bluuebunny/crossref_metadata_2025 huggingface dataset:
The critical ordering temperature in alloys. II. The existence of a two-phase region
Authors: Easthope C. E.
Published: November 2008

By using Bragg and Williams' approximations and Bethe's interaction energies between nearest neighbours, it has been shown that, for alloys of theAB3type, (1) the maximum value of the critical temperatureTcmust occur whenc= ½ (2) for all temperatures within the range 0 < T < (Tc)maxthere is always a concentration range within which the alloy exists in a two-phase state, the one phase being partially ordered and the other disordered. This two-phase region is small at all temperatures and vanishes both atT= 0 and atT= (Tc)max. In particular forT/(Tc)max= 0·82, which is the value ofTcforc= ¼, the limits of the two-phase region arec= 0·245 andc= 0·256. The theoretical phase diagram has therefore qualitatively the shape, given in Fig. 5. For a fixed concentration there is thus a certain temperature range in which a disordered phase and a partially ordered phase exist in equilibrium with one another. The temperature at which the disordered phase completely disappears is that usually given by experimental determinations of the critical temperature. It is somewhat higher thanTc, but its maximum value also occurs whenc= ½, so that the discrepancy between theory and experiment for this type of superlattice still remains.
Note on the absorption limits for the primary β-particles of mesothorium 2 and uranium X2
Authors: Feather N.
Published: November 2008

The absorption limits for the primary β-particles of mesothorium 2 and uranium X2have been determined using modified tube counters of various types. Results in good agreement with previous single determinations with gold-leaf electroscopes, and at variance with deductions from expansion chamber observations, show the need for caution in fixing limiting energies in β disintegrations from measurements on a relatively small number of tracks.
On the energies associated with the two-centre problem
Authors: Monroe E.
Published: November 2008

A scheme is presented to show the limiting behaviour of the electronic energy levels in the diatomic one-electron problem for unequal centres, as the ions are separated. This generalizes the well known order of levels in the hydrogen molecule ion. Certain implications of the result for the properties of asymmetric diatomic molecules are briefly discussed.
Contributions to the statistical theory of adsorption
Authors: Cernuschi F.
Published: November 2008

Considerable progress has been made recently in the study of the adsorption of gaseous molecules in a monolayer on a solid surface, when the molecules are attached to fixed locations on that surface and allowance is made for interaction between the adsorbed molecules. It has been shown in particular that, if there is an attractive field between two adsorbed molecules so that two adsorbed molecules in neighbouring locations are more tightly bound than when they are adsorbed separately at a distance, the adsorption isotherm shows critical phenomena. This feature of the isotherm may be used to give a generally successful interpretation of the well-known critical condensation phenomena discussed first in this manner by Langmuir and Frenkel. A recent example of experiments on this phenomenon is to be found in the work of Cockcroft on the deposition of cadmium on copper. The work of Peierls, who uses Bethe's method for approximating to the partition function for the adsorbed layer, shows in the most convincing way how interactions between neighbours in the single layer lead to critical phenomena for the degree of completion of the layer.
The theory of adsorption when each particle occupies more than one site
Authors: Roberts J. K.
Published: November 2008

In order to be able to identify the type to which an adsorbed film belongs from its experimentally determined behaviour, it is necessary to know theoretically how various simple and general types, which are likely to occur frequently, behave. Films formed on a rectangular solid lattice with another atom at the centre of each rectangle are considered. Each adsorbed molecule is assumed sufficiently large to exclude the possibility of adsorption on the four neighbouring sites and, owing to the rectangular shape of the lattice, reacts appreciably with two and not with four neighbouring adsorbed molecules. The statistics and kinetics of the film are considered. It is shown that the variation of heat of adsorption with fraction of sites occupied is quite different from that of other types of film which have so far been treated. Immobile films of the same type are briefly discussed.
Production of the long-lived radioactive cobalt isotope from nickel bombarded by fast neutrons
Authors: Viktorin Otakar
Published: November 2008

It is shown that a long-lived radioactive cobalt isotope, which had been previously obtained by Livingood and Seaborg by bombardment of cobalt with slow neutrons, can be produced by irradiation of nickel with fast neutrons.
Notes on the two-centre problem in wave mechanics. I. The hyperbolic nodes of the wave equation
Authors: Marrack P. E.
Published: November 2008

Approximate expressions are obtained, when the distance R between the nuclei is very large, for that portion of the wave function in the two-centre problem which depends on the hyperbolic coordinate μ. From these expressions the number and the approximate position of the nodes in μ can be deduced and hence the rules can be found by which that state of the combined atom at R = 0 can be determined which corresponds to a given state of the atom when the nuclei are completely separated. These rules are also applicable to the cases where the two atoms which can be formed when the nuclei are completely separated have the same energy. The converse problem of finding what state of the completely separated atom corresponds to a given state of the combined atom at R = 0 can also be solved by the use of the rules.
Electronic states at the surfaces of crystals
Authors: Goodwin E. T.
Published: November 2008

