Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1952"

On Sound Generated Aerodynamically. I. General Theory

Abstract: A theory is initiated, based on the equations of motion of a gas, for the purpose of estimating the sound radiated from a fluid flow, with rigid boundaries, which as a result of instability contains regular fluctuations or turbulence. The sound field is that which would be produced by a static distribution of acoustic quadrupoles whose instantaneous strength per unit volume is ρv i v j + p ij - a 2 0 ρ δ ij , where ρ is the density, v i the velocity vector, p ij the compressive stress tensor, and a 0 the velocity of sound outside the flow. This quadrupole strength density may be approximated in many cases as ρ 0 v i v j . The radiation field is deduced by means of retarded potential solutions. In it, the intensity depends crucially on the frequency as well as on the strength of the quadrupoles, and as a result increases in proportion to a high power, near the eighth, of a typical velocity U in the flow. Physically, the mechanism of conversion of energy from kinetic to acoustic is based on fluctuations in the flow of momentum across fixed surfaces, and it is explained in § 2 how this accounts both for the relative inefficiency of the process and for the increase of efficiency with U . It is shown in § 7 how the efficiency is also increased, particularly for the sound emitted forwards, in the case of fluctuations convected at a not negligible Mach number.

Anomalous Paramagnetism of Copper Acetate

Abstract: The paramagnetic resonance spectrum of copper acetate is anomalous in that it resembles that of an ion of spin 1, and its intensity decreases as the temperature is lowered. The latter is correlated with the decreasing susceptibility found by Guha (1951). The following hypo­theses are suggested: (1) the crystalline field acting on each copper ion is similar to that in other salts such as the Tutton salts; (2) isolated pairs of copper ions interact strongly through exchange forces, each pair forming a lower singlet state and an upper triplet state, the latter only being paramagnetic. On this basis both the fine structure and the hyperfine structure of the spectrum have a simple explanation, and the theory also predicts a small initial split­ting of the triplet state of the same order as that found experimentally. The unit cell of the crystal contains two differently oriented pairs of ions, and, using an empirical value for the exchange parameter, fair agreement with the susceptibility measurements of Guha is obtained.

Supercooling of Liquids

Abstract: I shall concentrate upon reviewing the important recent change in our appreciation of the facts of supercooling which has been brought about particularly by the work of Turnbull at the General Electric Research Laboratory in Schenectady. I suppose that most of us, talking about supercooling a couple of years ago, would have divided substances into two classes, one with simple crystal structures like gold, and all the other ‘good’ metals on the one hand, and those with complex crystal structures, such as glycerol and the silicates on the other; saying that whereas the latter class can be very much supercooled, and will form glasses, the former class can only be supercooled a very few degrees. Then we would have added that there are some ‘ bad ’ metals, with moderately complex crystal structures, such as antimony or bismuth, which can be supercooled some tens of degrees, forming an intermediate class. I think we would then have added that this is quite comprehensible. In particular, that the X-ray diffraction patterns of the monatomic liquids show us that most of the atoms have the right numbers of nearest neighbours in a first co-ordination shell, all ready in place to start the growth of a crystal; which readily explains why these substances cannot be supercooled very much—a nice simple experimental fact, with a straightforward theoretical interpretation—and both are wrong.

Analysis of the Swimming of Long and Narrow Animals

Abstract: The swimming of long animals like snakes, eels and marine worms is idealized by considering the equilibrium of a flexible cylinder immersed in water when waves of bending of constant amplitude travel down it at constant speed. The force of each element of the cylinder is assumed to be the same as that which would act on a corresponding element of a long straight cylinder moving at the same speed and inclination to the direction of motion. Relevant aerodynamic data for smooth cylinders are first generalized to make them applicable over a wide range of speed and cylinder diameter. The formulae so obtained are applied to the idealized animal and a connexion established between B/$\lambda $, V / U and R$\_{1}$. Here B and $\lambda $ are the amplitude and wave-length, V the velocity attained when the wave is propagated with velocity U, R$\_{1}$ is the Reynolds number Ud$\rho $/$\mu $, where d is the diameter of the cylinder, $\rho $ and $\mu $ are the density and viscosity of water. The results of calculation are compared with James Gray's photographs of a swimming snake and a leech. The amplitude of the waves which produce the greatest forward speed for a given output of energy is calculated and found, in the case of the snake, to be very close to that revealed by photographs. Similar calculations using force formulae applicable to rough cylinders yield results which differ from those for smooth ones in that when the roughness is sufficiently great and has a certain directional character propulsion can be achieved by a wave of bending which is propagated forward instead of backward. Gray's photographs of a marine worm show that this remarkable method of propulsion does in fact occur in the animal world.

