1. Many investigations have been made to determine the wave resistance acting on a body moving horizontally and uniformly in a heavy, perfect fluid. Lamb obtained a first approximation for the wave resistance on a long circular cylinder, and this was later confirmed to be quite sufficient over a large range. In 1926 and 1928, Havelock (4, 5) obtained a second approximation for the wave resistance and a first approximation for the vertical force or lift. Later, in 1936(6), he gave a complete analytical solution to this problem, in which the forces were expressed in the form of infinite series in powers of the ratio of the radius of the cylinder to the depth of the centre below the free surface of the fluid. General expressions for the wave resistance and lift of a cylinder of arbitrary cross-section were found by Kotchin (7) using integral equations, and the special case of a flat plate was evaluated. He continued with a discussion of the motion of a three-dimensional body. More recently, Haskind (3) has examined the same problem when the stream has a finite depth.

An asymptotic expression is found for the lift distribution on a long, narrow, laminar wing, at incidence in a supersonic stream. The approximations of the linearized potential theory are used.

1. The following sentence occurs in a recent paper ((1), p. 54): we do not understand precisely what it means, but we shall show that the idea behind it can be put on a reasonably quantitative basis: ‘In the vocabulary of solvers of simultaneous equations, the operator (1, −2, 1) tends to lead to ill-conditioning when the number of equations is large.’

The aim of this article is to discuss a paradox of importance in the quantum theory of measurement. The paradox was first propounded by Einstein, Podolsky and Rosen in 1935(1), and was discussed subsequently by Bohr(2), Furry(3), Schrödinger(4, 5), as well as in texts (6, 7). We begin by formulating the paradox.

The exact (Navier-Stokes) equations for the flow of a viscous compressible fluid are examined in a search for simple solutions, in which the equations reduce to ordinary differential equations. Such solutions are found for the uniform shearing motion in the Couette flow between both parallel flat plates and coaxial circular cylinders in relative motion. For a compressible fluid there are no simple solutions corresponding to certain other flows (such as the Poiseuille flow between fixed parallel flat plates) described by simple solutions for an incompressible fluid. A solution is found for the flow of a gas over an infinite porous plate with uniform suction through the plate (gravity neglected), and the relation of the solution to asymptotic solutions of the boundary-layer equations is discussed. Ordinary differential equations are obtained for the problems of (i) the flow due to the rotation of a horizontal disk in a gas (dissipation neglected), and (ii) the steady circulatory flow round a porous circular cylinder with uniform suction (gravity neglected). Discussion of the gas-dynamical problems is preceded by a description of the state of a horizontally stratified gas in static and thermal equilibrium.

1. It is frequently required to find the numerical value of the definite integral It is, however, often found that even if the analytical expression of f(x) is given, it cannot be integrated in terms of known elementary functions. The elliptic integrals are perhaps the best known examples of functions of this type; and more common are cases where f(x) is not even defined by an analytical expression but merely specified by a table of numerical values. In the two cases mentioned one has to resort to a numerical integration procedure in which the integral is evaluated in terms of the values of f(x) at a finite number of the arguments x1, x2, …, xn. There are essentially two different types of quadrature formulae available, those which depend on the use of finite differences and the tabulation of f(x) at equidistant points, and those which depend on the values of f(x) at suitably selected points xi. These latter, so called mean-value methods, will be discussed in the present paper. For a recent treatment of such methods see, for example, Beard (5).

In an earlier paper (1), which will be referred to as A, the present author has demonstrated the relativistic invariance, for general transformations of coordinates, of the Einstein-Bose and Fermi-Dirac quantizations of linear field equations derived from higher order Lagrangians. The proof consisted of the identification of the commutation relations with the generalized Poisson brackets introduced by Weiss (2) and proving the invariance of the latter.

The addition theorem for Legendre functions leads, as is well known, to a useful expansion formula of importance in the theory of electrostatic potentials,

or, in an alternative notation,

In this paper, a family of exact solutions of the problem of two-dimensional flow of a compressible perfect fluid about a cylinder is found, the solutions being generalized from those for the flow of an incompressible fluid about an elliptic cylinder of arbitrary eccentricity and angle of attack. The circulation is taken to be zero and the speed of the fluid at infinity subsonic. This analysis is an application of the general theory given by T. M. Cherry (1, 2); it was done to exhibit the details of the analysis for a flow other than that corresponding to the low-speed flow past a circular cylinder.

The grand partition function is derived by averaging over all partitions and numbers of molecules of a grand ensemble. The statistical parameters that arise are interpreted thermodynamically. A formula for the mean fluctuation of the numbers of molecules is deduced, and it is shown that the latter can become very great for singularities of the partial potentials. In that case, the system becomes statistically indeterminate, and separates out into several phases. Conversely, in the statistics of actual systems, the search for singularities, at which the partial potentials become independent of the composition, leads to the conditions under which a phase-change takes place. It is shown that, on certain hypotheses, the grand partition function is the generating function of the ordinary partition function, and that the latter can always be represented in product-form. The generating function of a factor in this product-form is called a reduced grand partition function. The significance of its parameters is discussed, and thoir use in the statistical thermodynamics of liquid mixtures is reviewed.