The method of the previous paper is extended to determine the surface states of a simple cubic crystal on the approximation of tight binding. It is also applied to the case of a semi-infinite linear chain when the atomic s- and p-states are to be regarded as degenerate, the existence of surface states being again predicted.
Quantum electrodynamics. I
Authors: Hoyle F.
Published: November 2008

In this paper the usual theory of quantum electrodynamics is outlined and an alternative method of quantization is introduced. Some of the difficulties of this theory are discussed and a modification is introduced to meet them. The theory is developed so as to satisfy the principle of relativity, but is not quantized. In the following paper an extension of the theory is given which removes this restriction.
The cut-off characteristic of the single anode magnetron
Authors: Harvey A. F.
Published: November 2008

For the purpose of the present discussion the magnetron will be defined as a diode thermionic tube, having concentric cylinders as electrodes, in which there is a uniform magnetic field parallel to the axis of the electrodes. An electron emitted from the cathode travels towards the anode in a path which is bent by the action of the magnetic field. If the anode has radius a and is at potential V with respect to the cathode of radius b, then it is well known that electrons emitted without velocity from the cathode will just graze the anode when the uniform and axial magnetic field has a value H such that
Note on a Divergent Series
Authors: Hardy G. H.
Published: November 2008

1. The series

is the simplest and most familiar power series whose radius of convergence is zero. It is natural to regard it as a development of the function G(z) defined, when

by

For

say;and

or

according as x is positive or negative. Thus the series (1·1) is an asymptotic series for G(z) in the sense of Poincaré.
The Stability of Crystal Lattices
Authors: Fürth R.
Published: November 2008

A short survey of Born's theory of the thermodynamics and melting of crystals is given. It is shown that Lindemann's and Grüneisen's law for the normal melting temperature can be deduced from this theory, and that the dependence of the melting temperature on pressure, and of the compressibility and the elastic constants on pressure and temperature, as predicted by the theory, are in good agreement with experiment. Several connexions between breaking and melting, suggested by the fundamental ideas on melting and stability of crystals, are discussed and verified. Finally a relation between the heat of melting and the heat of sublimation is deduced and compared with experiment.
On Hausdorff's methods of summability. II
Authors: Rogosinski W. W.
Published: November 2008

A general theory of ‘strength’ for Hausdorff methods T of summability was given in the first part of this paper, to which we shall refer in the following as I. Those methods T which are regular (that is, at least as strong as ordinary convergence C), are defined by the linear transforms

of a given sequence (sk). Here ø(t) (that is, each of its real and imaginary parts) is of bounded variation (b.v.) in the closed interval 〈0, 1〉 [see I, (4)]. The additional condition for consistency (with C) is that ø(t) is continuous at t = 0, and that ø(1) − ø(0) = 1 [I, H. 3]. We write also T ˜ ø(t).
On certain expansions of the solutions of Mathieu'S differential equation
Authors: Erdélyi A.
Published: November 2008

1. There are several known types of expansions of Mathieu functions, i.e. mod 2π periodic solutions of Mathieu's equation ((9), chap. 19),

The simplest expansion is the Fourier series

Almost equally well known are Heine's expansion ((4), p. 414; see also (5) and (6))

and
On the stability of crystal lattices VIII. Stability of rhombohedral Bravais lattices
Authors: Peng H. W. , Power S. C.
Published: November 2008

The main purpose of the paper is an investigation of the stability of a certain class of Bravais lattices, namely, those with a rhombohedral cell of arbitrary angle. The potential energy is assumed to consist of two terms, each proportional to a reciprocal power of the distance. In the continuous series of lattices obtained by changing the rhombohedral angle, there are included the three cubic Bravais lattices, the simple (s), the face-centred (f) and the body-centred (b) lattices. It is shown that (f) and (b) correspond to a minimum of the potential energy, and (s) to a maximum. A method for calculating the potential energy for the intermediate rhombohedral lattices is developed, and, with the help of a certain characteristic function, it is shown by numerical calculation that the (f) lattice corresponds to the absolute minimum of potential energy, and that no extrema, other than (f), (s) and (b), exist. In the last section, the case of a compound (non-Bravais lattice) is considered, and it is shown that the equilibrium and stability conditions for the law of force assumed can be divided into one set for change of volume, and an independent set for change of shape.