Edge Waves on a Sloping Beach

Abstract: The set of eigenfrequencies of a mechanical system forms its spectrum. A discussion is given of systems with discrete, continuous and mixed spectra. It is shown that resonance occurs at discrete points of the spectrum, and at cut-off frequencies (end-points of the continuous spectrum). The motion in a semi-infinite canal of finite width closed by a sloping beach has a mixed spectrum. The inviscid theory predicts that at a discrete frequency the resonance is confined to the neighbourhood of the beach (inviscid edge wave), while at a cutoff frequency the resonance extends a long way down the canal. The latter resonance is confined to the neighbourhood of the beach (viscous edge wave) by viscosity which is important near a cut-off frequency. Especially large resonances are predicted for a series of critical angles, of which the largest is 30°. The theory is verified experimentally in the frequency range 100 to 17c/min for the angles 37⋅6 and 29⋅5°.

The effect of homogeneous turbulence on material lines and surfaces

Abstract: A material line consisting always of the same particles of fluid is lengthened by the convective action of turbulent motion, at a rate which is shown to be exponential at a sufficient time after the initial choice of the line. The area of material surface elements also increases exponentially, although with a different exponential coefficient. It is then a consequence of conservation of mass of the fluid that the normal distance between two neighbouring parallel material surfaces decreases at an exponential rate which is simply related to the rate at which material lines lengthen. These kinematical results are relevant to the convective action of the turbulence on the distribution of two kinds of local property of the fluid, viz. quantities, represented by Ɵ , of which the total amount in a material volume of the fluid remains constant (e. g. mass density of a foreign substance), and quantities, represented by F, of which the total flux across a material surface remains constant (e. g. vorticity). F is proportional, at all times, to the vector representing a material line element and G, = ∇ Ɵ , is proportional to the vector representing a material surface element, and hence they both increase exponentially in time. The last two sections of the paper are concerned with the combined effect on F and G of convection and molecular diffusion, and a condition for the effect of the convection to be dominant is obtained. In the case of F, convection is found to be dominant if the appropriate molecular diffusivity is small compared with the kinematic viscosity of the fluid, and F 2 will then increase indefinitely, unless some additional effect comes into play (as it does in the case of magnetic field strength). In the case of G, convection will be dominant if the initial distribution of G has a large enough length scale; the effect of the convection is to decrease this length scale so that ultimately the two effects reach an equilibrium. The length scale of the ultimate distribution of G, as obtained herein, is different from that put forward recently by Obukhoff.

Interaction of electrons with lattice vibrations

Abstract: Using methods of modern field theories a canonical transformation of the Hamiltonian of free electrons in the field of the lattice vibrations is performed. This transformation takes account of the bulk of the interaction of the electrons with the vibrational field and leads to a renormalization of the velocity of sound and of the interaction parameter F . An objection of Wentzel’s against the use of large F is removed in this way. Even in the case of weak interaction the transformed Hamiltonian contains already in zero order terms which require a modification of the usual procedure in the theory of metals, and which at low temperatures lead to an increase of the effective mass of the electrons. Treatment of strong interaction requires the development of a new method.

The Action of Waving Cylindrical Tails in Propelling Microscopic Organisms

Abstract: The action of the tail of a spermatozoon is discussed from the hydrodynamical point of view. The tail is assumed to be a flexible cylinder which is distorted by waves of lateral displacement propagated along its length. The resulting stress and motion in the surrounding fluid is analyzed mathematically. Waves propagated backwards along the tail give rise to a forward motion with velocity proportional to the square of the ratio of the amplitude of the waves to their length. The rate at which energy must be supplied to maintain the waves against the reaction of the surrounding fluid is calculated. Similar calculations for the case when waves of lateral displacement are propagated as spirals show that the body is propelled at twice the speed given it by waves of the same amplitude when the motion is confined to an axial plane. An externally applied torque is necessary to prevent the reaction of the fluid due to spiral waves from causing the cylinder to rotate. This is remarkable because the cylinder itself does not rotate. A working model of a spermatozoon was made in which spiral waves could travel down a thin rubber tube without rotating it. The torque just referred to was observed and was balanced by an eccentric weight. The performance of the model while swimming freely in glycerine was compared with the calculations. The calculated speed of the model was higher than was observed, but this discrepancy could be accounted for by the fact that the model has a body containing its motive power while the calculations refer to a disembodied tail.

