Let L be a Lie ring and denote the product of x and y in L by [x, y]. The ring L is said to satisfy the Engel condition (cf. (1)), if for every pair of elements x, yεL there is an integer k = k(x, y)such that

If k(x, y) can be taken equal to a fixed integer n for all x, y ε L then L is said to satisfy the n-th Engel condition.

This paper describes a generalization of the inverse of a non-singular matrix, as the unique solution of a certain set of equations. This generalized inverse exists for any (possibly rectangular) matrix whatsoever with complex elements. It is used here for solving linear matrix equations, and among other applications for finding an expression for the principal idempotent elements of a matrix. Also a new type of spectral decomposition is given.

1. Let H = (aij) be a positive-definite Hermitian matrix of order n. For any k distinct integers i1, i2, …, ik between 1 and n, we shall use the symbol (i1, i2, …, ik) to denote the k-rowed principal submatrix of H corresponding to the rows and columns with indices i1, i2, …, ik. It is well known thatM

and more generally,

A number of methods are at present available for the numerical solution of linear ordinary differential equations over a restricted range of the independent variable. Of these perhaps the most commonly used are methods based on finite differences which have as their aim the construction of a table of values of the dependent variable at stated intervals of the independent variable. The appropriate method to be used depends upon the boundary conditions. Thus for second-order equations, for example, if boundary conditions occur at each end of the range of integration the relaxation method of Southwell (6) is suitable, whereas if this is not the case resort may be made to step-by-step methods or the recently proposed method of Allen and Severn(1). This method, applicable to equations of any order, brings the problem within the scope of normal relaxation methods at the expense of an increase in the order. In all such methods the accuracy attainable for a given (small) amount of labour is a worth-while consideration, and methods have been suggested by Fox (3) and Fox and Goodwin (4) for improving this by the use of comparatively large intervals of the independent variable.

Certain stochastic processes with discrete states in continuous time can be converted into Markov processes by the well-known method of including supplementary variables. It is shown that the resulting integro-differential equations simplify considerably when some distributions associated with the process have rational Laplace transforms. The results justify the formal use of complex transition probabilities. Conditions under which it is likely to be possible to obtain a solution for arbitrary distributions are examined, and the results are related briefly to other methods of investigating these processes.

1. A particle executes a random walk over the possible positions x = 0,1,2,…, its initial position being x = d ≥ 0. At the nth step it occupies the position x with probability pn (x | d)and is in the state (n, x). The transition from (n, x) to (n + 1, y) has the probability px, v given by

The vector notation commonly employed in elementary physics cannot be applied in its usual form to spaces of other than three dimensions. In plane dynamics, for instance, it cannot be used to represent the velocity (– ωx2, ωx1) at the point (x1, x2) due to a rotation ω about the origin, or the (scalar) moment about the origin of the force (F1, F2) acting at (x1, x2). In relativity physics the symbol ⋅ is often used to denote the scalar product of two vectors, it is true, and the tensor aαbβ – aβbα is sometimes denoted by a × b, but there exists no body of rules for the manipulation of these symbols that enables one to dispense with the suffix notation as in the case of vectors in three-dimensional space.

From the idea of the continuous development of an investigation, as expressed in the use of indefinitely continuing sequences of procedures, the concept of a measurement together with its result is formulated. The resulting theory is initially very general, but it is possible to restrict it in such a way as to make the construction of a consistent world possible. The physical condition providing the restrictions that make this construction possible is found in the use of the general idea of a test-particle in all fundamental investigations, which is shown to be a case of the use of the theory-languages of the previous paper (1). Finally, the theory is applied to the solution of the problem of preferred inertial frames.