We take this opportunity of expressing our sincere thanks to Prof. Born for his interest in our work, and for much valuable advice.
The quantum mechanics of a bounded linear harmonic oscillator
Authors: Auluck F. C.
Published: November 2008

1. The present paper deals with an enclosed (linear) harmonic oscillator. The usual boundary condition that the wave function vanishes at infinity is here replaced by the condition that the wave function vanishes at the walls of the enclosure. The problem has been treated before (1, 2, 4). However, the present discussion goes much further than that given in the papers cited*.
On the reflexion of surface waves by a submerged plane barrier
Authors: Dean W. R.
Published: November 2008

1. The reflexion of waves on the surface of water by a thin plane vertical barrier is considered and the coefficient of reflexion (the ratio of the amplitudes, at a great distance from the barrier, of the reflected and incident waves) is calculated. If the top edge is at a depth a below the surface, it is found that the coefficient of reflexion is about ¼ when where T is the period of the incident waves, so that the condition that the coefficient may exceed ¼ is a .
On the scattering of fast neutrons by protons
Authors: Pais A.
Published: November 2008

1. Introduction. Recent experiments by Amaldi and his collaborators on the scattering of high-energy neutrons (of 10–15 MeV.) by protons(2) have disclosed a considerable anisotropy in the angular distribution of the scattered particles. Theoretical discussions of this problem show an interesting feature in that the results depend sensitively on the basic assumptions involved with regard to the charge dependence of the neutron-proton interaction. This can be seen in particular from calculations by Rarita and Schwinger(3) and by Ferretti(4). The former authors started from the assumption of a distance dependence of this interaction represented by a square well, while the angular and spin dependence included terms of the axial dipole type. If the charge dependence was further assumed to be of the ‘symmetrical’ type, they found a value for the anisotropy in strong disagreement with experiment, whereas the total cross-section agreed with the measured value; a ‘neutral’ theory, on the other hand, yielded agreement as regards anisotropy, but a total cross-section too large by a factor of the order of 1·5. Ferretti investigated the scattering on Bethe's neutral meson theory(5) and found satisfactory agreement with regard to both angular distribution and total cross-section. It should be stressed that all calculations mentioned were performed in the approximation in which only the contributions of the S- and P-waves are considered.
An explanation of anomalous thermionic emission current constants
Authors: Nien-T'ai Sun , Band William
Published: November 2008

It is shown that anomalously large and anomalously small values of the constant A in the thermionic current formula for metals can be explained by the usual statistical equilibrium theory provided proper account is taken of the fact that free electrons may be shared by two competing overlapping energy bands. When this theory is developed there arises a comparatively large temperature variation of the statistical parameter ζ which occurs in the Fermi function, and when the exact formula is forced into the empirical form with a constant work function, the temperature factor appears as a multiplying factor of the current constant, this factor being greater or less than unity depending upon the nature of the overlapping bands. In particular the observed values of the constant A for nickel and for hafnium are well explained.
The complete system of the binary (3, 1) form
Authors: Todd J. A.
Published: November 2008

1. The only double binary forms for which the complete systems of invariants and covariants are known appear to be the (1, 1), (2, 1) and (2, 2) forms, for which the complete systems were determined sixty years ago by Peano. In the present paper we determine the complete system of the binary (3, 1) form and establish its irreducibility. The system proves to contain twenty forms, and is thus scarcely more complicated than that of the (2, 2) form which includes eighteen concomitants, and much simpler than that of the (2, 1, 1) form derived by Gilham. It is hoped in a subsequent note to interpret the system geometrically.
On the generation of sets of four tetrahedra of which any two are mutually inscribed
Authors: Rao C. V. Hanumanta
Published: November 2008

1. A paper with the same title was published with Professor H. F. Baker in 1920; that paper was severely compact and would seem to deserve elucidation, more detail, and some revision. We use a symbolism which, while leaving the proofs geometrical, readily converts itself into ordinary analysis, as we shall show in § 12.
The elastic stresses produced in a thick plate by the application of pressure to its free surfaces
Authors: Sneddon I. N.
Published: November 2008

In a recent paper (1) an analysis was given of the distribution of stress in a semi-infinite elastic medium deformed by the pressure of a rigid body on part of the plane boundary, the remainder of the plane being free. In that form of the problem—the so-called ‘Boussinesq problem’—the normal displacement of a point within the pressed area was prescribed and the distribution of pressure over that area determined. In this paper the corresponding analysis is given for the case in which the pressed area and the distribution of pressure over it are both prescribed and the normal displacement of a point on the free surface is determined.
Statistical mechanics and the partitions of numbers
Authors: Auluck F. C. , Kothari D. S.
Published: November 2008

1. The properties of partitions of numbers extensively investigated by Hardy and Ramanujan (1) have proved to be of outstanding mathematical interest. The first physical application known to us of the Hardy-Ramanujan asymptotic expression for the number of possible ways any integer can be written as the sum of smaller positive integers is due to Bohr and Kalckar (2) for estimating the density of energy levels for a heavy nucleus. The present paper is concerned with the study of thermodynamical assemblies corresponding to the partition functions familiar in the theory of numbers. Such a discussion is not only of intrinsic interest, but it also leads to some properties of partition functions, which, we believe, have not been explicitly noticed before. Here we shall only consider an assembly of identical (Bose-Einstein, and Fermi-Dirac) linear simple-harmonic oscillators. The discussion will be extended to assemblies of non-interacting particles in a subsequent paper.
The variation of the dipole moment of adsorbed particles with the fraction of the surface covered
Authors: Miller A. R.
Published: November 2008

