This note is a sequel to two in earlier volumes of the Proceedings, the first by myself and the second by Wilton†.

Suppose that

for |z| <1; that x > 0; that

forr ≥ 0; and that

where τ(x) is to mean 0 if x is not an integer. Thus

where the dash shows that the last term is to be halved when x is an integer;

forr > 1; and

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

It is shown that slight departures from independence and equal weight of the sample observations introduce a bias into Fisher's z, but that the distributions of the variances “within and between groups”, and of Fisher's z would appear to maintain their form to the first order, although the extent to which the approximations hold in practice has not been considered.

The present author recently published what he believed to be a proof and extension of Campbell's theorem on fluctuations. More recently a paper has been published containing a criticism of this proof. It therefore seems desirable that some parts of the proof be amplified. Further explanation of the foundations of the theory seems required, in order that it shall not appear that the theory ignores the randomness of the events, and a more detailed account of the derivation of the mean value of functions of pairs of events (i.e. sums of the type ) is necessary.

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 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.

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 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.

The wave equation for the deuteron in its ground state is solved on the assumption that the mutual potential energy of a neutron and a proton is of the form r−1e−λr. The binding energy of the hydrogen isotope H3 is calculated approximately by the variation method.

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

In the last chapter of his Introduction to statistical mechanics Gibbs introduces the idea of the grand canonical ensemble. He had previously determined the properties of an assembly or a phase containing a given number of systems by averaging the properties of the assembly over an ensemble of examples canonically distributed in phase, keeping the number of systems in the assembly fixed. This means of course constructing what we now call the partition function for the assembly by summing or integrating e−E/kT over the whole of the accessible phase space.

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

The essential difference between mobile and immobile adsorbed films of gases on solids when there is repulsive interaction between adsorbed particles is discussed. Ordinary statistical methods can be applied to the equilibrium configurations of the particles in a mobile film but not to immobile films. Films formed by the adsorption of diatomic molecules with dissociation are then considered. It is pointed out that for a mobile (equilibrium) film there are differences between the true heat of adsorption and the apparent heat obtained from the temperature variation of the evaporation rate. A detailed comparison is made by obtaining a formula for the apparent heat of evaporation and comparing it with that for the true heat obtained earlier by Wang, and the physical reasons for the differences are discussed. A model suitable for studying the pseudo-equilibrium states in immobile films is described and formulae and diagrams in connexion with the kinetics of the formation of such films and the variation of heat of adsorption with fraction of surface covered are given.

The rates of condensation and evaporation of adsorbed particles when they interact with long-range forces are deduced from kinetic considerations, including also the case of adsorption of diatomic molecules with dissociation. The rate of evaporation is compared with experiment, and the variation of the dipole moment with θ required to make the theory agree with experiment is discussed.

Finally the formulae for the rates of condensation and evaporation are applied to the problem of the diffusion of gases through metals, and it is found that for the processes considered the effect of interaction does not alter the conclusion arrived at in an earlier paper concerning the diffusion equation at small p and at large p.

When the approximate methods of Bragg and Williams or Bethe are used in theoretical work on cooperative phenomena it is customary to suppose that only the configurational energy of the assembly depends on the molecular arrangement; the contribution which a molecule makes to the partition function, apart from its share in this configurational energy, is supposed to be independent of the positions of the other molecules.

The disintegrations by which Ac B passes into the inactive AcPb are accompanied by a γ-radiation very weak compared with the intense γ-emission in the case of Ra or Th-active deposit. The analysis of the secondary β-ray spectrum of actinium-active deposit has revealed the existence of at least five γ-rays (1) (see Table I), of which the ray with energy 0·349 × 106 e.V. definitely belongs to the disintegration Ac C—C″ and is associated with the fine-structure of α-particles of Ac C. According to the measurements of Surugue the two rays of 0·4038 × 106 and 0·4257 × 106 e.V. energy are to be attributed to the disintegration AcB—C, whereas the origin of the 0·829 × 106 e.V. ray is less definite. The fit with experiments is best if this ray is assumed to be emitted from Ac B—C; but it may also (within experimental error) be attributed to the disintegration Ac C″—Pb. Experiments on the absorption coefficient of the γ-radiation of RaAc and its disintegration products prove that the 0·829 × 106 e.V. ray is the hardest γ-ray emitted by the active deposit of actinium(2). The value for the absorption coefficient between 4·6 and 10·6 cm. of lead was found to be μ/ρ = 0·76, in good agreement with the value found in previous experiments at smaller absorptions in aluminium (3). There is, on the whole, excellent agreement concerning the energies of the γ-components as measured according to different methods by different observers.

The method of coincidence counting has been applied to an investigation of the possible production of positron-electron pairs by the high energy β-particles from a source of uranium X in absorbers of aluminium, brass and lead. The results are not inconsistent with the high values recently reported for atomic cross-sections for the effect, nor with the suggestion that the atomic cross-section is proportional to the first power of the atomic number rather than the second. Suggestions are made for the use of the β-particles from artificially radioactive substances in an attempt to increase the sensitivity of the method.

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 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 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.