bluuebunny/crossref_metadata_2025 huggingface dataset:
Concerning some Solutions of the Boundary Layer Equations in Hydrodynamics
Authors: Goldstein S.
Published: November 2008

The boundary layer equations for a steady two-dimensional motion are solved for any given initial velocity distribution (distribution along a normal to the boundary wall, downstream of which the motion is to be calculated). This initial velocity distribution is assumed expressible as a polynomial in the distance from the wall. Three cases are considered: first, when in the initial distribution the velocity vanishes at the wall but its gradient along the normal does not; second, when the velocity in the initial distribution does not vanish at the wall; and third, when both the velocity and its normal gradient vanish at the wall (as at a point where the forward flow separates from the boundary). The solution is found as a power series in some fractional power of the distance along the wall, whose coefficients are functions of the distance from the wall to be found from ordinary differential equations. Some progress is made in the numerical calculation of these coefficients, especially in the first case. The main object was to find means for a step-by-step calculation of the velocity field in a boundary layer, and it is thought that such a procedure may possibly be successful even if laborious.

The same mathematical method is used to calculate the flow behind a flat plate along a stream. The results are shown in Figures 1 and 2, drawn from Tables III and IV.
A property of continuous arcs
Authors: Verblunsky S.
Published: November 2008

A continuous arc joining two points (x1, y1), (x2, y2) is the aggregate of points given by

where f and φ are continuous functions, and where
Enumerative Formulae for Surfaces
Authors: Roth L.
Published: November 2008

The formulae considered in this paper deal with the multiple tangent lines and planes to surfaces in three dimensions; the functional method employed is that of a previous paper, in which a more restricted discussion was given for surfaces in higher space.
Note on the Theory of Measure
Authors: Young L. C.
Published: November 2008

A function of sets γ(E) absolutely additive is termed by Radonée Poussin continuous if γ(E) vanishes whenever E consists of a single point only. It is known that if a function of sets is not continuous there exists a countable set ε1 and a continuous function of sets such that
An optical illusion due to contrast
Authors: Thomson G. P.
Published: November 2008

In the course of a long series of experiments on the scattering of positive rays I have had occasion to take some hundreds of photographs in which positive rays diverging from an approximately linear source pass through a metal slit about ·5 mm. wide parallel to the source and then strike the sensitive surface of the plate. The change caused by the actual impact of the rays can be developed like the ordinary latent image produced by light. The pattern thus formed on the negative is a rectangular strip about 1·8 mm. × 20 mm. answering in shape to the shape of the slit, and bordered by a much less intense edge (of the order 10%) due to the scattered rays whose investigation was the object of the experiment. It was found that, in a majority of the photographs, the rectangular strip, which should have been approximately uniform in blackness, at least across, had its longer edges apparently much blacker than its centre. So marked is this effect that it was supposed to be due to some cause affecting the distribution of the rays. However, careful measurements with a microphotometer showed that the effect was unreal, the edges always being less black than the centre. The effect of the positive rays is thus merely to produce a certain distribution of blackening on the plate, and the increased blackness seen at the edges is purely subjective.
Improvements in the apparatus for measuring the velocity of very rapid chemical reactions
Authors: Hartridge H. , Roughton F. J. W.
Published: November 2008

During the last year the homogeneous reactions of haemoglobin, and the heterogeneous reactions of the red blood corpuscle, have continued to furnish us with abundant opportunity for the application of our method for measuring the velocity of very rapid chemical reactions*. The recent types of apparatus, which we have devised, have not departed in essentials from the general principle previously employed, but a number of modifications and extensions have been introduced to meet with greater efficiency the special conditions which were imposed upon us. Since these developments would seem to be of much wider application than the special purposes for which they were in the first place designed, we think that it may be useful to workers in other fields to give a brief summary of their general features here. In the present paper we shall accordingly describe the methods we have adopted for 1. Very fast reactions. 2. Very slow reactions. 3. Very dilute solutions. 4. Small quantities of fluid. 5. Reactions following one another in rapid succession. 6. Reactions involving short-lived transient compounds. 7. The detection of concentration gradients in heterogeneous reacting solutions. Special apparatus has been devised to deal with each of these cases which will be briefly described in the following sections.
On the remarkable quadrics of a space cubic curve
Authors: Vaidyanathaswamy R.
Published: November 2008

In the geometry of the cubic curve Γ in space, we have to study the quadrics Q which stand in certain special relations to the curve. We shall find it convenient to refer to these relations as A, B, B′, C, C′; these are defined below.
An unsuccessful attempt to influence the normal decay of a weak source of polonium
Authors: Feather N.
Published: November 2008

Several suggestions have recently been made that the course of radioactive transformation is not, as is usually believed, independent of such changes of the physical environment as may be effected on earth. To the particular locality, to the concentration of the source, and to the action of intense γ radiation have been attributed effects which their investigators have been unable to ascribe to other contingencies. In particular, Pokrowski has developed a tentative hypothesis to explain the complicated anomalies which he had already observed.
The rotational specific heat of a polyatomic molecule for high temperatures—correction
Authors: Viney Irene E.
Published: November 2008

I am indebted to Mr L. Kassel of the Department of Commerce, U.S.A., for pointing out an error of computation in a recent paper in these Proceedings.

