are essentially the same for all values of n greater than 2. In Part III the theory is applied to the classical problem of a particle subject to an attraction of Newtonian type to two fixed centres, a problem of some interest in relation to the general problem of three bodies, and to some current questions in atomic dynamics. The problem is of course not a new one, but it seems that a satisfactory classification of all the possible orbits has not previously been given. It must be confessed that the complete classification for this problem is very laborious.
Current measurement with a Compton quadrant electrometer
Authors: Watson E. E.
Published: November 2008

The purpose of the present paper is to illustrate the use of the Compton Quadrant Electrometer for current measurement by the rate of deflection method. There are several points in the theory and technique for this instrument which do not appear to have been mentioned in any previous papers. With the Compton Electrometer the rate of deflection method gives accurate measurements of currents of 10−14 amperes in less than a minute. The instrument is of course more difficult to use than the Dolezalek type, and it is for this reason that an attempt is made here to discuss and illustrate its use. The theory is given in Compton's original paper and there is also an important note on the subject by Cox and Grindley.
Thermal Properties of an Incompletely Degenerate Fermi Gas
Authors: Mott N. F.
Published: November 2008

The purpose of this note is to calculate the specific heat and paramagnetic susceptibility of an electron gas obeying the Fermi-Dirac statistics for all temperatures, including those temperatures for which the gas is partially degenerate. The results are applicable to the electrons in a metal, whether free or moving in a periodic field, provided only that the number of electronic states per gram atom with energy between E and E + dE can be expressed in the form

as for free electrons.
A generalization of Abel summability
Authors: Kuttner B.
Published: November 2008

The Abel sum of the series can be written in the form

where

This suggests that we should define the “Abel limit”, as u → ∞, of any function A(u) as being given by the expression (1) whenever this exists. We shall, however, in this paper, restrict ourselves to functions A(u) which are bounded in any finite interval. Since we are concerned only with the behaviour of A(u) as u → ∞, this does not involve any serious loss of generality, while we avoid difficulties arising from the divergence of integrals at finite points. We note that the expression (1) can be written in the form

where

A1(u) may conveniently be described as the “Abel transform” of A(u).
The second order electrical effects in metals
Authors: Wilson A. H.
Published: November 2008

The theory of transport phenomena in metals depends upon the solution of an integral equation for the velocity distribution function f of the conduction electrons. This integral equation is formed by equating the rate of change in f due to external fields and temperature gradients to the rate of change in f due to the mechanism which produces the resistance. If this latter rate of change is denoted by [∂f/∂t]coll it happens with some mechanisms that

where f0 is the equilibrium distribution, and ℸ is the time of relaxation which does not depend on the external fields. When equation (1) is true, the problem is comparatively simple, but in general [∂/∂t]coll is an integral operator and it is not possible to define a time of relaxation and a free path. It is known that at high temperatures, such that (Θ/T)2 can be neglected, where Θ is the Debye temperature, a free path exists; but, in general, special methods have to be used to solve the integral equation.
The expansion of an arbitrary function in a series of conal or toroidal functions
Authors: Banerji D. P.
Published: November 2008

1. Particular series of conal and toroidal functions have been studied by various authors*, but so far no method has been given of expanding an arbitrary function in such a series. Ordinary methods fail, since conal functions (and also toroidal functions) are not orthogonal, although they form a complete set. In this paper I give an expansion theorem.
Contributions to the statistical theory of adsorption. II
Authors: Dube G. P.
Published: November 2008

The adsorption of gaseous molecules in a monolayer on solid surfaces has been studied theoretically by several workers, namely Fowler, Peierls and Wang. They have shown that the adsorption isotherm exhibits critical phenomena if there is assumed to be an attractive force between the neighbouring adsorbed molecules, and this has provided an explanation of the critical condensation phenomena observed in deposition experiments, for example those of Cockcroft on the deposition of cadmium on copper. The actual critical phenomena observed consist in the deposition of many layers which are formed if the first layer has fairly started. It is of interest, therefore, to consider if there are other ways in which critical phenomena can arise than through interactions in the first layer. The general problem of adsorption of an arbitrary number of layers is too complicated to study because of the mathematical difficulty in constructing the partition function. We therefore confine ourselves to the consideration of adsorption in two layers only.
Lorentz-invariance in quantum theory
Authors: Eddington Arthur
Published: November 2008

The formal conditions of Lorentz-invariance apply to coordinates x1, x2, x3, x4 which form a 4-vector, and to expressions associated with them. In this paper attention is called to the frequent disregard in quantum investigations of the proviso that the coordinates form a 4-vector.

In most quantum investigations with a practical application the coordinates employed are relative space coordinates ξ, η, ζ coupled with a progressive time coordinate, so that (ξ, η, ζ, it) is not a 4-vector. Nevertheless, conditions of Lorentz-invariance are applied by many authors. Alternatively, they attempt to base the investigation on wave functions of non-relative coordinates x, y, z, t. It is here pointed out that such wave functions give no information about the eigenstates, and that there is no means of deriving wave functions of ξ, η, ζ from those of x, y, z.

