General methods of successive approximations for problems in the theory of large elastic deformations have been proposed by Signorini (9), Misicu (5), Rivlin (6), Green and Spratt (4) and Rivlin and Topakoglu (7). Rivlin has applied his method to the problem of torsion of cylinders as far as terms of the second order. A solution of the problem of second order effects in the torsion of incompressible cylinders of arbitrary cross-section has been given by Green and Shield (3) in terms of complex potential functions, and the complete details of the stress distribution can be obtained once certain integral equations are solved. Blackburn and Green (l) have considered the second order effects in the deformation of compressible cylinders by forces on the ends. For the problems of torsion and bending by couples they obtained details of the displacements and stress distributions in terms of two complex potential functions which satisfy a certain boundary condition. The second order torsion problem has also been discussed by Sheng(8).

This paper describes a method for computing the coefficients in the Chebyshev expansion of a solution of an ordinary linear differential equation. The method is valid when the solution required is bounded and possesses a finite number of maxima and minima in the finite range of integration. The essence of the method is that an expansion in Chebyshev polynomials is assumed for the highest derivative occurring in the equation; the coefficients are then determined by integrating this series, substituting in the original equation and equating coefficients.

Comparison is made with the Fourier series method of Dennis and Foots, and with the polynomial approximation method of Lanczos. Examples are given of the application of the method to some first and second order equations, including one eigenvalue problem.

Solutions of Fock's equations for the self-consistent field with exchange have been carried out for Ne+4, using the EDSAC at the Mathematical Laboratory, Cambridge. The initial estimates for the calculation were made by a simplified version of a method previously suggested for interpolating atomic wave functions with respect to atomic number. This gave good estimates for Ne+4, and it is probable that estimates for Ne+3, obtained similarly, are already accurate enough for practical use. Such wave functions for Ne+3 are given. The results have application in astrophysical contexts.

A direct method of integrating the equation y″ + g(x) y = h(x), with the two-point linear boundary conditions y′(a) + αy(a) = A, y′(b) + βy(b) = B, is based on the factorization of the equation into two first-order linear equations v′ − sv = h and y′ + sy = v, where s is a solution of the Riccati equation s′ − s2 = g. The first-order equations for v and y are integrated in succession, one in the direction of x increasing, and one in the direction of x decreasing, one boundary condition being used in each of these integrations. The appropriate solution of the Riccati equation is determined by the boundary condition at the end of the range from which the integration of the equation for v is started. The process is compared with the matrix factorization method of Thomas and Fox, and its stability discussed.

The four first-order ‘coupled’ equations governing the propagation of electromagnetic waves in the ionosphere, previously obtained in symbolic matrix form (Clemmow and Heading (4)), are expressed explicitly in terms of the ionospheric parameters. The physical significance of the equations is illustrated by considering the energy flux in one characteristic wave when coupling and damping are neglected. Three special cases are then discussed for which second-order coupled equations are also given, namely, the cases of (a) vertical incidence with oblique magnetic field, (b) oblique incidence with vertical magnetic field, (c) horizontal magnetic field in the plane of incidence. For case (a) the second-order equations are those previously derived by Försterling(5).

The form of the coupled equations is physically illuminating and, in principle, suitable for solution by successive approximations. Extensive numerical work has indeed been carried out on the second-order coupled equations in case (a) (e.g. Gibbons and Nertney(6)), and it is probable that the first-order coupled equations would prove more advantageous. The present authors, however, feel that better methods are available for purely numerical work (e.g. Budden(3)), and that the chief interest of the coupled form is that it shows the scope and limitations of the physical conception of characteristic waves.

Proof of convergence of the Crank-Nicolson procedure, an ‘implicit’ numerical method for solving parabolic partial differential equations, is given for the case of the classical ‘problem of limits’ for one-dimensional diffusion with zero boundary conditions. Orders of convergence are also given for different classes of initial functions. Results do not support the validity of so-called h2-extrapolation in some cases.

A discussion is given of the convergence of the eigenvalues Λ (N) of two-dimensional finite-difference equations towards the eigenvalues Λ of the corresponding second-order differential equation, and it is shown that

where h = N−1 and ν2 is a constant. As in our previous paper (4), this can be used to make an accurate estimate of λ by extrapolating to h = 0. After an account of the relaxation method used for computing Λ(N) and a discussion of the residual vector, results are presented for an approximation to the lowest spatially symmetric and antisymmetric states of two electrons in a sphere, interacting through their Coulomb potential.

It is shown that a combination of the methods developed in the two previous papers in the series allows the reformulation of general relativity as a technique for extrapolating physical theory to distant regions, whose accessibility to experiment is limited. General relativity is thus shown to be one of a number of extrapolation theories allowed by the spread theory of Paper II.

A scheme is proposed for solving a class of integral equations by electronic analogue computing techniques in times as short as one-tenth of a second. The scheme utilizes a recently developed high-speed analogue function store for carrying out a special iterative procedure which is shown to be more efficient than the classical Neumann process. The problem of the kernel generation at high repetition rates is considered and a novel method based on pivotal function generators is described. Likely errors are analysed and an overall accuracy of the order of 1% is shown to be attainable with known techniques.

