1. Let f(x) be a real function of period 2π, integrable L over (0, 2π), and let

By sn(x) and σn (x) we denote respectively the partial sums and the first arithmetic means of the series (1·1). Similarly, by and we denote the partial sums and the first arithmetic means of the series

conjugate to (1·1). By we mean the function conjugate to f(x), that is

where the integral is taken in the principal-value sense.

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.

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

The technical importance of constructing valve amplifiers disturbed by as little background noise as possible has led to a very searching investigation of the causes of this defect. Some causes—bad contacts, changes in the surface of the filament, secondary ions from the grid, and so forth—though technically speaking not trivial, are of minor physical interest; but it has long been recognized that there are two which cannot be dismissed so easily.

13. Formulae have been obtained for the moment of the fluid forces which act on a plane plate which is gliding on the surface of a stream of finite or infinite depth. It has been shown that the formulae for the lift and moment forces acting on a plate which is gliding on a stream of infinite depth may be obtained by a limiting process from the formulae for the general case of gliding on a stream of finite depth. Also, by another limiting process, the lift and moment forces for the Rayleigh flow past a flat plate in an infinite stream have been obtained from the lift and moment forces for gliding on a stream of infinite depth.

Numerical work has been carried out for the case when the angle of incidence of the plate to the stream is 5° and this has shown the variation of the lift and moment forces and the position of the centre of pressure with a range of values of the length of the plate, the depth of the stream and the height of the trailing edge of the plate above the bed of the stream.

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 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.

An elementary theory of the diffraction of light by supersonic waves was put forward by Sir C. V. Raman and myself (4) to explain many of the important features of the phenomenon observed by Bär and others. Recently Mueller (2) pointed out the essential idea that the optical diffraction effects in solids may be interpreted on the basis of this theory if due account is taken of the photoelastic effects arising from the periodic strains caused by the sound waves. He was thus able to give an explanation of the results of Schaefer and Bergmann (5, 6). Mueller's idea, however, required a still further refinement, since it did not consider the phase relationships between the differently polarized components of the diffraction orders. This refinement has, however, been recently incorporated by Mueller and myself (3). This led us to offer an explanation of the diffraction effects observed by Hiedemann and Hoesch (1) in glass blocks.

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.

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.

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.

A three-filament mechanism is described which will enable multimolecular films to be studied by the method of contact potentials and this method is applied to the study of the evaporation of sodium films of various thickness from one to twenty layers. It is shown that the vapour pressure and heat of evaporation both increase as the thickness increases from one to three layers and thereafter become practically independent of any further increase in the film thickness. The vapour pressure for a thickness of four layers is within experimental error the same as that for bulk sodium.

A difficulty due to the effect of the evaporating stream of atoms annulling the space charge around the emitter filament is discussed, and attempts are made to use this effect to measure the rate of evaporation at temperatures from 500° K. to 825°K.

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.

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.

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.

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 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.

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.