The Generation of Noise by Isotropic Turbulence

Abstract: A finite region, with fixed boundaries, of an infinite expanse of compressible fluid is in turbulent motion. This motion generates noise and radiates it into the surrounding fluid. The acoustic properties of the system are studied in the special case in which the turbulent region consists of decaying isotropic turbulence. It is assumed that the Reynolds number of the turbulence is large, and that the Mach number is small. The noise appears to be generated mainly by those eddies of the turbulence whose contribution to the rate of dissipation of kinetic energy by viscosity is negligible. It is shown that the intensity of sound at large distances from the turbulence is the same as that due to a volume distribution of simple acoustic sources occupying the turbulent region. In this analogy, the whole fluid is to be regarded as a stationary and uniform acoustic medium. The local value of the acoustic power output P per mass of turbulent fluid is given approximately by the formula P = ─3/2 α d u 2 ¯ /d t ( u 2 ¯ / c 2 ) 5/2 , where α is a numerical constant, u 2 ¯ is the mean-square velocity fluctuation, t is the time, and c is the velocity of sound in the fluid. The constant α is expressed in terms of the well-known velocity correlation function f(r) by assuming the joint probability distribution of the turbulent velocities and their first two time-derivatives at two points in space to be Gaussian. The numerical value α ~ 38 is then obtained by substituting the form of f(r) corresponding to Heisenberg’s theoretical spectrum of isotropic turbulence. It is found that the effects of decay make only a small contribution to the value of α, and that the order of magnitude of α is not changed when widely differing forms of the function f(r) are used.

The Commutation Laws of Relativistic Field Theory

Abstract: A definition of Poisson brackets is given which is related to the action principle, but does not require the introduction of canonical variables. This permits the laws for forming both the commutators of canonical theory and the anticommutators of Fermi-Dirac particles to be stated in a manifestly covariant way. Examples of the use of this method are given. The last section discusses tentatively the extension to the case of equations which cannot be written in canonical form.

The anomalous skin effect

Abstract: The anomalous skin effect arises in good conductors at low temperatures and high frequencies when the electronic mean free path becomes comparable with or greater than the classically calculated skin depth. Measurements have been made on a number of metals at frequencies of 1200 and 3600 Mc/s, and the form of variation of r.f. surface conductance with d.c. conductivity agrees well with that predicted theoretically by Reuter & Sondheimer, assuming that the electrons are scattered diffusely when they hit the surface of the metal. From the results, estimates are made of the effective value of $\sigma $/l, the ratio of d.c. conductivity to mean free path, and hence of the free surface area of the occupied region of k-space. The estimate for copper agrees well with that expected theoretically; those for silver and gold are rather lower than the theoretical values. For the other metals investigated, tin, cadmium, lead and aluminium, no theoretical estimates are available. The results are very sensitive to the presence of surface imperfections; the effect of these is discussed.

Some Magnetic Properties of Metals. II. The Influence of Collisions on the Magnetic Behaviour of Large Systems

Abstract: A discussion of the effect of collisions on the magnetic properties of a large system of free electrons shows that the non-periodic term in the susceptibility is hardly affected, but that the periodic terms are reduced in magnitude by a factor exp (- hp /┬βH ), where p is the harmonic considered, ┬ is the mean collision time, and β = eh /2π mc .

A New Radio Interferometer and Its Application to the Observation of Weak Radio Stars

Abstract: A new type of radio interferometer has been developed which has a number of important advantages over earlier systems. Its use enables the radiation from a weak ‘point’ source such as a radio star to be recorded independently of the radiation of much greater intensity from an extended source. It is therefore possible to use a very much greater recorder sensitivity than with earlier methods. It is, in addition, possible to use pre-amplifiers at the aerials, and the resolving power which may be used is therefore not restricted by attenuation in the aerial cables.