1. A particle with mass m and coordinates x1x2, x3 relative to a set of rectangular axes fixed in Newtonian space is moving in a field of conservative forces with a potential energy V(x1, x2, x3) and a kinetic energy

The equations of motion, written

(representing the three equations i = l, i = 2, i = 3 in a way to be used in this paper), constitute, as they stand, a sufficient condition in order to ensure

in the sense that the Hamiltonian integral has a stationary value if the actual motion is compared with neighbouring motions with the same terminal positions and the same terminal values of the time as in the actual motion.

In his paper on ‘The propagation of weak spherical shocks in stars’ Whitham(1) uses as the basis of his non-linear theory the solution of the linear equation

where the suffixes denote partial derivatives and A, B, C are functions of r only. In obtaining his solution he assumes (i) that

where ξ = t – ∫dr/A and fn(ξ) is the nth integral of f with respect to ξ, and (ii) that the function f may be identified with the corresponding function which occurs when gravity is neglected, i.e. when A becomes the constant A0, B = −2/r and C = 0. Having assumed that (2) exists, Whitham is able to show that it will hold asymptotically for Aξ/r ≪ 1 and A′ = O (A/r).

Before the war, self-consistent field calculations for the Au+ ion had been carried out by W. Hartree but were left still unpublished at his death (see prefatory note in (5)). These results have been used by Brenner and Brown (1) in a relativistic calculation of the K-absorption edge for gold, and they were also used in obtaining initial estimates for the partial self-consistent field calculations for thallium of which results are given in §§3–5 of the present paper. In the meantime an independent calculation for Au+ has been carried out by Henry (6), and his results agree closely with those of W. Hartree. However, it still seems desirable to publish the latter, since they give directly the radial wave function P(nl; r) at exact values of r, whereas Henry used log r as independent variable, as had been done for similar calculations for Hg(4), and has tabulated r½P(nl; r) which is the natural dependent variable to use with log r as independent variable (2); in some applications it is more convenient to have the radial wave functions themselves.

Detailed energy calculations for CH4, SiH4 and GeH4 have been carried out using the Thomas-Fermi approximation together with a nuclear field which is averaged over all orientations about the central nucleus. The energy, including exchange, has been calculated using the unmodified Thomas-Fermi electron densities as variation functions, and in this way bond lengths and force constants for the three molecules have been obtained. The effect of exchange is very marked, agreement with experiment being quite reasonable when it is included in this way.

In order to test the accuracy of this procedure, a scale factor has been introduced and used as a variational parameter. In this refined approximation, calculations have been made for SiH4, thus enabling improved charge distributions to be obtained. Energy curves are also presented for this molecule, showing separately the variations of kinetic, potential and exchange energies with internuclear distance. It is shown that whilst the scale factor has an appreciable effect on the separate energy curves, the total energy is virtually unchanged.

Finally, the reason why the exchange energy is of considerable importance in the theory is briefly discussed, and some observations are made concerning the rather abnormal bonding in GeH4.

In Parts I and II of this series (March(6); Ballinger and March(1)), we considered in detail the application of the Thomas-Fermi (T.F.) approximation to molecules with tetrahedral and octahedral symmetry. In these papers, following the work of Buckingham, Massey and Tibbs(2), who obtained results for CH4, we averaged the nuclear field over angles (the ‘smoothing approximation’) and considered the electrons as though they moved in the resulting central field. In this way, it was possible in (6) to carry through self-consistent (T.F.) calculations giving the electron distributions and potential fields in tetrahedral and octahedral molecules.

The readings Φ(x) of an instrument with the apparatus function g(x) are related to the true values f(x)of the observed function by the equation

A three-point interpolation operator can be chosen to simulate the spreading action of a class of functions g(x). Under certain conditions the interpolation operator has an exact inverse which, when applied to Φ(x), gives a function which is a better approximation to f(x). The accuracy obtained is discussed, and the effect of the method on the standard errors in the measurement is found.

To obtain certain results of their perfect cosmological principle, Bondi and Gold made use of a part of the theory of kinematic relativity. The same results are here deduced directly from the perfect cosmological principle.