The statistical treatment which was given previously to determine the variation of the heat of adsorption of polar molecules is extended to show how the variation of the dipole moment with the fraction of the surface covered can be taken into account. The equations have been used to determine the variation of the heat of adsorption of ammonia on a non-conducting surface. The contributions to the heat of adsorption due to the van der Waals and to the electrostatic forces are of the same order of magnitude and of opposite signs so that the resultant variation in the heat of adsorption is very much less than would be expected from a consideration of forces of one kind only. The contributions to the heat of adsorption, which are made by the electrostatic forces when the dipole moment is constant and equal to M0, when it is constant and equal to M1, and when its variation with the fraction of the surface covered is taken into account, are compared for the whole range of values of θ. It is shown that the mutual depolarizing action of the molecules reduces the magnitude of the total variation in the heat of adsorption by a considerable amount. The heat curve which is calculated from the equations given by the statistical analysis is compared with that which is obtained when it is assumed that the particles form a random distribution over the surface, and the effect of the clustering of the adsorbed particles on the surface on the variation of the heat of adsorption is determined.
The number of configurations of molecules on a lattice
Authors: Miller A. R.
Published: November 2008

The number of configurations of mixtures of dimer and single molecules, of trimer and single molecules and of trimer and dimer molecules is examined by using the Bethe technique when there may be some vacant sites. This condition provides a check on the internal consistency of the Bethe method through the integrability of the resulting partial differential equations.
A note on the distribution of the different orderings of n objects
Authors: Haden H. G.
Published: November 2008

Suppose that we have n objects, which may conveniently be represented by the integers 1, 2, 3, …, n. The total number of ways in which they can be ordered is n!. Let the order 1, 2, 3, …, n, be termed the normal order. Any other order can be classified according to the minimum number of interchanges (of one number with an adjacent one) required to restore the normal order.
Time effects in the magnetic cooling method. II
Authors: Temperley H. N. V.
Published: November 2008

Some data obtained at Leiden on relaxation effects in a diluted potassium chrome alum are discussed in the light of a paper by Temperley (10), and excellent agreement is found with the conclusions of that paper. The primary effect of diluting the salt is to reduce the magnetic interaction, and the observed effects are a big increase in the relaxation time in zero field, and a decrease of the relaxation time with increasing field instead of an increase. It is predicted that a decrease should set in with an ordinary salt at liquid helium temperatures at fields somewhat higher than have so far been used. Van Vleck's criticisms of Temperley's paper are discussed, and it is concluded that they are not valid in their present form.
Rectilinear plastic flow of a Bingham solid
Authors: Oldroyd J. G.
Published: November 2008

The velocity distribution in steady rectilinear plastic flow of a Bingham solid between coaxial circular cylinders in relative motion can be obtained exactly by simple arguments. On the basis of the general equations of plastic flow, an approximate velocity distribution is derived below for the case of eccentric circular cylinders as rigid boundaries and zero pressure gradient by making use of a conformal transformation. This approximation is valid only if a dimensionless parameter S (which is proportional to the yield value) is sufficiently small, and there is flow in the whole region between the cylinders. The differences in velocity distribution in the cases of Bingham solid (S finite) and Newtonian liquid (S zero) are illustrated by a particular example. It is shown that the effect of a finite yield value is to cause the contours of equal velocity in a normal section to become more nearly concentric with the inner boundary as the yield value increases from zero.
Prime ideals and postulation formulae
Authors: Goddard L. S.
Published: November 2008

1. In this paper we find explicitly the base for the prime ideal associated with any irreducible Vd−1 on a Segre variety, or a Veronesean variety, Vd. This work extends that of an earlier paper (1) in which the base was found when the Vd−1 is a complete intersection of Vd with one primal. As particular examples we can write down the base for any irreducible curve on a quadric surface, a Veronese surface or a Del Pezzo surface. We also show how the base for any prime ideal changes under a Veronesean transformation.
Large sample tests of statistical hypotheses concerning several parameters with applications to problems of estimation
Authors: Radhakrishna Rao C.
Published: November 2008

If the probability differential of a set of stochastic variates contains k unknown parameters, the statistical hypotheses concerning them may be simple or composite. The hypothesis leading to a complete specification of the values of the k parameters is called a simple hypothesis, and the one leading to a collection of admissible sets a composite hypothesis. In this paper we shall be concerned with the testing of these two types of hypotheses on the basis of a large number of observations from any probability distribution satisfying some mild restrictions and their use in problems of estimation.
Laminar flow behind a two-dimensional grid
Authors: Kovasznay L. I. G.
Published: November 2008

The production of a ‘wake’ behind solid bodies has been treated by different authors. S. Goldstein (1) and S. H. Hollingdale (2) have discussed the laminar wake behind a flat plate, while the wake of a two-dimensional grid has been treated for the turbulent case alone using L. Prandtl's ‘Mischungsweg’ theory by E. Anderlik and Gran Olsson (3), (4).
Some inequalities in the theory of functions
Authors: Hayman W. K.
Published: November 2008

In this paper we investigate the following problem.