The problem considered in this paper is the fluid motion arising from a thin semi-infinite plate started to move impulsively from rest (in viscous incompressible fluid at rest) with a velocity, subsequently maintained uniform, parallel to the edge. Two solutions are given, one obtained in polar coordinates is in the form of an infinite series, whilst the other, derived operationally in parabolic coordinates, leads to a single integral for the velocity distribution. The former is convenient for computation in the vicinity of the edge, the latter being more convenient elsewhere.

A hypothesis originally introduced by Rayleigh after he had discussed the corresponding flow for an infinite plane has been used here to draw deductions about the steady flow past a quarter-plane whose leading edge is normal to the direction of flow and also to obtain approximate expressions for the effect of the edges on the skin friction of a sufficiently broad rectangle whose length is parallel to the incident stream.

In many technical problems on conduction of heat involving convection, radiation, or evaporation at the surface of a body, the flux of heat at the surface is known empirically as a function of the surface temperature with reasonable accuracy. The thermal properties of the body also vary with the temperature, but in many cases the nature of this variation is completely unknown, and in others it is slight over the range of temperature involved. Thus it seems worth while studying problems on conduction of heat in a medium with constant thermal properties and with heat transfer at its surface a given function of the surface temperature. Mathematically such problems occupy an interesting position between the classical linear theory and the general case in which both the differential equation and the boundary conditions are non-linear.

In considering the equilibrium of stars of high density the effects of the Pauli Exclusion Principle must be taken into account. For large values of the degeneracy parameter, which will be denoted by λ, an explicit formula for the partition function may be obtained, from which we may easily find the pressure and density in terms of λ. When λ ≪ 1 the relations for a Fermi-Dirac gas reduce to those for a Boltzmann gas. For λ of the order of, but less than, unity, series expansions for the relevant physical quantities can be found, but for λ of the order of, but greater than, unity, a set of numerical quadratures must be performed at intervals (in λ) close enough for interpolation purposes. The method for this is discussed in § 2 and the numerical results are given at the end of the paper.

A method is given for the calculation of the surface waves of small amplitude generated on deep water by a normal velocity distribution of period 2π/σ prescribed over a submerged circular cylinder. The method of solution involves a system of linear equations in an infinite number of unknowns; this system always possesses a solution. The unknowns may be obtained as power series in a parameter Ka, convergent for sufficiently small values of the parameter. When the parameter is not small, the equations can be solved by infinite determinants. It is shown that the reflexion coefficient of waves incident on a fixed circular cylinder vanishes, as was first shown by Dean. The pulsations of a submerged cylinder are discussed when the normal velocity is the same at all points of the cylinder at any given time.

In Fart 1 the oscillations are examined which may develop in supersonic flow through the divergent part of a nozzle. Hooker's theory is found to be confirmed by observations on a steam nozzle.

In Part 2 it is shown that Taylor's approximate theory of flow between circular arcs can also analyse the throat conditions when the velocity of approach to the throat is supersonic. Limiting symmetrical flow occurs when the velocity at the centre of the throat just exceeds the local velocity of sound, no matter what the radii of the arcs and the ratio of the specific heats of the gas may be. The unique asymmetrical case is merely the reverse of Taylor's solution for the flow when the velocity of approach is subsonic.

A numerical comparison has been made of the velocities across the throat as calculated by Taylor's method and by Fox and Southwell's iterative process when the velocity of approach is subsonic. The agreement is good for limiting symmetrical flow but not so satisfactory for the unique asymmetrical case.

In the first part of this paper a method was given for constructing a wave potential when the normal velocity is a prescribed function of the angular variable on a submerged circular cylinder. It was shown that the method breaks down for values of the parameters Ka and Kf for which a certain infinite determinant vanishes. The vanishing of this determinant implies the existence of a non-trivial velocity potential, such that the normal velocity vanishes on the cylinder and both velocity components vanish at infinity. In this part of the paper it is shown that there can be no non-trivial solution of this kind; in other words the infinite determinant does not vanish. In the absence of a general uniqueness theory for surface waves it seems worth while to establish this particular result.

The absolute measurement of fast neutron flux presents several difficult problems. Few methods have yet been described in the literature, although the experimental techniques developed by several authors for the detection of fast neutrons (Baldinger, Huber and Staub(7), Barshall and Kanner(9), Amaldi, Bocciarelli, Ferretti and Trabacchi (3), Gray (19), Barshall and Battat(8)) may easily be adapted to this type of measurement. It is, however, most important to have available methods of measuring fast neutron flux to permit the determination of cross-sections for nuclear processes induced by fast neutrons, and several such methods have been developed in the Cavendish Laboratory in recent years. They are the subjects of separate papers (Bretscher and French (13), Kinsey, Cohen and Dainty (21), Allen (l), Allen and Wilkinson (2)). The main purpose of the present paper is to describe the results of experiments carried out to compare these methods in order to test the validity of the assumptions implicit in the individual methods.

This paper is an application of O. G. Sutton's theory of eddy diffusion to a case for which it was not originally developed; the theory is used to predict the dispersion of a cloud of falling droplets released from a point source at great height above the surface of the earth. Some experimental and theoretical values are given for the widths of ground area covered by liquid drops released from sources situated at heights between 1000 and 5000 ft., and agreement between theory and experiment is seen to be quite reasonable. This paper therefore provides further support for the use of O. G. Sutton's theory in problems of this type. The theory has practical applications, e.g. in the spraying of crops from aeroplanes.