1. The general theory underlying sampling inspection methods introduced during the war under the name of sequential analysis, applicable in cases where the sampling units can be taken serially, has mainly been developed by Wald and is now published (5). The original purpose of the present investigation was to exhibit the main structure of the distributional theory relating to the size of sample required by noting its relationship with the classical ‘random-walk’ problem. Thus in Part I of this paper are derived the distribution and characteristic functions of the absorption time for one-dimensional random-walk theory with constant drift and absorbing barriers. Attention is confined to the asymptotic case of numerous independent displacements, for which it is well known that the unrestricted total displacement becomes Gaussian; the motion may also be treated as continuous. Since these results were obtained Wald has independently given an alternative and more general discussion of this problem((4); see also Tweedie (3)), but the extension here of the method of images used by Chandrasekhar (1) still appears of interest, and has advantages over alternative direct methods of solution for the distribution function involving Fourier expansion (cf. 2).

The equations of motion in a surge-tank fitted to a hydraulic pipe-line are non-linear. Various methods of solving them were described by Durand: (1) simplification of the equations and the use of empirical coefficients, (2) simplification by taking the friction forces as proportional to the first power of the velocity, (3) a step-by-step method of solution. As a revolt against the serious expense of time involved in (3) (which was later developed by Jakobsen and by Cole), he was led to devise other methods including the use of models, which he dealt with subsequently in more detail. Nevertheless, there remains a need for a quick approximate method of calculating the maximum surges which may be used with confidence at least in the early stages of design. A promising line of attack was suggested by the success which has attended the approximate analysis of mechanical vibrating systems having non-linear characteristics. This method has been set out by Den Hartog, who dismisses step-by-step solutions as ‘too laborious’.

The paper considers the solution of the general variational equation of relativistic dynamics using the methods of Mathisson with a generalization at one stage. The solution gives us two equations describing the translational motion and rotational motion. The rotational equation is also investigated by setting up a variational equation using the conservation of angular momentum, and this is found to be of the same form as that obtained from Mathisson's variational equation.

Rayleigh's method of deducing the probability distribution of the amplitude of the sum of n equal vibrations of random phase is generalized to the case when the amplitude of each vibration is a definite function of its phase. The same method is applied to the shot effect and it enables the distribution of random noise to be obtained. Campbell's Theorem and its generalizations can then be deduced from this.

Few papers presenting a theory of operation of thermostats can be found. This may be explained by mathematical difficulties which are met when considering the rather complicated construction, from the thermal point of view, of most thermostats. There is, however, a relatively great number of papers describing various mechanical constructions of thermostats. In many cases designers obtain very good practical results by making use of some approximate theories only.

In developing the theory of radiative transfer in expanding atmospheres (Chandrasekhar, 1945 a, b) the author has recently encountered certain novel types of boundary-value problems in hyperbolic equations which appear to merit consideration for their own sake. As related to the equation

the boundary-value problems which occur are of the following general type

i.e. along AB in Fig. 1. Along AD (x = 0 and 0 ≤ y ≤ l2) and BC (x = l1, 0 ≤ y ≤ l2) we are further given that

and

where φ(y) and ψ(y) are two known functions. The problem is to solve equation (1) in the rectangular strip ABCD satisfying the stated boundary conditions.

The combustion of wood presents an interesting problem in non-steady heat flow. When wood is heated, the temperature distribution at a given time may be calculated by means of the well-known conduction equation together with the relevant boundary conditions, provided that the temperature is nowhere sufficiently high to cause appreciable thermal decomposition. When this condition does not apply, the calculation becomes much more complicated, since, as has been recognized for a considerable time, the decomposition is exothermic. The general problem, therefore, is to calculate temperatures and rates of decomposition inside a mass of material, the thermal breakdown of which is accompanied by a heat change, given an initial set of conditions, and a known rate of supply of heat to the surface. The theoretical part of this paper aims at solving this problem for the case of sheets of wood heated in a comparatively simple manner. The treatment is, however, general and may be applied to other materials which undergo thermal changes without melting, if the relevant thermal and other constants are inserted.