The asymptotic expansion for the partition function should read

where
The optical properties of metals
Authors: Mott N. F. , Zener C.
Published: November 2008

Three main stages may be marked in the development of the theory of the optical properties of metals. First, there is Drude's original theory, based on Maxwell's equations; in this theory the current densityjat any point in a metal is supposed to be equal to the product of the electric vector of the light and of the conductivity of the metal. The theory yields the well-known Hagen-Rubens formula for the reflecting power, which appears to be in agreement with experiment for very long wave-lengths (λ > 10μ), but leads to completely incorrect results in the optical region. Various investigators ‡ have therefore modified the theory to take account of the finite mass of the electron; the formulae obtained pass over into the Drude formulae for sufficiently long wave-lengths. Finally a quantum theory of the phenomenon has been given by Kronig§, the electrons being treated as moving in a periodic field due to the crystal lattice in the manner originated by Bloch; this theory, in its turn, becomes identical with the modified classical theory if the periodic lattice is neglected.
The stability of viscous fluid flow between rotating cylinders
Authors: Goldstein S.
Published: November 2008

The stability of the motion of viscous incompressible fluid, of density ρ and kinematic viscosity ν, between two infinitely long coaxial circular cylinders, of radiiaanda+d, whered/ais small, is investigated mathematically by the method of small oscillations. The inner cylinder is rotating with angular velocity ω and the outer one with angular velocity αω, and there is a constant pressure gradient parallel to the axis. The fluid therefore has a component velocityWparallel to the axis, in addition to the velocity round the axis. A disturbance is assumed which is symmetrical about the axis and periodic along it. The critical disturbance, which neither increases nor decreases with the time, is periodic with respect to the time (except whenW= 0, when the critical disturbance is a steady motion). As Reynolds number of the flow we take ||d/ν, whereis the average value ofWacross the annulus, and we denote bylthe wave-length of the disturbance along the axis, by σ/2π the time period of the critical flow, bycthe wavelength of the critical flow, byωcthe critical value of ω, and we put

approximately, if α is not nearly equal to 1.
A note on the criterion of maximum overlapping of wave functions
Authors: Coulson C. A.
Published: November 2008

The criterion of maximum overlapping, as developed by Pauling and Slater, is shown to be true only under certain conditions, depending on the relative positions of three term values, and the relation of the true energy to these three subsidiary ones is discussed in some detail.
A note on the absorbability in cadmium of neutrons slowed down in water and boric acid solutions
Authors: Westcott C. H.
Published: November 2008

Measurements have been made on the proportion of the effect induced in silver which is due to neutrons strongly absorbed by cadmium, using cylinders of water and boric acid of different sizes to slow down the neutrons. In addition to furnishing data for use in the foregoing paper, the results show that practically all the neutrons of groups A and B must have energies greater than any of those of group C.

The author desires to thank Lord Rutherford for his interest in the work and for the use of a Ra + Be source, and Dr Oliphant for discussion and advice. He is also indebted to the Department of Scientific and Industrial Research for a Senior Research Award.
On the magneto-resistance effect in single crystals of cadmium
Authors: Milner C. J.
Published: November 2008

A method of growing single-crystal resistance specimens free, to a large extent, from strains and impurities is described. The magneto-resistance effect in cadmium single crystals has been studied in some detail at the temperature of liquid nitrogen, using sufficiently high magnetic fields to observe the linear effect found by Kapitza.

Although it proved impossible, in general, to determine the orientation of individual crystals, the experiments suggest that the “critical field” of the linear effect is dependent on the orientation of the crystal with respect to the current and magnetic field, and not on the perfection of the crystal lattice.
On two classical lattice point problems
Authors: Ingham A. E.
Published: November 2008

1. Let r(n) be the number of representations of n as a sum of two squares, d(n) the number of divisors of n, and

where γ is Euler's constant. Thus P(x) is the error term in the problem of the lattice points of a circle, and Δ(x) the error term in Dirichlet's divisor problem, or the problem of the lattice points of a rectangular hyperbola.
Deuteron bombardment of the heavy elements.
Authors: Krishnan R. S. , Nahum E. A.
Published: November 2008

The results of the bombardment of mercury, lead and thallium by 9 M.e.V. deuterons are reported. The following radioactive isotopes have been detected: 5·5 min., 48 min., 36 hr., 60 day mercury isotopes; 4·4 min., 10·5 hr., 44 hr., and 13 day thallium isotopes; 10·25 min., 2·75 hr., and 54 hr. lead isotopes; 18 hr. and 6·35 day bismuth isotopes. The 10·25 min. lead isotope is positron emitting, an interesting result in an element of high atomic number. Absorption measurement have been made of the radiations emitted by many of these isotopes and assignments have been made in most cases.