Diffraction by a semi-infinite metallic sheet

Abstract: In spite of the considerable attention which has been focused on diffraction by perfectly conducting structures, little success has so far been achieved when finite conductivity is introduced. It is now shown that with the assumption of suitable boundary conditions, the problem of diffraction at a metal sheet is capable of exact solution. Corresponding to each of two fundamental polarizations, a pair of Wiener-Hopf integral equations is derived from which to determine the electric and 'magnetic' currents present in the sheet. One of these equations is subjected to a rigorous solution, and from it the solutions of the other three are deduced by symmetry considerations. Use of the generalized method of steepest descent then serves to determine the diffracted fields. The case of a circularly polarized incident wave is also briefly discussed and a comparison presented between the theoretical and experimental forms of the scattered field; good agreement is obtained.

Some Magnetic Properties of Metals. I. General Introduction, and Properties of Large Systems of Electrons

Abstract: A general introduction surveying the problems to be examined in a series of papers is followed by a detailed treatment of the magnetic behaviour of a large system of electrons. The Schrodinger equation is solved on the assumption that the system is unbounded, and the modifications caused by the finite size of the system are then determined for the limiting case in which the system is much larger than the electronic orbits. An expression is then obtained for the density of states, and the free energy of the system found assuming that k T E 0 , where E 0 is the degeneracy parameter. The magnetic susceptibility, thermodynamic potential and specific heat are discussed for the two cases N constant and E 0 constant. Explicit formulae are given for the temperature-dependence of the field-independent term in the susceptibility. In the final section the corrections due to electron spin are introduced.

Studies in jj-Coupling. I. Classification of Nuclear and Atomic States

Abstract: Stimulated by the many successes of the recent jj-coupling shell-model of nuclear structure which is based upon the assumption of strong spin-orbit coupling, the present work aims at obtaining allowed terms and energies of systems of many equivalent particles in jj-coupling for various types of two-body interaction. In this first paper the methods of group theory are employed to enumerate and classify the states of the nuclear configuration (l$_{j}$)$^{n}$ Classification of states according to the irreducible representations of groups of linear transformations leads naturally to the introduction of certain quantum numbers which then characterize the states. Invariance under the group of unitary transformations in the space of the 2j+1 individual-particle functions introduces the isotopic spin, while invariance under the subgroup of symplectic transformations introduces the concepts of seniority and reduced isotopic spin. The complete classification and enumeration of the states of the nuclear configurations j$^{n}$, carried through for j = $\frac{3}{2}$, $\frac{5}{2}$ and $\frac{7}{2}$, is presented in table 6; the states of the corresponding atomic configurations are included, these being states of maximum isotopic spin. As a result of the classification, certain degeneracies observed in the energy matrix of the electrostatic interaction in jj-coupling are shown to be intimately connected with the properties of the symplectic groups. Finally, the spins of nuclear ground states are briefly discussed in the light of the jj-coupling shell-model.

Cross-linking of polythene by pile radiation

Abstract: Polythene subjected to irradiation in the Harwell B.E.P.O. pile becomes cross-linked, and a new type of plastic is produced which does not melt at about 115° C, nor dissolve in hot organic compounds. The mechanical properties are also altered, especially above 115° C, when the plastic shows rubber-like elasticity. The paper is mainly confined to a study of the relationship between the degree of cross-linking and the amount of incident radiation causing cross-linking. Possible mechanisms of cross-linking are briefly considered. Cross-linking is shown to arise primarily from the fracture of C H bonds, and the liberation of hydrogen. The weight changes Δ M of a specimen of weight M , and surface area A , subjected to radiation R is found to be represented by the equation Δ M = —α 1 MR +α 2 MR 2 +β 1 AR 2 —β 2 AR 2 . These terms are considered to arise from hydrogen evolution from the bulk of the polymer, methane, ethane, etc., evolution from near the surface, and surface oxidation. From the hydrogen-loss term α 1 the efficiency of cross-linking is deduced as 1 % of carbons cross-linked per unit radiation defined as 10 17 thermal neutrons/cm 2 , and the associated fast neutrons and gammas. Microchemical analysis reveals a reduction in H/C ratio with radiation. The cross-linking ratio deduced is 1.1 % of carbon cross-linked per unit radiation. The corresponding figure deduced from the amount of radiation required to render cross-linked polythene insoluble is 1.1 to 1.4 %. For paraffin wax the figure is 0.9 %. From the volume of gas evolved a value of 1.16% is obtained. The effects of different forms of radiation are considered. It is concluded that γ-radiation as well as fast and thermal neutrons are responsible. The energy required to break a C—H bond is found to be of the order of 25 eV. The value of this physical method of producing cross-linking in polythene and in other long-chain polymers under accurately controllable conditions, without the incorporation of other chemical compounds and without heat treatment, is discussed. Since the mechanism of polymerization is different, a range of new polymers can be envisaged, of which the physical properties can be studied as a function of the degree of cross-linking.