Special attention is directed to the misapplication of Lorentz-invariance in Dirac's theory of the hydrogen atom, and in the calculation of degeneracy energy of ionized material.
The production of 8Li by bombardment of boron with neutrons
Authors: Lawrance A. M.
Published: November 2008

The decay period of the α-activity produced when boron is bombarded with neutrons from a (Li + D) source has been measured, and found to agree with the known decay period of 8Li. It has been shown that no activity is produced when (Be + D) neutrons (maximum energy 4·5 × 106 e.V.) are used. The results are explained by postulating the formation of radioactive 8Li according to the process

A short investigation of the number-range distribution for the α-particles emitted by 8Li indicated that a maximum occurred in the distribution at a range of about 6 mm.
Note on General Bi-Harmonic Analysis for a Plate Containing Circular Holes
Authors: Green A. E.
Published: November 2008

In a previous paper a general solution was given for problems of stress distributions in a plate containing circular holes of varying sizes arranged in any manner. This work was a generalization of special methods used by various writers for particular arrangements of holes. The types of stress distributions were, however, confined to those which produce zero force resultants on each hole and the solutions were therefore independent of the elastic constants. Bickley has studied distributions of stress round one circular hole in an infinite plate when the force resultants on the hole are no longer zero, and a few other problems of this type have been dealt with by other writers.
On sets of fractional dimensions (II)
Authors: Best E.
Published: November 2008

1. Any number x between 0 and 1 may be expressed uniquely in the form

where xr is a non-negative integer less than r (r = 2,3,…). We consider the set E of numbers x for which

We establish an inequality connecting the dimensional number of the set E with certain constants of the series

in particular we show that, when ξτ = rθ, the dimensional number of E is θ. We are concerned with the measure of Hausdorff.
The action in a uniform field
Authors: Pars L. A.
Published: November 2008

We consider the motion of a particle in a plane field of force. We take rectangular cartesian coordinates (x, y) in the plane, and denote by V(x, y) the potential of the field, and by h the (constant) energy of the motion. If A and B, whose coordinates are (x0, y0) and (x1, y1), are any two points on an orbit, the orbit is characterized by the property that

taken along the orbit is stationary as compared with the integral taken along a neighbouring curve joining the same points. Here s denotes the length of the are measured from A to B. This is one form, sometimes called Jacobi's form, of the principle of least action. The value of the integral, taken along the orbit, and expressed in terms of the coordinates of the termini and the constant of energy,

is the action function for the field V.
On the stability of crystal lattices
Authors: Fürth R.
Published: November 2008

It has been shown by M. Born (1) that the equation of state and other thermo-dynamical properties of crystals, for example the melting temperature and its dependence on pressure, can be derived by a logical development of the lattice theory of crystals. I was able to show in a recent paper (6) that the results of a very large number of experiments of various kinds on the mechanical and thermal behaviour of solids are in satisfactory agreement with this theory. In the present paper I wish to show that new experiments of Bridgman (7) on the compression of solids under very high pressure can also be used for a comparison with the same theory, and that the results of this investigation are in favour of the theory. Some considerations on the range of stability of a lattice under a high uniform negative pressure are added, which support the idea that there is a close relation between the theory of stability of crystal lattices and the theory of melting.
A note on k-connexes
Authors: Hodge W. V. D.
Published: November 2008

The expression

for the number of linearly independent line complexes of degree n in space of r dimensions was derived by Sisam as a particular case in the determination of the number of linearly independent hyperconnexes of given degrees. The purpose of this note is to generalize Sisam's result by determining the number of linearly independent algebraic forms of a special type to which we give the name k-connex.
On the solutions of the wave equation with discontinuous derivatives
Authors: Friedlander F. G.
Published: November 2008

The differential equation of wave propagation is

where u is a variable describing the state of a medium, x, y, z are rectangular coordinates, and r is the time (the time scale is assumed to be such that waves are propagated with unit velocity).
On the reflexion of a spherical sound pulse by a parabolic mirror
Authors: Friedlander F. G.
Published: November 2008

This paper is an attempt to apply a simple series solution of the wave equation to the reflexion of a sound pulse. The reflector is a paraboloid of revolution, and the incident pulse is spherically symmetrical and comes from the focus of the reflector, so that the wave fronts of the reflected pulse are planes at right angles to the axis of symmetry of the reflector. If the incident pulse consists, at any point, of a discontinuous pressure rise followed by constant excess pressure, the series reduces to a power series in the time counted from the onset of the reflected pulse; for other forms of the incident pulse it can be interpreted as the result of the superposition of elementary pressure pulses which are constant in a small time interval and vanish outside it. No convergence proof is given, so that the interest of the investigation is physical rather than mathematical; but the numerical results indicate that the convergence of the series is unsatisfactory except near the vertex of the reflecting paraboloid. The coefficients of the series are obtained with the aid of recurrence formulae, and the first seven coefficients have been calculated. The calculations become more laborious at each successive stage. A detailed numerical investigation of the reflexion of the ‘simple rectangular pulse’ referred to already (for which the series reduces to a power series) reveals that initially the maximum pressure on any plane section at right angles to the axis of the paraboloid, due to the reflected pulse alone, occurs at the axis and has the same value everywhere on it; but after some time a secondary pressure maximum is established over a circular ring at some distance from the axis. A consideration of the initial pressure gradient of the reflected pulse suggests that a similar state of affairs exists at all distances from the vertex, but the actual calculations only extend to a plane section whose distance from the vertex is four times the focal length. The unsatisfactory convergence of the series precludes the investigation of subsequent changes in the distribution of pressure. It is finally pointed out that these results apply to a certain extent to a finite parabolic mirror.
On the representations of a number as a sum of squares and certain related identities
Authors: Rankin R. A.
Published: November 2008