In conclusion we wish to thank Dr N. Feather for valuable discussions, and also for making for us a Ra E source. We are indebted to Dr Lewis for advice in setting up the thyratron scale of eight counter. This paper would be incomplete without a sincere acknowledgement of our indebtedness to the hard work of past members of this laboratory who have been mainly responsible for setting up the cyclotron.

One of us (R. S. K.) is grateful to the Royal Commissioners for the Exhibition of 1851 for the grant of an overseas scholarship which made this work possible.
The Fixed Part of the Canonical System
Authors: Bronowski J.
Published: November 2008

On an algebraic surface f of order n in space of three dimensions, the canonical system | k | of curves is traced by all those surfaces π of order n − 4 which fulfil certain conditions of adjunction at the singularities of f: for example, which pass simply through the double curves and through the isolated tacnodes of f; which pass doubly through the isolated fourfold points of f; and so on. The assigned fixed points and curves of the adjoint surfaces ø at these singularities of f are not taken to be part of the canonical system | k |; but | k | may have unassigned fixed parts e. Three cases are usually (3) distinguished.
On average values of arithmetic functions
Authors: Fogels E.
Published: November 2008

The problem considered in this paper is that of finding the least possible h = h(x) such that a given arithmetic function a(n) should keep its average order in the interval x, x + h, i.e. that we have

and

as x → ∞.
Surface motion of sputtered particles
Authors: Roulston K. I.
Published: November 2008

The deposits formed by allowing sputtered material to fall on a glass plate, through a fine grating of wires placed close to the plate, are studied in detail. Evidence obtained showing the presence of material in the shadow of the wires and other less direct evidence gives strong support for the view that the sputtered material is able to move about on the surface. It is also shown that when the grating is very fine a considerable fraction of the incident material is not condensed.
A method of calculating the integral breadths of Debye-Scherrer lines: generalization to non-cubic crystals
Authors: Stokes A. R. , Wilson A. J. C.
Published: November 2008

Stokes and Wilson(1,2) have shown that the integral breadths of the Debye-Scherrer lines produced by small or imperfect crystals of the cubic system are given by

or, what is the same thing, that the apparent particle sizes are given by

In these equations λ is the X-ray wave-length, θ is the Bragg angle, Vt is the volume common to the crystal and its ‘ghost’ shifted a distance t in the hkl direction(1), and Jt is the mean value of the product FF of the structure amplitudes of two cells separated by a distance t in the hkl direction (2). The purpose of this note is to outline an argument which establishes the same result for crystals of any symmetry.
Quadratic and higher reciprocity of modular polynomials
Authors: Pocklington H. C.
Published: November 2008

1. We first compare the λ-ic characters of two polynomials with respect to each other and the prime p, the polynomials being irreducible with respect to p. This is done by relating each to the resultant of the polynomials. We then show in § 6 that the ordinary quadratic character of one prime with respect to another is also the resultant of two polynomials.
On the origin of the solar system
Authors: Hoyle F.
Published: November 2008

3. The essential requirement that the nova must satisfy in the above theory is that the total mass in the form of diffuse gaseous material must be of the order of 1/10 the solar mass, which requirement seems to be consistent with the observations of novae. Thus, although the theory applies explicitly to novae that break into two pieces of stellar mass having a gaseous filament drawn out between them (the pieces may have a mass ratio as great as 5/1), it seems clear that the discussion could be adjusted to include the case of novae following other models.

The process described above can be applied to the formation of non-solar planets. The mass of available planetary material depends upon such factors as the mass of the nova and its separation from the companion before outburst. It is to be expected that variations in these factors can lead to planetary masses that vary over a fairly wide range. It follows that since an appreciable fraction of the stars of mass comparable or greater than the solar mass are known to be members of binary systems, the number of planetary systems must be at least of the same order as the number of novae that have occurred in this class of star.
Some further results concerning the integrals of moduli of regular functions along curves of certain types
Authors: Garriel R. M.
Published: November 2008

In a paper* of some time back I gave the following theorem:

If f(z) is a regular function of z for z inside and on a circle Τ, and C is any convex curve inside Τ, and λ>0, then
Some Quartic Primals and Associated Cremona Transformations of Four-Dimensional Space
Authors: Babbage D. W.
Published: November 2008

The object of this paper is to draw attention to a series of five types of rational quartic primal in [4]. Two of these are already known. The greater part of this paper deals with one of the other three types of primal and with a new symmetrical quarto-quartic Cremona transformation of [4] determined by a homaloidal system of primals of this type passing through a certain surface of order nine. It is hoped in a subsequent paper to continue the investigation into the remaining two types of primal.
Adsorption Isotherms. Critical Conditions
Authors: Fowler R. H.
Published: November 2008