Certain equivalence relationships between different problems occurring in diffraction theory are established. The method used is essentially Schwarz's principle of analytic continuation by reflexion across a straight boundary. An explicit solution is also obtained by this method for a plane wave normally incident on a T-shaped structure.

In some recent work Kodis* considers the scattering of a plane harmonic wave by a circular cylinder and uses variational methods to deduce an asymptotic formula for the scattering coefficient in the case of high-frequency waves. The scattering coefficient, σ, is denned as the total energy flux outward from the cylinder in the scattered wave divided by the energy flux in the beam of the incident wave which falls on the cylinder. In the limit, of geometrical optics, the scattered wave is made up of a wave reflected from the forward half of the cylinder with energy flux equal to that in the incident beam, and a wave behind the cylinder cancelling the incident wave there to form the shadow. Thus σ → 2 as ka → ∞, k being the wave number of the incident wave and a the radius of the cylinder. The next term in the asymptotic expansion for σ is proportional to . It has been determined already from the exact series expression for the field by White (6), Wu and Rubinow (7) and Kear (3). But Kodis gives an approximate method which also supplies this result and, since the coefficient is in good agreement with the values obtained from the exact solution, concludes that his approximate procedure could also be used for more general obstacles. In the present paper, we propose an alternative approximate method which is related to the one given by Kodis but is very much simpler. Also we go on to obtain the correction terms for general obstacles in detail. Various writers (see, for example, Keller (4) and van de Hulst(8)) have suggested previously that the correction is still proportional to , where a is some length defined by the body shape. However, the determination of a and of the numerical coefficient has been limited to two-dimensional obstacles.

The power and appropriateness of renewal theory as a tool for the solution of general problems concerning counters has been amply demonstrated by Feller (7), who considered a variety of counter problems and reduced them to special renewal processes. The use of what may be called renewal-type arguments had certainly been made by authors other than Feller (e.g. in § 3 of Domb (3)), but it was only in (7) that the simplicity of the renewal approach to counter problems was recognized and systematically applied. More recently, Hammersley (8) was concerned with the generalization of a counter problem previously studied by Domb (2). This problem may be introduced, mathematically, as follows. Let {xi}, {yi} be two independent sequences of independent non-negative random variables which are non-zero with probability one (i.e. two independent renewal processes). The {xi}, are distributed in a negative-exponential distribution with mean λ-1, and we write Eλ for their distribution function and say ≡ {xi} is a Poisson process to imply this special property of ; the {yi} have a distribution function ‡ B(x) with mean b1 ≤ ∞. Form the partial sums and define ni to be the greatest integer k such that Xk ≥ t, taking X0 0 and nt = 0 if x1 > t. Then define the stochastic process

Hammersley'sx counter problem concerns the stochastic process

1. Introduction. Among the more notable attempts to derive a generalization of Einstein's gravitational theory is the recent one of Einstein and Schrodinger ((1)–(8)). This was formulated by dropping the symmetry of the fundamental tensor gμν and the components of the affine connexion. The most serious defect of these non-symmetric theories is that the field equations, in their original form, do not determine the motion of electrically charged particles in an electromagnetic field, as has been proved by Infeld(9), Callaway (10) and Bonnor (n). Together with the lack of an energy-momentum tensor and a geometric description of the paths of charged particles, this seems to indicate that the concept of motion is missing in this type of theory. It is clear that one of the most important results which should follow from a generalization of Einstein's gravitational theory is the correct equations of motion of charged particles in an electromagnetic field.

A theoretical treatment is given of the absorption of longitudinally polarized sound in dielectric crystals of high symmetry, due to interaction with the thermal phonous. The Q value for the absorption is calculated in terms of the deviation from Hooke's law, the result being applicable at low temperatures when the mean free path of the phonons is greater than the wavelength of the acoustic wave. The theory is applied to cubic crystals, and in particular to solid argon, where a minimum Q value of about 108–109 is calculated. It is suggested that measurement of the absorption (at present somewhat beyond experimental techniques) would supply information on the deviation from linear elasticity, as well as providing a satisfactory verification of the basic interaction theory. The effect of isotopes, and, in more complicated structures, of optic modes, is shown to be negligible.

A brief account is given of the fundamental properties of a new generalization ((1), (2)) of Einstein's gravitational theory. The field equations are then solved exactly for the case of a static spherically symmetric gravitational and electric field due to a charged particle at rest at the origin of the space-time coordinates. This solution provides information about the gravitational field produced by the electric energy surrounding a charged particle and yields the Coulomb potential field. The solution satisfies the required boundary conditions at infinity, and it reduces to the Schwarzschild solution of general relativity when the charge is zero.

Various definitions of uniform acceleration are possible in relativity theory. Two of these are considered with particular reference to their physical realizations.