We suppose given a sequence of complex values wn, defined for n = 0, 1, 2, …, and for n = ∞, and such that

while at least one wn differs from zero and ∞. We consider functions f(z), which are regular in | z | < 1, and take none of the sequence of values wn, and we investigate the effect of this restriction on the rate of growth of the function, as given by the maximum modulus
Some probability distributions connected with recording apparatus
Authors: Domb C.
Published: November 2008

1. The problem to be discussed in the present paper arises when the finite resolving time of a recording apparatus is taken into account. Events are divided into two classes, recorded and unrecorded. Any recorded event is followed by a dead interval of a certain length of time, during which any other event which occurs will be unrecorded. A typical example is an α-particle counter; a recorded particle causes the chamber to ionize, and no other particle can be recorded until the chamber has deionized. In the case when the dead interval is of constant length τ, if the observation time t lies between (n − 1)τ and nτ, the number of recorded events must be 0, 1, 2, …, n − 1 or n. We shall assume that the probability of an event occurring in the interval [t, t + dt] is λdt, λ being constant; this is the case of most practical interest. The probability distribution of recorded events for dead intervals of constant length has been determined by Ruark and Devol (2). The explicit expression of this distribution is fairly complicated, and it is therefore difficult to manipulate. The method employed in the present paper leads naturally to the Laplace transform of the distribution with respect to the time t, and this is relatively simple. The method can easily be generalized to deal with the case of dead intervals which are not all equal, but follow a probability distribution u(τ) dτ. Finally the Laplace transform is very convenient for determining the asymptotic behaviour of the distribution of recorded events for large times of observation.
On the quantization of field theories derived from higher order lagrangians
Authors: de Wet J. S.
Published: November 2008

Heisenberg and Pauli (1) have shown how to quantize field theories derived from a Lagrangian containing first derivatives of the field quantities only. The present paper extends the theory of quantization of fields to the case of higher order Lagrangians, i.e. Lagrangians in which higher derivatives than the first appear. It is shown how such field equations can be put into Hamiltonian form and how the quantization can subsequently be carried out. Both the cases of Einstein-Bose and Fermi-Dirac quantization are discussed. It is established that the quantization is relativistically invariant and consistent with the field equations. An interesting feature of the present theory is that the Hamiltonian proves to be different, in general, from the integral of the 4–4 component of the energy momentum tensor.
Surface effects in co-operative phenomena
Authors: Chang T. S.
Published: November 2008

It is shown how the configurational partition functions of certain assemblies can be rigorously constructed. From them it is proved that the terms in the expression for the free energy of the assembly depending on the surface enclosing the assembly are of a smaller, order than N, N being the total number of atoms. This justifies the usual ignoration of surface effects in studying the bulk properties of a large assembly of atoms.
The evaluation of integrals of the form
Authors: Goodwin E. T.
Published: November 2008

It is well known to computers that the approximate formula

yields a surprising degree of accuracy even for quite large values of the interval h; for example, if h = 1 the error is one unit in the fourth decimal.
The interaction between floating particles
Authors: Nicolson M. M.
Published: November 2008

A quantitative study of the law of force between bubbles floating on a fluid has been made and the results exhibited in the form of a potential energy-distance curve. The main approximations made are that the inclination of the surface of the fluid is small (so that the equations for the surface become linear), and that the submerged portion of the bubble remains spherical. The latter assumption restricts the validity of the results to small bubbles of about 3·0 mm. diameter and less.

One object of the work was to determine the size of bubble which simulated most nearly a typical atomic interaction curve. This application arises in connexion with Prof. Bragg's ‘bubble’ model of crystal lattices. Exact simulation cannot be expected, for the bubble model is two-dimensional, and also does not exert forces analogous either to Coulomb repulsion or to the metallic bond. It can be shown, however, that the law of force between bubbles of a particular size is a remarkably close fit to the law of force between inert gas atoms, that is, to an interaction consisting of a van der Waals attraction and a short-range or ‘overlap’ repulsion. Fig. 4 illustrates this point. The two full curves are the energy-distance curves for krypton and argon atoms (plotted from Fowler, Statistical Mechanics, 2nd ed., Chapter x, Table 29). The circles mark the corresponding curve for bubbles with α = 1·0 (i.e. of diameter 3 mm. in soap solution). The units of energy and distance are chosen so as to bring the minimum of each curve to the point (1, − 50). It will be seen that the curves are remarkably similar over the range shown.
On the theory of statistical procedures
Authors: Pitt H. R.
Published: November 2008

The essential elements in any application of statistical theory are

(1) the set of hypothetical physical states θ which may give rise to the observed phenomena,

(2) the set of all possible observations or samples x,

(3) the set of actions α which may be taken when x is known.