1. In a recent note I showed that Langmuir's adsorption isotherm

where θ is the fraction of the surface covered by adsorbed gas, p the gas pressure in equilibrium with it, and A (T) a specified function of the temperature, can be derived as a theorem in statistical mechanics without any appeal to the mechanism of deposition and re-evaporation. Necessary and sufficient assumptions for the truth of (1) are that the atoms (or molecules) of the gas are adsorbed as wholes on to definite points of attachment on the surface of the adsorber, that each point of attachment can accommodate one and only one adsorbed atom, and that the energies of the states of any adsorbed atom are independent of the presence or absence of other adsorbed atoms on neighbouring points of attachment. Under these assumptions the explicit form of (1) is

where m is the mass of the adsorbed atom or molecule, bg(T) the partition function for its internal states in the gas phase, and vs(T) the partition function for its set of adsorbed states. These sets of states are to be so specified that the energy zero is assigned tot the lowest state of each set in constructing bg(T) and vs(T), and then X is the energy required to transfer a molecule from the lowest adsorbed state tot the lowest gas state. Quite another adsorption isotherm was shown to hold when adsorption of a molecule takes place as atoms and requires two or more points of attachment.
A theorem on algebraic correspondences
Authors: Hodge W. V. D.
Published: November 2008

In his chapter on correspondences between algebraic curves Prof. Baker has raised a problem concerning the possibility, when we are given the equations of Hurwitz for a correspondence between two algebraic curves, of obtaining therefrom a reduction of the everywhere finite integrals on either curve into complementary regular defective systems of integrals. The problem is stated as an unproved theorem, an exact formulation of which is given below. The object of the present note is to give a proof of this theorem on the lines of Prof. Baker's chapter.
On Ising's model of ferromagnetism
Authors: Peierls R.
Published: November 2008

Ising discussed the following model of a ferromagnetic body: Assume N elementary magnets of moment μ to be arranged in a regular lattice; each of them is supposed to have only two possible orientations, which we call positive and negative. Assume further that there is an interaction energy U for each pair of neighbouring magnets of opposite direction. Further, there is an external magnetic field of magnitude H such as to produce an additional energy of − μH (+ μH) for each magnet with positive (negative) direction.
The resolution of monoidal Cremona transformations of three-dimensional space
Authors: Babbage D. W.
Published: November 2008

A Cremona transformation Tn, n′ between two three-dimensional spaces is said to be monoidal if the surfaces of order n in one space which form the homaloidal system corresponding to the planes of the second space have a fixed (n − 1)-ple point O. If the surfaces of order n′ forming the homaloidal system in the second space have a fixed (n′ − 1)-ple point O′, the transformation is said to be bimonoidal. A particularly simple bimonoidal transformation is that which transforms lines through O into lines through O′, and planes through O into planes through O′. Such a transformation we shall call an M-transformation. Its equations can, by suitable choice of coordinates, be expressed in the form

where φn−1(x, y, z, w) = 0, φn(x, y, z, w) = 0 are monoids with vertex (0, 0, 0, 1).
Thermal equilibrium at temperatures below 1° absolute
Authors: Shire E. S. , Allen J. F.
Published: November 2008

By measuring the resistance of a phosphor bronze wire in thermal equilibrium via various substances with crystals of iron ammonium alum it is shown that the time for thermal equilibrium between the ionic magnets of the salt and its lattice vibrations is less than 0·5 sec. for all temperatures above 0·025° T*. When liquid helium or a german silver tube forms part of the cooled portion of the apparatus, the time for equilibrium is increased to a few seconds for temperatures below 0·4° K. It appears possible that the thermal conductivity of german silver is less than 10−8 cal. cm.−1 sec.−1 degree−1 below 0·05° T*, and there are indications that the thermal conductivity of liquid helium at temperatures below 0·3° K. is small compared with its value at 2° K.
The charge-dependence of nuclear forces
Authors: Kemmer N.
Published: November 2008

The theory of heavy electrons recently developed by several authors may be considered to give a satisfactory account of the empirically known neutron-proton interaction. However, it now seems well established that there exists a proton-proton interaction of comparable magnitude which is not accounted for equally well. Owing to the fact that the emission of a heavy electron involves the change of a neutron into a proton or vice versa, the first approximation of this theory gives only an exchange force between unlike particles, whereas a force between like particles must be due to double transitions and thus only appears in the second approximation. It is true that the expansion in terms of the number of particles emitted is actually so badly convergent that the second order proton-proton force at distances of about 10−13 cm. is found to be not essentially smaller than the first order neutron-proton force (see FHK), but nevertheless this does not explain experimental facts, since the calculated second order force is always repulsive.
Variation of ionization with range of α-particles, protons, deuterons and 3H particles
Authors: Bower J. C.
Published: November 2008

Photometrical determinations of the variation of density along photographs of expansion champer tracks have enabled the positions of maximum ionizing efficiency for single α-particles, protons, deuterons, and 3H particles to be determinated.

The maximum rate of ionization for an α-particle in air occure when it has a residual range of 3·7 mm. in standard air.

