Some years ago Evelyn and Linfoot found an asymptotic formula, with estimation of remainder, for the number of representations of a large integer n as the sum of s r-free integers. For s ≥ 4 their formula was subsequently sharpened by Barham and Estermann, and for s = 3 recently by me.

This paper is devoted to the classical problem of surfaces of minimum area subtending a given closed contour.

1. The problem to be discussed in the present paper arises when the finite resolving time of a recording apparatus is taken into account. Events are divided into two classes, recorded and unrecorded. Any recorded event is followed by a dead interval of a certain length of time, during which any other event which occurs will be unrecorded. A typical example is an α-particle counter; a recorded particle causes the chamber to ionize, and no other particle can be recorded until the chamber has deionized. In the case when the dead interval is of constant length τ, if the observation time t lies between (n − 1)τ and nτ, the number of recorded events must be 0, 1, 2, …, n − 1 or n. We shall assume that the probability of an event occurring in the interval [t, t + dt] is λdt, λ being constant; this is the case of most practical interest. The probability distribution of recorded events for dead intervals of constant length has been determined by Ruark and Devol (2). The explicit expression of this distribution is fairly complicated, and it is therefore difficult to manipulate. The method employed in the present paper leads naturally to the Laplace transform of the distribution with respect to the time t, and this is relatively simple. The method can easily be generalized to deal with the case of dead intervals which are not all equal, but follow a probability distribution u(τ) dτ. Finally the Laplace transform is very convenient for determining the asymptotic behaviour of the distribution of recorded events for large times of observation.

A rubber molecule containing n + 1 carbon atoms may be represented by a chain of n links of equal length such that successive links are at a fixed angle to each other but are otherwise at random. The statistical distribution of the length of the molecule, that is, the distance between the first and last carbon atoms, has been considered by various authors (Treloar (1) gives references). In particular, if the first atom is kept fixed at the origin of a system of coordinates and the chain is otherwise at random, it has been conjectured that the distribution of the (n + 1)th atom will tend, as n increases, towards a three-dimensional normal distribution of the form

where σ depends on n. Thus r2 (= x2 + y2 + z2) will be approximately distributed as σ2χ2 with three degrees of freedom.

In the dynamical theory of the motion of the Earth relative to its centre of mass, the planet is usually regarded as a rigid or at most only slightly deformable body, and moments of inertia are adopted that are taken to refer to the Earth as a whole, while the motion itself at any instant is assumed capable of representation by a single angular velocity vector. This procedure, however, appears to involve unwarranted assumptions the recognition and removal of which may lead to conclusions of considerable importance. For it is well known from the theory of earthquake waves that the material of the central core of the Earth behaves like a liquid in that it transmits only longitudinal wave vibrations, while there is also other evidence suggesting that the material of the core is a true liquid (1). There is accordingly no a priori reason for supposing that the core will behave like a rigid body firmly attached to the surrounding shell if more or less permanent shearing forces are applied to it. In particular, in respect of any couple known to act on the outer shell, it is not permissible to assume, without examination of the assumption, that its effect will be transferred immediately to the inner core in a way preserving rigid-body rotation of the whole. If the material of the core behaves like a liquid where wave-motion is concerned, this suggests that it will probably also behave like a liquid whatever shearing forces act on it, and the extent to which changes in the rotatory motion of the outer shell can be communicated to the core, and what effects direct gravitational forces acting on the core may have, must in the first instance be questions of hydrodynamics and not rigid dynamics.

It has been noticed by various authors that the use of the hodograph transformation for problems of compressible fluid flow leads to flows having singularities where the acceleration becomes infinite, streamlines have cusps and the solution of the equations of motion in the hodograph plane ceases to represent any possible physical flow. In this paper we investigate the nature of the singularities which can occur in the representation of one plane upon another and apply our results to the hodograph transformation. Finally, we discuss the significance for practical purposes of such singularities and in particular their connexion with the occurrence of shock waves.

The problem of the propagation of steady disturbance in a semi-infinite supersonic stream moving parallel to and bounded on one side by a subsonic stream is discussed and the complete solution obtained for any assumed form of incident disturbance. It is shown that a local initial disturbance generally produces a reflected wave extending to infinity both upstream and downstream of the initial disturbance. A (positive) pressure pulse as an incident wave produces pressure upstream and rarefaction downstream and the magnitude of the effects is calculated in a particular case. Conditions along the axis of symmetry are calculated for the subsonic region in this case.

