The conjugate g of a periodic and integrable function f is not necessarily integrable, even when f is monotone inside its fundamental interval. Paley and Wiener, however, proved that g is integrable if f is monotone and odd. A simpler proof was given later by Zygmund‡.

The famous theorem of Schnirelmann states that a constant c exists such that every integer greater than one may be expressed as the sum of at most c primes. Recently, Heilbronn, Landau and Scherk proved that this holds with c = 71. Probably the true value of c is 3. Another well-known theorem (Hilbert's solution of Waring's problem) is that every integer is the sum of a bounded number of positive kth powers. If we omit the restriction that all the kth powers are positive, the problem is referred to as the easier Waring problem and the proof of the result is then much simpler.

In the first part of this paper I announced some new Pflaster theorems for arbitrary r-dimensional closed sets lying in the Euclidean space Rn. In § III I proved them for the special case of an r-dimensional closed set F linked (rel a neighbourhood U) with an (n – r – 1)-dimensional spherical cycle. I shall now prove these theorems in the general case of a quite arbitrary closed set F.

The classical configuration connected with the names of Miquel and Clifford associates with a set of n straight lines in a plane a point called their Miquel point when n is even, and a circle, their Clifford circle, when n is odd. When the number of lines n in a set is even, the omission of these, one at a time, gives n subsets whose associated Clifford circles are concurrent at the Miquel point of the original set; when n is odd, the omission of the lines one at a time gives n subsets whose associated Miquel points are concyclic on the Clifford circle of the original set. To start the chain, it is only necessary to define the Miquel point of two lines as their common point.

The theory of surfaces and of manifolds of higher dimension is greatly handicapped by the lack of exact information concerning the extension of the theorem of Riemann-Roch for curves. Allied to this is the difficulty of finding an expression for the arithmetic genus Pa of primals in terms of their defining elements. This paper seeks to examine certain particular applications of the theorem.

We often wish to test the difference of the means of two series of measures of a quantity for a systematic difference. A suitable test, in terms of the theory of probability, can be obtained as follows. We suppose that in each series the probability of error is normally distributed with unknown standard error. Then if we have only one observation in each series, the difference of the two could equally well be interpreted as entirely due to the random error in one series, the other being right, or entirely due to systematic difference, both observations being free from random error. This suggests that we should take as one of our fundamental quantities the expectation of the difference of a single measure in one series from one in the other; the expectation of the square of the difference, however, appears still better.

The stability of the motion of viscous incompressible fluid, of density ρ and kinematic viscosity ν, between two infinitely long coaxial circular cylinders, of radii a and a + d, where d/a is 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 velocity W parallel 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 when W = 0, when the critical disturbance is a steady motion). As Reynolds number of the flow we take | | d/ν, where is the average value of W across the annulus, and we denote by l the wave-length of the disturbance along the axis, by σ/2π the time period of the critical flow, by c the wavelength of the critical flow, by ωc the critical value of ω, and we put

approximately, if α is not nearly equal to 1.

The following theoretical investigation of the two-dimensional flow of an inviscid fluid past a keel and rudder, and of the consequent lateral force, follows experiments performed by Prof. T. B. Abell in the Department of Naval Architecture of the University of Liverpool, and we wish to acknowledge our indebtedness to him for the information given in many discussions.

There exists a group of action functions, depending on a parameter γ giving for every γ ≥ 0 a static solution with central symmetry and finite energy, both for the electric and magnetic field, and going over into Maxwell's action function for weak fields. For γ = 1 we obtain Born's case distinguished by a perfect symmetry between electric and magnetic field. For γ = 0 the action function takes a particularly simple form and the comparison with the results of Heisenberg, Euler and Kockel allows of an approximate determination of the fine structure constant.

Born's electrodynamics, together with its quantization, has been developed in several papers by Born and Infeld†. It is a general concept, containing many different systems of electrodynamics, of which Maxwell's theory is one particular case. Each system is characterized by a Lagrangian (or Hamiltonian) function of two invariants F and G (or P and Q). These invariants are built up by means of two antisymmetrical tensors, which are the “field quantities”.

The vibrations of the particles which constitute a crystal can be represented in terms of normal vibrations subject to the condition that the energy of the crystal can be expressed as a homogeneous function of the second order in the displacements. This has been shown by Born and by Waller. It has been proved by Debye and, more generally by Born, that the density of the normal vibrations varies as ν2 when the frequency is very small. The spectrum has recently been worked out for a Born-v. Kármán lattice in the two- and three-dimensional cases. Otherwise practically nothing is known about the spectrum in individual cases, except perhaps in isolated cases where the frequency branches split up and the frequency spectrum shows certain gaps.

A new method is developed for evaluating certain integrals occurring in molecular problems, where distances measured from three different centres of force are involved. The convergence of the results is discussed numerically and theoretically.

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.

It has been shown by Mott that it is legitimate in calculating the probability of excitation of an atom by close collision with a slow α-particle to treat the α-particle as a moving centre of force. This method gives for the probability P of excitation

where

and

The paper describes calculations on the distribution in space of slow neutrons based on the application of diffusion laws to their motions. The effects of using spheres of different sizes, and of changing the composition of the hydrogenous slowing-down medium, are discussed. Curves are given which can be compared with the results of the experiments on the slowing down in different media carried out by the author in collaboration with Mr T. Bjerge, and a revision of the calculations based on these experiments leads to a value of 6 × 10−25 cm.2 for the total absorption cross-section of the water molecule, and indicates that most of this absorption must be attributed to the hydrogen atoms.

The author desires to acknowledge the receipt of a Senior Research Award from the Department of Scientific and Industrial Research.

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.

Measurements of the reflection coefficient at normal incidence on lead and tantalum mirrors in the supraconducting and non-supraconducting state did not reveal any change within the limits of accuracy (0·2−0·5% in R). This does not agree with a suggestion of R. de L. Kronig, according to which an increase in reflectivity of about 35% could have been expected.

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.

Absorption experiments have been carried out on the β- and γ-rays of RaC”. The range of the β-rays in aluminium corresponds to an energy limit of (1·95 ± 0·15) × 106 e.v.; the absorption coefficient of the γ-rays in copper corresponds to a fairly homogeneous radiation of quantum energy equal to (5 ± 1) × 106 e.v. The energy of disintegration is, within the limits of error, in agreement with the value 6·1 × 106 e.v. to be expected from the known energies of the other disintegrations in the RaC branches.

The actual distribution of a homogeneous group of electrons by semicircular magnetic focusing is calculated for different types of source.

The dependence of maximum intensity on the radius of curvature of the path of electrons is investigated and the approximate power laws assigned to different cases. The results can be used in the measurement of the intensity of β-ray line spectra.

Tables of numerical values of effective solid angle subtended by the photographic plate at the source are given for some special cases.

The results of the above calculations have been verified experimentally.