The distance of the position of maximum ionizing efficiency from the end of the particle range is shown to be approximately in the ratio of the particle mass in the case of the particles 1H, 2H, and 3H.

These experiments were carried out in the Cavendish High Tension Laboratory.
A note on the melting-point of paraffins and fatty acids
Authors: Moullin E. B.
Published: November 2008

The purpose of this note is to draw attention to a certain correspondence between the melting-points of normal paraffins and of fatty acids and to indicate a simple interpretation of this phenomenon. If the number of carbon atoms in a normal paraffin is plotted against the corresponding melting temperature, all the points in the diagram lie very close to a smooth curve drawn among them. If a similar diagram is made for the fatty acids, the points corresponding to an even number of carbon atoms lie on a curve of the same character as that found for paraffins. The points for an odd number of carbon atoms lie on a separate but similar curve, exemplifying the well-known alternation property.
On Euclid's algorithm in real quadratic fields
Authors: Heilbronn H.
Published: November 2008

The object of this paper is to complete the proof of the

Theorem.LetP(√d)be the quadratic field of discriminant d> 0.Then Euclid's algorithm does not hold inP(√d)if d is sufficiently large.
Uniqueness theorems and the maximum-minimum principle for a type of non-linear partial differential equations
Authors: Fuchs W. H. J. , Weiss P.
Published: November 2008

It is well known that solutions of partial linear differential equations of the second order and of elliptic type are uniquely determined by their boundary data, and that they assume their maximum and minimum values on the boundary. The usual proofs make use of the principle of superposition and are therefore not applicable to non-linear problems. But recently Pryce has proved the uniqueness theorem for the non-linear equations of minimal surfaces and of Born's electrostatics. These equations are the Euler equations of the variational problem

k = + 1 corresponds to the case of minimal surfaces in n + 1 dimensions; k = − b−2, n = 3 corresponds to Born's electrostatics. Pryce's procedure depends essentially on the notion of conjugate variables in the calculus of variations for multiple integrals and can therefore be extended to a wide class of differential equations arising from variational problems (for several functions of several variables) as we show in § 3.
On subordinate functions
Authors: Rogosinski Werner
Published: November 2008

The so-called Lindelöf principle is nothing more than a transformation and systematic application of the simple Schwarz lemma. Nevertheless, it is a very powerful tool for the solution of many questions in the theory of functions. It is based on the conception of “subordinate” functions.
The fine structure of the X-ray absorption edges of aluminium and zinc
Authors: Johnston J. E.
Published: November 2008

The fine structure of the Al L23 and Zn M23-absorption edges has been determined using the experimental methods described in an earlier paper(1).

A table containing numerical data for the Al and Zn edges, together with revised data for the edges of Li, Mg, Ni and Cu is included.
On the easier Waring problem for powers of primes. II
Authors: Erdös P.
Published: November 2008

In a previous paper I proved that the density of the positive integers of the form where the letters p, q, and later P, Q, r, denote primes, is positive. As indicated in the Introduction of I, I now give proofs of the following results:

The density of each of the sets of integers

is positive.
A note on tests of significance in multivariate analysis
Authors: Bartlett M. S.
Published: November 2008

Multivariate generalizations. In multivariate statistical analysis, common terms such as variances and correlation coefficients have received certain generalizations. Wilks (7) has called the determinant |V|, where V is the matrix of variances and covariances between several variates, a generalized variance; certain ratios of such determinants have been called by Hotelling(5) vector correlation coefficients and vector alienation coefficients. While these determinantal functions have properties which justify to some extent this kind of generalization, it sometimes seems more reasonable to leave any generalized parameters, or corresponding sample statistics, in the form of matrices of elementary quantities. This is stressed by the formal analogy which then often exists between the generalized and the elementary formulae.
A new notation for quantum mechanics
Authors: Dirac P. A. M.
Published: November 2008

In mathematical theories the question of notation, while not of primary importance, is yet worthy of careful consideration, since a good notation can be of great value in helping the development of a theory, by making it easy to write down those quantities or combinations of quantities that are important, and difficult or impossible to write down those that are unimportant. The summation convention in tensor analysis is an example, illustrating how specially appropriate a notation can be.
On a Hardy-Littlewood problem of diophantine approximation
Authors: Spencer D. C.
Published: November 2008

1. Let .

When r is a positive integer, various writers have considered sums of the form

where ω1 and ω2 are two positive numbers whose ratio θ = ω1/ω2 is irrational and ξ is a real number satisfying 0 ≤ ξ < ω1. In particular, Hardy and Littlewood (2,3,4), Ostrowski(9), Hecke(6), Behnke(1), and Khintchine(7) have given best possible approximations for sums of this type for various classes of irrational numbers. Most writers have confined themselves to the case r = 1, in which
On the theory of the liquid state
Authors: Fürth R.
Published: November 2008