Dielectric dispersion in pure polar liquids at very high radio frequencies - III.The effect of electrolytes in solution

Abstract: A description is given of measurements at millimetre and centimetre wave-lengths of absorption in electrolytic solutions, up to concentrations of about 3 normal, with water and methyl alcohol as solvents. The observations are analyzed in terms of Debye’s basic theory of dispersion in a polar dielectric, and it is shown what modifications to the theory are necessary to take account of the ionic conductivity produced by an electrolyte. Measure­ments on aqueous solutions of sodium chloride are discussed in relation to Huckel’s theory of electrolytic solutions.

The correspondence between the resonance and molecular orbital theories

Abstract: The resonance theory of organic chemistry is critically examined from a theoretical view­point. It is stressed that this theory is not rigorously founded on the quantum-mechanical valence bond method, but involves additional assumptions which cannot be defended. The practical success of the resonance theory must therefore be explained in some other way. It is here shown that a remarkable correspondence exists between the resonance theory and the molecular orbital method; and it is suggested that the resonance theory owes its success more to this correspondence than to the validity of its own premises.

Dielectric dispersion in pure polar liquids at very high radio-frequencies I. Measurements on water, methyl and ethyl alcohols

Abstract: The results are given of some measurements of the absorption coefficient and refractive index of water, methyl and ethyl alcohols at wave-lengths of 6·2 mm, 1·24 cm and 3·21 cm over the temperature range —10 to 50°C, including some observations on water in the supercooled state. The method used is based upon the fact that the rate of attenuation of radio-frequency energy along a wave-guide filled with the liquid is dependent upon both the absorption coefficient and the refractive index when the guide is operated near to the cut-off condition. The electrical characteristics of water vary in a continuous manner through the normal freezing-point of 0°C down to at least —8°C. The measurements indicate that both alcohols, like water, have relatively high atomic polarizations.

The Diffraction of Galactic Radio Waves as a Method of Investigating the Irregular Structure of the Ionosphere

Abstract: The amplitudes of the radio waves received from a radio star at two points separated by about 1 km have been studied and compared. The results indicate that the variations of phase and amplitude at one point can be ascribed to the steady drift of an irregular wave-pattern over the ground. It is shown how the structure and movement of the wave-pattern can be deduced from the observations. The irregular wave-pattern across the ground can be thought of as a diffraction pattern produced by the passage of the waves through a portion of the ionosphere which imposes irregular changes of phase on it, and it is shown how the ionospheric characteristics can be deduced. The ionospheric irregularities are shown to have a lateral extent of the order 2 to 10 km, and a variation of electron content of about 5 $\times $ 10$^{9}$ electrons per cm.$^{2}$ It is deduced that the irregularities are at a height of about 400 km. They are most pronounced around midnight and exhibit little annual variation. The irregular portion of the ionosphere moves with a steady wind-like motion with a velocity of the order 100 to 300 m/s. The velocity decreases after midnight, and large velocities are associated with periods of magnetic disturbance.

The theory of the magnetic properties of rare earth salts: cerium ethyl sulphate

Abstract: The theory of a Ce$^{3+}$ ion in a crystal is considered as that of a free ion with a single 4f-electron outside closed shells, and the effect of the surrounding lattice is included as a static crystalline electric field. The problem is solved when this field has C$\_{3h}$ symmetry. It is possible that this theory will be applicable to many cerium salts-it is here applied to the ethyl sulphate for which fairly extensive experimental data are available. It is found that in order to obtain agreement with experiment it is essential to consider small terms of C$\_{3v}$ symmetry, the presence of which is not incompatible with the crystal structure. It is then possible to determine values of the magnitudes of the crystal fields and to fit the magnetic resonance data accurately and the susceptibility data to within a small percentage.