The whole procedure in the common applications can be divided into two parts:

(A) the estimation of θ from the known observation x,

(B) action on the basis of this estimation.
Notes on the forces acting on a two-dimensional aerofoil in shear flow in the presence of a plane boundary
Authors: Coombs A.
Published: November 2008

1. The forces acting on the two-dimensional aerofoil in a bounded uniform stream have been found for a variety of cases, and in the present paper an attempt is made to extend the theory to include linear shear-flow. The special case of the symmetrical Joukowsky aerofoil in unbounded shear-flow has been solved by Tsien (6) using real-variable theory. A more satisfactory method is using complex-variable technique indicated in § 3, and is applicable to a more general shaped aerofoil. The effect on the lift and moment when a plane boundary is present is then considered. When the aerofoil is not too near the boundary, the lift and moment can be expanded in powers of the ratio of a typical length in the aerofoil to the height of the aerofoil above the boundary, by following exactly the technique used by Green (1) in a recent paper dealing with the same problem but with a uniform fluid flow. In the last section the limiting case of the flat plate touching the boundary with its trailing edge is discussed.
Axial symmetric stress distributions in aeolotropic hexagonal crystals. The problem of the plane and related problems
Authors: Elliott H. A.
Published: November 2008

The method of transforms (7), (2) is shown to be directly applicable to the case of axial symmetric stress distributions in hexagonal crystals. It is shown that solutions for problems of indentation of the hexagonal plane by rigid punches can be found for punches of arbitrary axial symmetric shape. Solutions are given in full for the cases of spherical, conical and circularly cylindrical punches.

The same method is used to find the solutions for a material containing disk-shaped cracks between hexagonal planes and the results for the isotropic case deduced from the general solution.
The invariants of a finite collineation group in five dimensions
Authors: Todd J. A.
Published: November 2008

In 1914 Mitchell (13) underook the determination of the finite primitive collineation groups in more than three dimensions which are generated by homologies. Among these groups are five which are remarkable as not being members of an infinitely extended series, and these five isolated groups fall naturally into two sets. One set comprises three groups of orders 27.34:5, 29.34.5.7 and 213.35.52.7, belonging respectively to space of five, six and seven dimensions, the centres of the generating homologies of the smaller groups being among those of the larger ones; these groups had been considered earlier by Burnside (3) as groups of linear substitutions with rational coefficients, and both the groups and the configurations formed by the centres of the generating homologies are closely related to polytopes in Euclidean space of six, seven and eight dimensions first described by Gosset (9) and later studied in detail by Coxeter (6, 7). The two remaining groups, of orders 26.34.5 and 28.36.5.7, belong respectively to four and five dimensions, and the centres of the forty-five generating homologies of the former appear among the centres of the 126 generating homologies of the latter. The smaller group is the well-known simple group of order 25920, and is probably the most extensively studied of all special groups of finite order. Its representation as a collineation group in four dimensions was first obtained by Burkhardt (2), who determined the invariants of the group; the simplest of these is a quartic primal with nodes at the 45 centres of homologies, which was later studied by Coble (5) and myself (14). The geometry of the configuration formed by these 45 nodes has been investigated in detail by Baker (1), whose results I have used elsewhere (15) to classify the operations of the group and obtain their distribution into conjugate sets.
Markoff chains with an enumerable number of states and a class of cascade processes
Authors: Foster F. G.
Published: November 2008

In § 1 a Markoff chain is defined, and a theorem of Kolmogoroff relating to its asymptotic behaviour is stated. Its stable distributions are examined in § 2 and some further results are obtained. These are then applied in §§ 3, 4 to a certain generalization of the cascade process, regarded as a Markoff chain with a special kind of matrix.
On Boltzmann's equation in the kinetic theory of gases
Authors: Wild E.
Published: November 2008

1. Boltzmann's differentio-integral equation for the molecular velocity distribution function in a perfect gas forms the natural starting-point for a mathematical treatment of the kinetic theory of gases. The classical results of Maxwell and Boltzmann in this theory are well known. They include the proof that, for simple gases, i.e. those in which the molecules have only the three translational degrees of freedom, the only stationary and spatially homogeneous solution is the one which corresponds to the Maxwellian distribution.
On multiplicative systems defined by generators and relations
Authors: Evans Trevor
Published: November 2008

It is the purpose of this paper to study the properties of multiplicative systems, for which the associative law is not assumed, when these systems are given in terms of generators and relations. We confine ourselves mainly to loop theory, although the general theory holds also for groupoids, groupoids with division on one side, and quasigroups. Throughout the paper we are guided by two main considerations, to discover how far the concepts and results of group theory carry over to the non-associative case, and to exhibit a specific example of some of the fundamental concepts of abstract algebra. In many ways, in the general theory, we are able to obtain more complete results than in group theory. There remain, however, many interesting analogues of group theoretical concepts. It is hoped to deal with some of these in a later paper.
On the statistical distribution of the widths and spacings of nuclear resonance levels
Authors: Wigner Eugene P.
Published: November 2008