The theory of holes in liquids, suggested in a previous paper, is developed by means of classical statistical mechanics, and it is shown that the principal thermodynamic properties of the liquid state can be derived in this way and that they are in numerical agreement with the experiments.
On the determination of intermolecular energies by inductive analysis
Authors: Orr William J. C.
Published: November 2008

For a direct comparison of the individual attractive and repulsive terms of an intermolecular potential determined by the inductive analysis of themodynamic data with the same terms calculated by quantal methods it is desirable to carry out the analyses, in the first approximation, with an intermolecular potential of the form ø(R) = Pe−aR − A1/R6 − A2/R8. For mathematical convenience, in place of the above expression, two potential functions,

and

are considered, the first being taken to be adequate in the range of values of R between 0 and R0 (the minimum of the potential function) and the second, in the range from R0 to ∞. By dividing the problem in this way it is possible to find substitutions which permit the integration of the classical expression for the second virial coefficients (and other appropriate thermodynamic data) directly in terms of fairly simple series in | ψ0 |, R0, a and r. Finally it is pointed out that for such simple atoms or molecules as the rare gases, oxygen, nitrogen and methane r may be taken as 0·15 throughout, which considerably simplifies the application of the method to the experimental data.
Note on an anomaly in the spectrum of O++
Authors: Jeffreys Bertha
Published: November 2008

There are two main anomalies in the spectrum of O++, for which so far no satisfactory explanation has been given: (1) the calculated values for the configuration (1s)2 (2s)2 (2p)2 of the ratio of the intermultiplet separations (1D−1S)/(3P−1D) is not in agreement with the observed value, (2) within the 3P term the ratio of the separations indicates a departure from Russell-Saunders coupling. We are here concerned with (1).
Momentum distribution in molecular systems. Part VI. Shape of the Compton line for methane CH4
Authors: Coulson C. A. , Duncanson W. E.
Published: November 2008

Calculations have been made of the momentum distribution and the shape of the Compton profile for gaseous methane, using the Dirac transformation theory and three distinct approximations (molecular orbital, electron-pair and self-consistent-field) to the true molecular wave functions. The bearing of the results upon the validity of the original wave functions is discussed; it is concluded that the exponents in molecular wave functions should be approximately 1·1 times greater than in corresponding atomic wave functions. Comparison is made with experimental electron scattering results.

Our thanks are due to Prof. Massey for providing us with the numerical tables of the self-consistent-field for methane in advance of publication, and to Mr E. G: Phillips of the Mathematics Department of Univeristy College of North Wales, for the loan of a calculating machine.
The elastic stresses produced by the indentation of the plane surface of a semi-infinite elastic solid by a rigid punch
Authors: Harding J. W. , Sneddon I. N.
Published: November 2008

During the course of some investigations on the distribution of stress in an elastic solid it was noticed by the senior author that a systematic application of the method of integral transforms to the problem of the indentation of the plane surface of a semi-infinite elastic solid by a rigid punch reduced the problem essentially to one of solving a pair of integral equations belonging to a class which has been studied by Titchmarsh and by Busbridge (4,5). This procedure allows one to obtain the solution for an arbitrary shape of punch by a general method which leads automatically to the solution and avoids the troublesome procedure, adopted by Love (2) in the case of a conical punch, of being obliged to guess appropriate combinations of solutions which will satisfy the prescribed boundary conditions in any special case. Moreover, it can easily be seen that an attempt to apply Love's method to more complicated shapes of punch will lead to considerable analytical difficulties.
Expansions of coordinates of points of a plane curve in terms of s or Ψ
Authors: Piaggio H. T. H.
Published: November 2008

If O is an ordinary point of a plane curve and the tangent and inward normal at O are taken as axes, then, under certain conditions, the coordinates x and y of a neighbouring point P on the curve can be expanded in powers of s, the arc OP, or of Ψ, the angle between the tangents at O and P. Lamb(3) gives these expansions as far as the terms in s4 and Ψ3. Dockeray(1) goes as far as the terms in s6, but there seem to be three errors in his results. No one, I believe, has given the general terms. I obtain these for both pairs of expansions. Fowler(2) gives the term in s2r+1 in the expansion of x in the special case where, owing to the vanishing of the curvature and of its first r − 2 derivatives, the expansion of y begins with the term in sr+1. I shall give reasons for disagreeing with this result.
Boussinesq's problem for a flat-ended cylinder
Authors: Sneddon Ian N.
Published: November 2008

1. In a recent paper(1) expressions were found for the elastic stresses produced in a semi-infinite elastic medium when its boundary is deformed by the pressure against it of a perfectly rigid body. In deriving the solution of this problem—the ‘Boussinesq’ problem—it was assumed that the normal displacement of a point within the area of contact between the elastic medium and the rigid body is prescribed and that the distribution of pressure over that area is determined subsequently. The solutions for the special cases in which the free surface was indented by a cone, a sphere and a flat-ended cylindrical punch were derived, but no attempt was made to give a full account of the distribution of stress in the interior of the medium in any of these cases.
Analysis of operation of a thermostat with contact thermo-regulator
Authors: Jelonek Z.
Published: November 2008