Dielectric dispersion in pure polar liquids at very high radio-frequencies II. Relation of experimental results to theory

Abstract: The results of the measurements at centimetre and millimetre wave-lengths on the dielectric properties of water, methyl alcohol and ethyl alcohol described in part I are analyzed. There is no evidence that, for any of these liquids, more than a single relaxation time as a function of temperature is required to account for the dispersion arising from dipole rotation. It is suggested that the observed behaviour of the two alcohols at wave-lengths near to 1 cm, which appears not to conform with the hypothesis of a single relaxation time, is in fact the consequence of resonance absorption. If it is supposed that in dipole rotation and viscous flow the molecules have to surmount potential energy barriers, then it appears that, in each of the liquids examined, the heights of the barriers concerned in the two processes are identical; but the ‘frequency factor’ associated with such processes is much larger for viscous flow than for dipole rotation.

The dielectric estimation of protein hydration

Abstract: Microwave dielectric measurements on six aqueous protein solutions are analyzed in terms of water ‘irrotationally bound’ to the protein, i. e. unable to rotate in the high-frequency field. Estimates are made of the maximum amount of bound water, i. e. water carried through the solution with the protein, as a function of axial ratio, assuming a spheroid molecule. Taking hydration estimates in conjunction with other data, we conclude inter alia: (1) That the β -lactoglobulin and egg-albumin molecules approximate to prolate rather than to oblate spheroids. (2) That a large proportion of the charged groups in horse methaemoglobin ‘protrude’ from the surface of the molecule.

Inelastic collisions between ions and atoms

Abstract: Charge-exchange cross-sections for H$^{+}$, D$^{+}$, O$\_{2}^{+}$, H$\_{2}^{+}$, O$^{+}$, CO$^{+}$ and N$\_{2}^{+}$ in A; D$^{+}$, O$^{+}$ and N$^{+}$ in Kr; D$^{+}$, C$^{+}$ and Br$^{+}$ in Xe, and O$^{+}$ in H$\_{2}$O have been measured between 25 and 4000 eV energy by a method previously described. The normal atomic collision cross-sections rise to a maximum at a voltage which depends on the value of $\Delta $E $\surd $M for the process, M being the atomic mass and $\Delta $E the energy defect of the reaction. Collisions between negative ions and atoms have been studied with the same apparatus, the cross-section of the detachment reaction X$^{-}$ + Y $\rightarrow $ X + e + Y - $\Delta $E being obtained. For S$^{-}$, Br$^{-}$, I$^{-}$, C$^{-}$, P$^{-}$, Li$^{-}$ in Ne, and H$^{-}$ in He, Ne, A, Kr, Xe, this rises with increasing energy of the incident ion. For O$^{-}$ in He, Ne, A, Kr, Xe, Cl$^{-}$ in He, Ne, A, Kr, Xe, and F$^{-}$ in Ne, Kr, Xe the cross-section at low energies is unexpectedly large for the value of $\Delta $E $\surd $M. This may be interpreted as being due to the presence of excited states of these ions, of low electron affinities, in the beam. With O$^{-}$, a low energy electron bombardment source gave smaller cross-sections, i.e. a smaller proportion of excited ions.

The Theory of the Double Compton Effect

Abstract: Using the methods of Feynman, the exact expression is derived for the differential cross-section that two photons of given energy are emitted into given directions, as the result of an unpolarized $\gamma $-ray striking an electron at rest. The probability for this process contains an extra factor e$^{2}$/$\hslash $c compared with ordinary Compton scattering. There is great preference for one soft and one hard photon to be emitted, rather than two photons of approximately equal energy. The hard photon is most likely to be ejected into a narrow cone in the forward direction and, as the bombarding energy increases, this cone becomes narrower. All these results were to be expected. However, the soft photon also has a considerable preference for small-angle scattering, and its angular distribution is not roughly isotropic as one might have expected.

Measurement of the Velocity Distribution along a Vertical Line through a Glacier

Abstract: Various theories of glacier flow have been used to predict widely different changes of velocity with depth. The paper describes a direct measurement of the vertical velocity distribution in a glacier in the Bernese Oberland. Using an electric heating element, a vertical hole 137 in long was melted from the glacier surface to the rock-bed. The hole was lined with a 3 in. steel tube which froze into the glacier and became tilted and bent according to the velocities prevailing at different depths. Using electrically recording pendulum instruments, the inclination of the tube at different depths was measured at the start of the experiment and again after one and two years. The rate of surface flow was found to be 35 m per year. Of this amount, roughly half seemed to be due to flow within the ice and the other half due to sliding of the glacier over its bed. The velocity decreased regularly from the glacier surface downwards, and there was no indication of a yield stress below which ice was completely rigid. If the assumption is made that the deformation of the ice in the glacier is due to simple shear stress, a tentative relationship between this stress and the rate of shear strain can be derived from the results. This shows that flow is not Newtonian, but approximates to a relation where γ = 10 -8 τ 1.5 (γ = rate of shear strain per second, τ = shear stress in bars), over a range of τ between 0.1 and 0.75 bars. Comparison with laboratory compression tests by Glen (1952) indicates that the exponent of τ increases at shear stresses above 1 bar.