If the average spacing of the resonance levels is very small as compared with the range of energy in which the spacing or width of the levels changes appreciably on the average, one can speak of a statistical distribution of the level spacings and widths. The question then comes up naturally, whether the ‘distribution law’ for width and spacing is different for protons, neutrons, etc., i.e. whether it depends on the long-range interaction of the scattered particle. It is pointed out that while the average width must depend on the long-range interaction (on account of the penetration factor) the distribution of the widths, if these are measured in terms of their average, can be expected to be independent therefrom. In the case of the level spacings, not only the distribution about the average, but the average itself is also independent of the long-range interaction. It is pointed out, incidentally, that the logarithmic derivative of the wave function at the surface of the nucleus will have, on the whole, a positive value between resonances because of the greater effect of the high-energy resonances than that of the bound states.
Factorization of the algebra of particles of half-odd spin
Authors: Le Couteur K. J.
Published: November 2008

It is proved that the matrix algebra for any relativistic wave equation of half-odd integral spin can be factorized as the direct product of a Dirac algebra and another, called a ξ-algebra. The structure and representation of ξ-algebras are studied in detail. The factorization simplifies calculations with particles of spin > ½, because the ξ-algebra contains only one-sixteenth as many elements as the original matrix algebra.
Some threefolds on which adjunction terminates
Authors: Roth L.
Published: November 2008

A well-known theorem, due to Castelnuovo and Enriques(1), states that a surface on which the process of successive adjunction, applied to any curve system, terminates, must be rational or scrollar; actually the result shows that, if the property in question holds for any one system of sufficiently general type, then it must hold for all systems; but this has not been established directly.
Wave functions for excited states of mercury and potassium
Authors: Mishra Brahmananda
Published: November 2008

In the course of some calculations on the lifetimes of excited states of mercury and potassium(9) and photo-ionization of excited potassium, wave functions were derived for various excited states of these atoms. Tables of these wave functions are presented in this paper.
Steady motion of conducting fluids in pipes under transverse magnetic fields
Authors: Shercliff J. A.
Published: November 2008

This paper studies the steady motion of an electrically conducting, viscous fluid along channels in the presence of an imposed transverse magnetic field when the walls do not conduct currents. The equations which determine the velocity profile, induced currents and field are derived and solved exactly in the case of a rectangular channel. When the imposed field is sufficiently strong the velocity profile is found to degenerate into a core of uniform flow surrounded by boundary layers on each wall. The layers on the walls parallel to the imposed field are of a novel character. An analogous degenerate solution for channels of any symmetrical shape is developed. The predicted pressure gradients for given volumes of flow at various field strengths are finally compared with experimental results for square and circular pipes.
Extremal lattices of convex bodies
Authors: Swinnerton-Dyer H. P. F.
Published: November 2008

Let C be a bounded closed convex body in n dimensions, symmetric about the origin. Any lattice Λ containing the origin but no other interior point of C is called admissible. There is a positive lower bound Δ(C) for the determinants of admissible lattices (since the origin is inside C); and any admissible lattice with determinant Δ(C) is called critical. Suppose that Λ is any admissible lattice, with determinant d(Λ). We may define A by a fixed set of generating points Li (i = 1,2, …, n); and we shall say that a lattice Λ′ lies in a small neighbourhood of Λ if Λ′ can be generated by a set of points L′i (i = 1,2, …, n) each of which lies in a small neighbourhood of the corresponding Li. We shall call Λ extremal if in a sufficiently small neighbourhood of Λ there are no admissible lattices Λ′ with d(Λ′) < d(Λ). Thus all critical lattices are extremal.
Micro-programming and the design of the control circuits in an electronic digital computer
Authors: Wilkes M. V. , Stringer J. B.
Published: November 2008

Experience has shown that the sections of an electronic digital computer which are easiest to maintain are those which have a simple logical structure. Not only can this structure be readily borne in mind by a maintenance engineer when looking for a fault, but it makes it possible to use fault-locating programmes and to test the equipment without the use of elaborate test gear. It is in the control section of electronic computers that the greatest degree of complexity generally arises. This is particularly so if the machine has a comprehensive order code designed to make it simple and fast in operation. In general, for each different order in the code some special equipment must be provided, and the more complicated the function of the order the more complex this equipment. In the past, fear of complicating unduly the control circuits of the machines has prevented the designers of electronic machines from providing such facilities as orders for floating-point operations, although experience with relay machines and with interpretive subroutines has shown how valuable such orders are.
Recurrence and first passage times
Authors: Bartlett M. S.
Published: November 2008