It is known as a result of various experiments on slow neutrons (1) that a heavy nucleus possesses an enormous number of energy levels which are very closely spaced if the nucleus is highly excited. Strong theoretical reasons for the existence of this great number of levels were given by Bohr (2) and since then various attempts have been made to calculate the number of energy levels of a heavy nucleus. In the first of these, due to Bethe(3), it was assumed that the interaction between the nucleons was small so that the nucleus could be treated as a gas. The alternative assumption, that we may consider the interaction to be large in comparison with the kinetic energy of the nucleons, was proposed by Bohr and Kalckar(4). In the present note we assume as our model a nucleus consisting of neutrons and protons independent of one another; we then have a neutron-gas in equilibrium with a proton-gas, Fermi-Dirac statistics being applied to both.

The secondary emission electron multiplier is chosen to illustrate the phenomenon of ‘cascade multiplication’. A method is given for deriving the semi-invariants of the probability distribution for the number of output electrons after any number of identical stages of multiplication, in terms of the corresponding semi-invariants for a single stage. The output distribution is not, in general, either of the Poisson or Gaussian types, though it tends to a limiting shape as the number of stages becomes very large. The special case in which each stage replaces a single primary electron by a Poisson distribution of secondaries is considered. The overall output distribution after many stages is still not Gaussian unless the mean amplification per stage is large compared with unity.

A method is presented for the accurate numerical treatment of molecular vibration problems in which the potential energy function is of the form

The treatment is carried through in detail for the case V(q) = Aq2 + Bq4, which, in most instances affords an adequate description of the potential. There is no restriction on the relative magnitudes of quadratic and quartic terms so that the method is equally applicable to the purely anharmonic oscillations occurring in types of ring bending (1), where V(q) = aq4.

An approximate formula is derived for the energy levels in the general case. The case V = aq4 affords an illustration of the accurate treatment; the first five eigen-values and eigen-functions are computed and from them the associated transitionprobabilities. Numerical results are presented in a form convenient for further application. On comparison the approximate formula is found to yield results in error by approximately 1 %, the error decreasing for the higher levels for which the formula tends to a B.W.K. expression. In conclusion a ground-state eigen-function of simple analytical form is found to approximate remarkably well to the case of purely quartic vibrations.

A simplified model of the ice crystal, equivalent to Barnes's, was used. By applying the symmetry operations of this model to the dynamical matrix, it was made to depend on six arbitrary constants. By assuming that the tetrahedra of oxygen atoms which form the lattice are regular, and hence applying a further symmetry transformation to a smaller unit of the crystal, the number of arbitrary constants was reduced to two. The elastic constants were then found in terms of these two atomic constants. Two of the experimental measurements of the elastic constants of polycrystalline (quasiamorphous) ice were used to calculate the atomic constants and hence the elastic constants for single crystals.

1. The half-life of Th C′ was determined by a new method of coincidence technique.

2. Previous investigators have recognized difficulties in regard to this investigation. These were: (a) random delays in the counters; (b) coincidences due to other disintegrations; (c) possibility of difference in delays in recording α- and β-particles. In this investigation the difficulties arising from these effects have been overcome.

3. From this type of experiment it is possible to determine the order in which the particles are emitted.

4. It would appear that the time of breakdown of a counter depends slightly upon the magnitude of the initialionization.

5. The value for the half-life of Th C′ was found to be 3.0 ± 0.15 × 10−7 sec.

A further study has been made of the disintegration of boron by slow neutrons, B10 (n, α) Li, in the cloud chamber. The total ranges of the α-particles and the lithium particles were measured and two groups found. 91·4 % of the disintegrations form the main group, which was previously measured, and 8·6 % a long-range group corresponding to the lithium being left in the ground state. The results of the previous measurements are corrected for the variation of stopping power with velocity and the ranges in standard air of the particles of the two groups are now given as 7·7 mm. α-particle and 4·8 mm. lithium particle, total 12·5 mm. for the main group, and 9·3 mm. α-particle and 5·6 mm. lithium particle, total 14·9 mm. for the long-range group.

These results, taken together with those of other workers, make it certain that in about 92 % of disintegrations the lithium is left in an excited state of about 0·47 MeV. energy and in the remaining 8 % in the ground state. The energies calculated from this scheme by using the known masses do not agree with those deduced from the measured ranges and the range-energy relation for a-particles. It is suggested that the rangeenergy relation is at fault and that these measurements provide two experimental points to which a range-energy relation should be fitted.

1. Shastri (1) has shown that if and then

But (Goldstein (2))

hence substituting from (1·2) in (1·1) and changing the order of integration, which we suppose to be permissible, we find that if

then

assuming of course that the integrals converge.