A Determination of the Frictional Forces in a Tidal Current

Abstract: The frictional forces in a tidal current have been determined from simultaneous observations of the surface gradients and the currents at various depths. The observations were made a few miles from the coast, off Red Wharf Bay, Anglesey. The gradients were derived from measurements of the surface elevations obtained by a pair of open-sea tide-gauges of the Fave type, modified to give increased accuracy. The Doodson electrically recording current meter was used for the current measurements. Each set of observations extended over 24 h, and they were subjected to harmonic analysis for the semi-diurnal constituent. The data from these analyses were then used in the dynamical equations, giving the amplitudes and phases of the frictional force at the bottom, and of the internal shearing stress in the water at various depths. Five complete sets of records were obtained, three of which are considered to have given significant results. Expressing the amplitude of the frictional force at the bottom in the form F = k$\rho $U$^{2}$, where U is the amplitude of the mean current from surface to bottom, and $\rho $ is the density of the water, the results give the coefficient k an average value of 1$\cdot $8 $\times $ 10$^{-3}$. The internal frictional stress in the water was found to increase approximately linearly with depth from the surface to the bottom, and the corresponding values of the mean eddy viscosity have been derived.

The Magnetic Susceptibilities of Platinum, Rhodium and Palladium from 20 to 290 degrees K

Abstract: An apparatus is described which makes possible the rapid determination of the temperature variation of the magnetic susceptibilities of small solid specimens. The deflexions of a beam of light passing through a Sucksmith balance are recorded photographically. The temperature of the specimen, measured by a Au-Co/Ag-Au thermocouple in contact with it, is initially reduced to 20° K and deflexions are recorded at small intervals of temperature as the temperature of the specimen rises to 290° K. The whole temperature range can be covered in 4 h. The mean deviation of single measurements from the best smooth curve passing through them is less than 0.2 % of the mass susceptibility of the specimens. The apparatus has been used to measure the temperature variation of the susceptibilities of spectrographically standardized specimens of platinum, rhodium and palladium. Measure­ments on three pure palladium specimens show a maximum in the susceptibility temperature curve at 80° K. The experimental results for platinum show a change in the curvature of the susceptibility temperature curve below 90° K, the susceptibility increasing more rapidly as the temperature decreases. The experimental results for rhodium show a regular increase of susceptibility with increase of temperature. The results for palladium are discussed in relation to the collective electron treatment of metals, and assuming a parabolic distribution of energy states in the d band, estimates are made of the effective Fermi-Dirac degeneracy temperature θ F , the number of holes per atom, q , and the exchange interaction energy θ9 . The estimated values are θ F = 1020° K, q = 0.29, θ9 = 554.0° K. Similar estimates are made in the case of platinum, the values being θ F =1750° K, q = 0.25, θ9 = 770° K.

On the Diffusion Theory of Rectification

Abstract: mental characteristics can be utilized to test for the validity of the Einstein equation. When this is done experiment suggests different values of v, D and v/D for forward characteristics from those it does for reverse characteristics. This result has been discussed, taking into account Fermi-Dirac statistics, in the light of the formal theory of conduction. This has been shown to lead (a) to the diffusion equation and (b) to a slight modification of Einstein's relation, provided v and D are regarded as averages for the barrier. Strictly, v and D depend on the concentration distribution of conduction electrons in the barrier. The shapes of the experimental curves are in agreement with theory. The purpose of this paper is to suggest a method of analyzing direct current-voltage curves of contact rectifiers by a uniform method which is based on both forward and reverse characteristics. Utilizing published experimental curves of a copper oxide rectifier, we add to recently given analyses of its reverse characteristics (Landsberg I95I a and b, to be referred to as I and II) an analysis of its forward characteristics. While there is good agreement as regards the shapes of the curves, the mobility (v) and the diffusion coefficient (D) appear to have different values for forward and reverse directions. This necessitates an inquiry into the original meaning of these quantities. Wagner (I93I) established the diffusion equation