Few papers presenting a theory of operation of thermostats can be found. This may be explained by mathematical difficulties which are met when considering the rather complicated construction, from the thermal point of view, of most thermostats. There is, however, a relatively great number of papers describing various mechanical constructions of thermostats. In many cases designers obtain very good practical results by making use of some approximate theories only.
Derivation of Dirac's equation for a free particle
Authors: Kai-Chia Cheng
Published: November 2008

If we assign to the relativistic energy of a free particle H the meaning of an energy operator, it can be shown that the phenomenon of ‘Zitter-Bewegung’ (and hence Dirac's equation) is obtainable from quantum mechanics alone.
The gravitational field of a rotating cohesive system
Authors: Clark G. L.
Published: November 2008

In this paper a periodic solution of the gravitational equations is examined. The solution, which is valid throughout all space, is such that the internal and external fields and their derivatives are continuous at the boundary of a spheroid rotating about an axis other than that of symmetry. At great distances from the system the solution has the same form as the field due to a rotating rod and can be applied to problems like the determination of the velocity of propagation of gravitational waves and the loss of energy of a rotating cohesive system.
A variational method for determining eigenvalues of the wave equation applied to tropospheric refraction
Authors: MacFarlane G. G.
Published: November 2008

A simple and direct variational method is described for finding both complex and real eigenvalues of the wave equation of anomalous propagation in a horizontally stratified atmosphere. It may be looked upon as an extension of Rayleigh's method to complex eigenvalues. In this paper it is illustrated by an example, taken from the duct theory of super-refraction, in which the refractive index of the air varies with height according to a power law. Numerical agreement in the values for the lowest order eigenvalues with those obtained by the differential analyser is better than ½%.
Minimum variance and the estimation of several parameters
Authors: Rao C. Radhakrishna
Published: November 2008

With the help of certain inequalities concerning the elements of the dispersion matrix of a set of statistics, and of the information matrix, the following results have been proved. Some of these inequalities are extensions of results given by Fisher (1) in the case of a single parameter.

(i) Efficient statistics are explicit functions of the minimal set of sufficient statistics.

(ii) Functions of the minimal set of sufficient statistics, satisfying the property of uniqueness defined in the text, are best unbiased estimates. Under certain conditions estimates possessing exactly the minimum possible variance can be obtained by the method of maximum likelihood.

(iii) In large samples maximum likelihood estimates supply efficient statistics in the case of several parameters.

(iv) The importance of replacing the sample by an exhaustive set of sufficient statistics (referred to in this paper as the minimal set) as a first step in any methodological problem has been stressed by R. A. Fisher in various articles and lectures. The above discussion supplies a formal demonstration of this view so far as the problem of estimation is concerned.
Random associations on a lattice
Authors: Moran P. A. P.
Published: November 2008

Wishart and Hirschfeld have considered the following problem (1). Given n points in order on a line, suppose that they may be ‘black’ or ‘white’ independently with probability p and q = 1 − p. What is the probability distribution of the number of black-white joins? They give an exact solution to this problem and prove that it tends to normality as n increases. This result is of interest in several branches of science (Mood (2)). In the present note we consider the analogous problem in more than one dimension. This has applications in physics and agriculture (compare, Ising (3), Cochran (4) and Todd (5)). Another problem of similar type occurs when we consider the number of black points to be fixed and their arrangement to be random, but we do not consider this here. A problem of arrangements of similar type has been considered by Onsager.
The distribution of projected areas of fragments
Authors: Walters A. G.
Published: November 2008

The drag on a fragment moving through the air at supersonic velocity tends to be proportional to the projected area of the fragment perpendicular to the line of flight and to be independent of the Reynolds number, depending only on the Mach number. Also the minimum velocity at which the fragment perforates a target is a function of the projected area at the moment of strike. Hence the probability that a fragment, with a given initial velocity, perforates the target at a given distance, depends on the mode of rotation and the distribution of projected areas of the fragment. It is customary to assume that the fragment rotates randomly so that we require to know the distribution of the projected areas when all orientations are equally likely. The distributions for fragments in the form of cylinders and rectangular parallelepipeds are derived in this paper.
An equation of quantum mechanics
Authors: Littlewood D. E.
Published: November 2008

The complete set of matrix representations of a set of n quantities β1, β2,…,βn satisfying

is obtained for all values of n. It is found that if n = 2v or n = 2v + 1, there are v + 1 irreducible representations with the respective degrees 1, but if n = 2v + 1 there are in addition two conjugate representations of order , the symbol denoting the binomial coefficient. The explicit representations are given for n = 2, 3, 4, 5.
On the geometry of the wave equation
Authors: Watson A. G. D.
Published: November 2008