The theoretical problem of the probable time of recurrence of a given state of a physical system which changes according to a specified random or stochastic process is in general not a very tractable one, but many situations are covered by the more particular methods and formulae given in this paper. Several of these are believed new, but others are also included for reference. We shall usually limit our attention to homogeneous processes, i.e. the transition probabilities do not depend explicitly on the time. Even so, a generally useful solution of the complete distributional problem of recurrence times would appear rather unlikely; for example, it would have to include as a special case the recurrence theory for trigonometrical or power series, problems of some complexity which have separately engaged the attention of mathematicians (see, for example, Kac (7)). Thus a manageable simplification of the complete distributional problem emerges only in particular but important classes of physical problems classifiable theoretically as Markoff processes or, somewhat more generally, as ‘renewal’ processes (cf. Feller (4)). In other cases, if we are content with mean values (which do not always exist), and are dealing with stationary processes with complete ergodic properties, such as a statistical mechanical system in equilibrium, the simple but powerful arguments originally used by Smoluchowski (11) are available. We first formulate the probability problem in symbolic terms, being more concerned in this paper with general structural relations than with special points of rigour, or specific applications.
The couple on a rotating spheroid in a slow stream
Authors: Sowerby L. , Rosenhead L.
Published: November 2008

The problem considered here is to determine the couple required to maintain the steady rotation about its axis of symmetry of an oblate spheroid which is placed in a uniform stream of viscous fluid moving parallel to this axis. The conditions of flow are assumed to be such as to permit the use of Oseen's approximation in the equations of motion. An expression in terms of spheroidal wave functions is derived for the transverse component of velocity in the fluid, and from this the expression for the couple is deduced. The disk rotating about the normal through its centre is considered as a special case, and some numerical results are given for certain values of a Reynolds number. A comparison is made, and agreement found, with results obtained by Jeffery in a previous paper which treated the case in which the mainstream was at rest.
Multipole expansions in the theory of surface waves
Authors: Thorne R. C.
Published: November 2008

Problems dealing with the generation of surface waves in water involve the consideration of singularities of different types in the liquid. In the case when bodies are present in the liquid, waves may be either generated by the movement of the body, or reflected from the body. The two cases are essentially equivalent, and the resulting motion can be described by a series of singularities placed within the body. The boundary conditions on the surface of the body give equations from which the exact form of the potential can be obtained. Ursell (10) has solved in this manner the problem, earlier discussed by Dean(1), of the generation of surface waves by a submerged circular cylinder. In this two-dimensional problem he used a series of complex potential functions arising from multipoles at the centre of the cylinder, but the velocity potential of the motion could have been described, without the introduction of the stream function, in terms of the velocity potentials of the multipoles.
The asymptotic distribution of serial correlation coefficients for autoregressive processes with dependent residuals
Authors: Walker A. M.
Published: November 2008

In a recent paper (3) Diananda has given an extension of the central limit theorem to stationary sequences of random variables {Yt} (t = 1,2,3,…), which arem–dependent, i.e. are such thatm+ 1 is the smallest integerrhaving the property that two sets of the Y's are independent whenever the suffix of any member of one set differs from that of any member of the other set by at leastr. As Diananda points out, this can be used to show that for linear autoregressive processes withm–dependent residuals, the joint distribution of any finite number of the forms which occur in Bartlett and Diananda's goodness of fit test (see (1)) is asymptotically multivariate normal, provided also that the fourth moment of the residuals is finite. These forms are linear functions of the sample serial correlations rt, and if the autoregressive process is defined by

they may be denoted by, where

Etbeing the usual shift operator such that for any function ftdefined for integral t,In particular, when, as for a moving average process, the joint distribution of any finite number of the rt, is asymptotically multivariate normal.
Pseudo-Abelian varieties
Authors: Roth L.
Published: November 2008

It is a familiar fact that the Picard surface (or hyperelliptic surface of rank 1) admits a completely transitive permutable continuous group of ∞2 automorphisms. There are, however, other non-scrollar surfaces which possess continuous groups of automorphisms, namely, the elhptic surfaces. Every elliptic surface V2 contains a pencil of birationally equivalent elhptic curves, which are the trajectories of the group in question; it also contains a second, elliptic, pencil of birationally equivalent curves; the intersection number of the two pencils is an important character, known as the determinant of V2. Just as any Picard surface can be mapped on a multiple Picard surface of divisor unity, so V2 can be mapped on a multiple elliptic surface of determinant unity, the branch curve (if any) corresponding to a certain number of trajectories.
The numerical solution of Poisson's and the bi-harmonic equations by matrices
Authors: Cornock A. F.
Published: November 2008

This note describes a simple method of solving the matrix forms of the customary families of finite difference equations which approximate to Poisson's equation (in three dimensions as well as two) and to the bi-harmonic equation. It is intended for use where the boundary of the region over which a solution is wanted is comparatively simple, that is to say, can be subdivided into a comparatively few rectangular areas or rectangular parallelepipeds.