LetRc(n) be the number of representations of any non-negative integernas the sum ofcsquares; i.e.Rc(n) is the number of different solutions (x1,x2, …,xc) of the equation

where thexμare integers, and may be positive, negative, or zero. Two such solutions, (x1,x2, …,xc) andare only considered to be the same solution if
Covariant line complexes of a pair of quadric surfaces
Authors: Todd J. A.
Published: November 2008

1. In the complete algebraic system of concomitants of a pair of quaternary quadrics given by Turnbull† there appear sixteen covariant line-complexes, eight of the second order and eight of the third. In a later paper‡, the same author gave some geometrical interpretations of these and other concomitants. None the less, no systematic geometrical account of the relations of these complexes seems to have been made. The purpose of the present paper is to discuss the geometrical relations systematically, and make their origin clearer. Incidentally, the syzygies which exist between the complexes are obtained explicitly.
On the summation of certain trigonometric series
Authors: Goddard L. S.
Published: November 2008

In the present note, which is introductory to the following paper, closed expressions, suitable for computational purposes, are found for the sums of the series

where α > 1, t = 1, 2, 3, …, and n is a positive integer. In each case a recurrent relation is found giving the values of and for t > 2 in terms of and the series Θκ(α) (κ = 1, 2, …, t), where

When κ is even the last series is expressed in closed form in terms of the Bernoullian polynomial φκ(l/α) and, when κ is odd and α is rational, a closed form is found involving the polygamma function Ψ(κ)(z), where The general expressions for and involve Ψ(z) and Ψ′(z) when α is rational, but for special values of α they reduce to a form independent of the Ψ-function. and are independent of n and are expressible as simple rational functions of α.
On the classical theory of radiating electrons
Authors: Eliezer C. Jayaratnam , Mailvaganam A. W.
Published: November 2008

1. In Dirac's classical theory of radiating electrons, the relativistic equations of motion of a point-electron in an electromagnetic field are

where (x0, x1, x2, x3) denote the electron's coordinates in flat space-time, dots denote differentiation with respect to the proper time , and the external electromagnetic field is described by the usual field quantities Fμν. The units are chosen so that the velocity of light is unity. These equations, which are derived from the principles of conservation of energy and of momentum, are the same as those obtained by Lorentz, when he used the spherical model of the electron and included radiation damping in an approximate way. But Dirac's method of derivation suggests that this treatment of radiation damping, and therefore the resulting scheme of equations, is exact within the limits of the classical theory.
A general form of the covering principle and relative differentiation of additive functions. II
Authors: Besicovitch A. S.
Published: November 2008

In my paper under the same title, to which I shall refer in future as I‡, I generalized the Vitali covering principle from the case of Lebesgue measure to the case of any non-negative additive function. This allowed me to establish the relative differentiation of additive functions. The convergent sequences of sets in this generalized form of the covering principle were restricted to sequences of concentric circles, and therefore the differentiation arrived at was that in the symmetrical sense. In the present paper, I extend the principle to the case of any regular convergent sequences of covering sets; and then establish the relative differentiation of additive functions in the general sense, and in particular the differentiation of indefinite integrals with respect to any measure function. This problem has a complete solution. It is established that indefinite integrals are differentiable at almost all points. In the case of the general measure function, it is not true that the derivative is equal to the integrand at almost all points, but necessary and sufficient conditions are given under which this is true.
A note on the Hamiltonian equations of motion
Authors: Chang T. S.
Published: November 2008

The Eulerian equations obtained by varying a Lagrangian containing certain field quantities qα and their derivatives of any order are cast into canonical forms. A modified canonical form for the equation of motion is given when the conjugate variables are not independent. Finally a special case is discussed.
The classical equations of motion of an electron
Authors: Eliezer C. Jayaratnam
Published: November 2008

A set of relativistic classical motions of a radiating electron in an electromagnetic field are derived from the principle of conservation of energy, momentum and angular momentum. It is shown that these equations lead to results more in harmony with the usual scheme of mechanics than do the Lorentz-Dirac equations. When applied to discuss the motion of the electron of the hydrogen atom, these equations permit the electron falling into the nucleus, whereas the Lorentz-Dirac equations do not allow this. When applied to consider the motion of an electron which is disturbed by a pulse of radiation, the solution is in a more symmetrical form. For scattering of light of frequency ν the expression for the scattering cross-section is found to be the same as the classical Thomson formula for small ν, and to vary as ν−4 for large ν.