In this paper the condition for the existence of a secondary flow in a straight non-circular pipe is determined according to the modified vorticity transfer theory, with Goldstein's assumed form for the tensor It is shown that a secondary motion arises if the mixture length is not constant on the curves along which | grad u | is constant, u being the velocity parallel to the pipe axis.

In problems of turbulent flow treated by means of the modified vorticity transfer theory, the quantity where is the mean value of the square of the velocity fluctuation and p the mean pressure, plays a part analogous to the pressure in laminar flow. In two-dimensional flow through a channel the theory shows the existence of a gradient of across the channel from the central plane to each wall. Qualitative arguments such as are used to explain the existence of a secondary flow for laminar motion in a curved pipe are applied here to show that a secondary flow is to be expected near the short sides in the turbulent flow through a straight rectangular pipe of large length/breadth ratio.

The equations to determine the secondary flow through an almost circular elliptic pipe are discussed, the mixture length being assumed constant on ellipses similar to and concentric with the pipe section. For a first approximation the problem is reduced to the numerical solution of three simultaneous ordinary linear differential equations.

The measurements of Hall and Hislop (see the preceding paper) were made in a wind tunnel considerably less turbulent than has usually been used in such measurements.

The distributions of velocity and temperature in the wake behind a heated body of revolution have been determined in a low-turbulence wind tunnel. Difficulty was experienced in obtaining a truly symmetrical wake, and observations have been reduced to mean values, curves of which are given.

It has been shown by Kulakoff that if G is a group, not cyclic, of order pl, p being an odd prime, the number of subgroups of G of order pk, for 0 < k < l, is congruent to 1 + p (mod p2); and by Hall that if G is any group of finite order whose Sylow subgroups of G of order pk, p being odd, are not cyclic, then, for 0 < k < l, the number of subgroups of G of order pk is congruent to 1 + p (mod p2). No results were given for the case p = 2. In the present paper it is shown that analogous results hold for the case p = 2, but that the role of the cyclic groups is played by groups of four exceptional types: the cyclic groups themselves, and three non-Abelian types. These groups are defined as follows:

(1) The dihedral group, of order 2k, generated by A and B, where

(2) The quaternion group, of order 2k, generated by A and B, where

(3) The "mixed" group, of order 2k, generated by A and B, where

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.

The radiations from Na24 have been examined by the absorption and coincidence methods and evidence has been obtained for the conclusion that the energy spectrum of the β-particles is a simple continuous spectrum with an upper limit at 1·40 × 106 e.V., and that the emission of γ-radiation after the particle disintegration takes place almost always as a cascade process in which at least two quanta are involved, and very rarely by a single transition to the ground state.

Photometrical measurements of the photographic images of cloud tracks of the disintegration of boron by slow neutrons,

have enabled the ranges of both the He and the Li particle to be determined. We find that the He particle has a range of 7·0 ± 0·3 mm. and the Li particle a range of 4·3 ± 0·2 mm. in standard air.

We deduce that the 7Li nucleus is usually formed in an excited state, with energy of excitation of either 0·5 or 0·8 M.e.V. The wide latitude in excitation energy is due to the uncertain state of our knowledge of the energy-range relation for α-particles.

The negative coefficient of thermal expansion of He ii is shown to be a consequence of the disordering process which takes place as the temperature rises. The conclusion is reached that the thermal energy of He ii is mainly energy of disorder; the vibrational energy of the sound waves amounts only to a few per cent of the whole energy. A theory of heat conduction considered as a flow of disorder is put forward and is shown to be not incompatible with the experimental data at present available.

Calculations are made to investigate the behaviour of ordering near the critical temperature for alloys with a superlattice of the type AB for various compositions. No fluctuation among the sites of the same kind, which may lead to the possible existence of some other types of superlattice, say, AB3, is considered. Results showing the change of ordering with T near Tc are plotted.

The theory is also applied to the adsorption of gases upon a solid surface, where the sites of accommodation are regularly arranged. It is shown that under certain conditions, a superlattice formation in the adsorbed layer is possible.

On any algebraic variety of d dimensions there exist certain systems of equivalence which are relative invariants under birational transformation. These systems include the canonical systems (Todd(3,4)) which can be defined for each dimension from zero to d − 1, and the systems defined by the intersections of these canonical systems among themselves. It is natural to enquire what is the precise behaviour of these invariant systems under birational transformations. Relatively few results of this kind are known. For threefolds, Segre has recently (2) investigated the transformations undergone by the invariant systems in any algebraic (not necessarily birational) transformation whose fundamental points and curves are of general character. More recently, A. Bassi (1) has obtained by topological methods the relations between the Zeuthen-Segre invariants of two Vd in an (α, α′) correspondence. I have recently (6) discussed the transformation of the invariant systems on a Vd for birational transformations on the assumption that the fundamental points are isolated and of general character.