An account is given of the transformation of coordinates and of absolute axial frames in Euclidean space and Galilean space-time. The connexion with Eddington's group of E-numbers is shown. The geometrical properties of Dirac's wave equation and of analogous equations in three dimensions are discussed in terms of absolute axial frames.
Particles of half-odd integral spin
Authors: Le Couteur K. J.
Published: November 2008

In three papers (1, 2, 3) Bhabha has set up a new theory for relativistic particles of any spin, and has suggested that a particular wave equation of half-odd spin, classified as (, ½), may be applicable to the proton. It is therefore of interest to study this equation and to calculate results which could perhaps be compared with experiment; since in the non-relativistic limit the particle obeys Dirac's equation, it is sufficient to consider the behaviour at energies large compared to the rest mass. Here the important problem is the calculation of scattering cross-sections for cosmic-ray processes.
On the matrix equation (AX)n = ∥A∥I
Authors: Grimshaw M. E.
Published: November 2008

It was proved by J. Williamson (1) that when C is any circulant matrix of order n and Ω is a certain diagonal matrix then

where ∥ C ∥ is the determinant of C and I is the unit matrix of order n. A simpler proof was given by U. Wegner (2), and later the characteristic equation of ΩC was discussed independently by L. Toscano (3). In the present note we prove the more general theorem that there corresponds to any matrix A of order n with simple elementary divisors a solution X of the equation (AX)n = ∥ A ∥ I. The theorem has a simple geometrical interpretation; the proof is almost immediate.
On the dynamical theory of the rotation of the earth
Authors: Bondi H. , Lyttleton R. A.
Published: November 2008

In the dynamical theory of the motion of the Earth relative to its centre of mass, the planet is usually regarded as a rigid or at most only slightly deformable body, and moments of inertia are adopted that are taken to refer to the Earth as a whole, while the motion itself at any instant is assumed capable of representation by a single angular velocity vector. This procedure, however, appears to involve unwarranted assumptions the recognition and removal of which may lead to conclusions of considerable importance. For it is well known from the theory of earthquake waves that the material of the central core of the Earth behaves like a liquid in that it transmits only longitudinal wave vibrations, while there is also other evidence suggesting that the material of the core is a true liquid (1). There is accordingly no a priori reason for supposing that the core will behave like a rigid body firmly attached to the surrounding shell if more or less permanent shearing forces are applied to it. In particular, in respect of any couple known to act on the outer shell, it is not permissible to assume, without examination of the assumption, that its effect will be transferred immediately to the inner core in a way preserving rigid-body rotation of the whole. If the material of the core behaves like a liquid where wave-motion is concerned, this suggests that it will probably also behave like a liquid whatever shearing forces act on it, and the extent to which changes in the rotatory motion of the outer shell can be communicated to the core, and what effects direct gravitational forces acting on the core may have, must in the first instance be questions of hydrodynamics and not rigid dynamics.
Boussinesq's problem for a rigid cone
Authors: Sneddon Ian N.
Published: November 2008

1. The problem of determining the distribution of stress in a semi-infinite solid medium when its plane boundary is deformed by the pressure against it of a perfectly rigid cone is of considerable importance in various branches of applied mechanics. It arises in soil mechanics where the cone is the base of a conical-headed cylindrical pillar and the semi-infinite medium is the soil upon which it rests (1). In this instance the distribution of stress in the soil is known to be more or less similar to that calculated on the assumption that the soil is a perfectly elastic, isotropic and homogeneous medium, at least if the factor of safety of a mass of soil with respect to failure by plastic flow exceeds a value of about three (2). The same problem occurs in the theory of indentation tests in which a ductile material is indected by cylindrical punches with conical heads (3).
Sufficient statistics and minimum variance estimates
Authors: Rao C. Radhakrishna
Published: November 2008

Let the probability density of observations be denoted by φ(x | θ), where x stands for the variables and θ for the parameters. A function t of the observations is called an unbiased estimate of the function ψ(θ) of the parameters if

where dx stands for the product of differentials.
Some new methods for the numerical integration of ordinary differential equations
Authors: Fox L. , Goodwin E. T.
Published: November 2008

The choice of a numerical method for the solution of ordinary differential equations depends on the associated boundary conditions. When all the boundary conditions are specified at one end of the range of integration, one of the well-known step-by-step methods will generally be used, while the method of relaxation is reserved for the case in which boundary conditions are specified at more than one point (1). In the latter, simple but inaccurate finite-difference formulae are used to provide a first approximation to the required solution; this approximation is then used to give an estimate of the errors involved in the use of the inaccurate formulae, and successive corrections are obtained until the full, accurate finite-difference equations are satisfied (1). The same principle is followed in this paper with regard to step-by-step methods, the main difference being the way in which approximate solutions are obtained. In relaxation methods simultaneous equations are solved, while in the use of the step-by-step methods suggested here successive pivotal values are built up by the use of recurrence relations. All the methods of this paper follow this principle, differing only in the method of obtaining a recurrence relation, and consequently in the form of the correction terms.