In a recent paper (1) an analysis was given of the distribution of stress in a semi-infinite elastic medium deformed by the pressure of a rigid body on part of the plane boundary, the remainder of the plane being free. In that form of the problem—the so-called ‘Boussinesq problem’—the normal displacement of a point within the pressed area was prescribed and the distribution of pressure over that area determined. In this paper the corresponding analysis is given for the case in which the pressed area and the distribution of pressure over it are both prescribed and the normal displacement of a point on the free surface is determined.

It is shown that anomalously large and anomalously small values of the constant A in the thermionic current formula for metals can be explained by the usual statistical equilibrium theory provided proper account is taken of the fact that free electrons may be shared by two competing overlapping energy bands. When this theory is developed there arises a comparatively large temperature variation of the statistical parameter ζ which occurs in the Fermi function, and when the exact formula is forced into the empirical form with a constant work function, the temperature factor appears as a multiplying factor of the current constant, this factor being greater or less than unity depending upon the nature of the overlapping bands. In particular the observed values of the constant A for nickel and for hafnium are well explained.

If we assign to the relativistic energy of a free particle H the meaning of an energy operator, it can be shown that the phenomenon of ‘Zitter-Bewegung’ (and hence Dirac's equation) is obtainable from quantum mechanics alone.

1. The properties of partitions of numbers extensively investigated by Hardy and Ramanujan (1) have proved to be of outstanding mathematical interest. The first physical application known to us of the Hardy-Ramanujan asymptotic expression for the number of possible ways any integer can be written as the sum of smaller positive integers is due to Bohr and Kalckar (2) for estimating the density of energy levels for a heavy nucleus. The present paper is concerned with the study of thermodynamical assemblies corresponding to the partition functions familiar in the theory of numbers. Such a discussion is not only of intrinsic interest, but it also leads to some properties of partition functions, which, we believe, have not been explicitly noticed before. Here we shall only consider an assembly of identical (Bose-Einstein, and Fermi-Dirac) linear simple-harmonic oscillators. The discussion will be extended to assemblies of non-interacting particles in a subsequent paper.

A set of relativistic classical motions of a radiating electron in an electromagnetic field are derived from the principle of conservation of energy, momentum and angular momentum. It is shown that these equations lead to results more in harmony with the usual scheme of mechanics than do the Lorentz-Dirac equations. When applied to discuss the motion of the electron of the hydrogen atom, these equations permit the electron falling into the nucleus, whereas the Lorentz-Dirac equations do not allow this. When applied to consider the motion of an electron which is disturbed by a pulse of radiation, the solution is in a more symmetrical form. For scattering of light of frequency ν the expression for the scattering cross-section is found to be the same as the classical Thomson formula for small ν, and to vary as ν−4 for large ν.

The paper deals with the motion of a Gaussian wave-packet in a parabolic potential field, contrasting the quantum-mechanical behaviour of the packet with that which follows from pure classical mechanics.

The behaviour is entirely classical only if the initial phase of the space-wave-function is linear in the space coordinate; in which case the wave-packet is initially a ‘best-possible’ packet according to Heisenberg's uncertainty relationship.

The paper concludes with a brief reference to the similar problem for a free particle.

Note added in Proof, 16 February 1946. Through the courtesy of Mr M. S. Bartlett, we have been privileged to see the manuscripts of some work by Mr J. E. Moyal and himself, in which the motion of a probability wave-packet is discussed in more general terms than our own: and in which it appears that the motion of our wave-packet, which we have described as non-classical, is indeed actually classical provided that a certain correlation exists between momentum and position in the p, q-distribution function. With this knowledge we now find that classical and quantal predictions agree precisely provided that the initial classical probability function is given by

this is a particular case of their more general results.