Suppose that we have a set of observed frequencies classified in groups, the number in a typical group being nr, and that we wish to estimate the expectations in the groups according to some law with adjustable parameters. If these expectations are mr we take

Σ denoting summation over the groups, and according to the law in question the probability that one observation should come in the rth group is mr/N. The joint probability, for given mr, that there should be n1 observations in the first group, n2 in the second and so on, is therefore

Actinium emanation was introduced into a cloud chamber for the purpose of making a study of the secondary β-rays. The energies were in agreement with those given by Surugue using the magnetic spectrum method, except that his strong line at 105 kV. was absent. By comparing the numbers of β-rays with the relative intensities of the groups of α-particles, measured by Rosenblum, it was found that the internal conversion coefficients were in general agreement with previous experimental values in the same region of energy. Tracks of Auger electrons associated with the secondary β-rays were found in the photographs. In general, the results from the cloud chamber method agreed with accepted theories and confirmed the results obtained by other methods.

The previous measurements made on the surface tension of liquid helium by Onnes(1) were carried only as far down in temperature as 1·5° K. Since we had apparatus set up suitable for such measurements and capable of reaching temperatures nearly half a degree lower, we considered it worth while to repeat the measurements. Not only was this done for the purpose of extending them to lower temperatures but also to investigate carefully the variation of surface tension in the region of the λ-point (2·186° K.), the transition temperature between Hei and Henii.

The paper is concerned with the influence on the effects produced by slow neutrons, of the temperature of the medium in which they are slowed down, attention being focussed on those neutrons which pass freely through cadmium. Results obtained by Arsenjewa-Heil, Heil and Westcott (10) had appeared to show that the groups A and B responsible for activating silver had energies low enough to be affected by temperature. In the present work, a confirmation of the change for group B has been obtained, and a similar effect has been observed with a boron detector. It is suggested that the explanation of the results is that cadmium transmits neutrons of energies down to about 0·2 e. V., and that “group B” is inhomogeneous. During the course of the work, a study of the conditions under which reproducible results could be obtained was undertaken.

1·1. Es bezeichne für nichtnegative ganze n und k

das n-te Cesàro-Mittel der Ordnung k der Reihe μ0 + μ1 + … Insbesondere ist also die n-te Partialsumme dieser Reihe.

This note describes measurements of the velocity of sound in liquid nitrogen by a method which has previously been used in the case of liquid oxygen(1). The method depends on the diffraction of light by suprasonic waves.

The method of the self-consistent field is applied to a discussion of the energy and wave-functions of the ground state of the hydrogen molecule; an analytical expansion of the wave-function is given in terms of spheroidal coordinates, and the distribution of charge is determined. Two simpler, though less accurate, wave-functions for the molecule are also included, and the possibility of using the method for more complex molecules is discussed.

The calculations of the velocity and temperature distributions for the turbulent flow in axially symmetrical and plane jets are described.

For axially symmetrical jets Tollmien's results derived from calculations based on the momentum theory are stated; the calculations on the basis of the modified vorticity transfer theory and on the vorticity transfer theory with symmetrical turbulence are carried out. Unfortunately it seems to be true that none of these theories provides results in good agreement with experiment for both the velocity and temperature distributions.

For plane jets the results obtained by Tollmien and based on the momentum transfer theory are stated; the calculations on the basis of the vorticity transfer theory are carried out. The velocity distributions on both theories are identical and in fair agreement with experiment. In order to test the validity of the theories it is necessary to compare the theoretical results for the temperature with experiment. No results for the latter seem to have been published.

The statistical theory of long-range interactions between adsorbed particles on a plane lattice is worked out approximately, by treating in detail the distribution of adsorbed particles among a few sites inside and on the boundary of a circular region, and regarding the distribution outside the circle as uniform and continuous with a density Kθ per unit area, where K is the number of lattice points per unit area and θ is the fraction of surface covered by adsorbed particles. The continuous distribution begins at a distance ρ from the centre of the circle, ρ being determined by the condition that the probability of occupation of a first shell site is equal to the probability θ of occupation of the central site. Using this method, general formulae for the adsorption isotherm and the heat of adsorption are obtained. Numerical applications for dipole interactions and for quadratic and hexagonal lattices are worked out in detail and the case in which the dipole moment varies with θ is discussed.