Suppose that we have n objects, which may conveniently be represented by the integers 1, 2, 3, …, n. The total number of ways in which they can be ordered is n!. Let the order 1, 2, 3, …, n, be termed the normal order. Any other order can be classified according to the minimum number of interchanges (of one number with an adjacent one) required to restore the normal order.

1.1. There are two familiar methods of summation of divergent series usually called the methods (R, 1) and (R, 2). If sn = u0 + u1 + … + un and, as it will be convenient to suppose throughout, u0 = 0, then

when h → + 0§: the convergence of the series for small positive h is presupposed.

We call a point set in a complex K a 0-cell if it contains just one point of K, and a 1-cell if it is an open arc. A set L of 0-cells and 1-cells of K is called a linear graph on K if

(i) no two members of L intersect,

(ii) the union of all the members of L is K,

(iii) each end-point of a 1-cell of L is a 0-cell of L

and (iv) the number of 0-cells and 1-cells of L is finite and not 0.

A method is described in general terms for finding the function of a variate of which the mean is a given function of a parameter of the population. This can sometimes be used for finding unbiased estimates and for finding the moments and moment-generating functions of a statistic when another statistic based on the same observations has a constant value. It is always available when the latter statistic is a ‘sufficient statistic’ for estimating the parameter, which requires the frequency function to be of a certain form. A number of examples are given.

This paper is concerned with methods of evaluating numerical solutions of the non-linear partial differential equation

where

subject to the boundary conditions

A, k, q are known constants.

Equation (1) is of the type which arises in problems of heat flow when there is an internal generation of heat within the medium; if the heat is due to a chemical reaction proceeding at each point at a rate depending upon the local temperature, the rate of heat generation is often defined by an equation such as (2).

Many writers have studied special cases of the Saint-Venant torsion and flexure problem for an isotropic beam of constant cross-section. An excellent summary of this work has been given by Stevenson (4) who also greatly extended the field of application of the theory by showing that the general Saint-Venant problem can be reduced to the solution of boundary problems involving six simple ‘canonical’ flexure functions. Miss Rosa Morris (2, 3) has applied the Saint-Venant theory, as extended by Stevenson, to the solution of the flexure and torsion problem for beams possessing two fairly general types of cross-section, her results including many previous solutions as special cases.

The possibility of the propagation of disturbances along a free surface with a velocity exceeding the velocity of sound in the liquid is examined. Simple necessary conditions for the occurrence of such ultrasonic propagation are found in terms of the boundary condition. It is shown that the conditions are not satisfied in the case of gravity waves, and a detailed examination is made of the pressure and surface waves due to a localized disturbance. Capillary waves are shown to satisfy the conditions for ultrasonic propagation, the existance of a ‘faster-than-sound’ wave is proved, and its main properties are examined.

The method for the solution of problems of surface waves on water with which this note is concerned can be sufficiently illustrated by a particular example given by Lamb (1). In Lamb's example it is supposed that a steady stream in which the x, y components of the velocity are U, 0, where U is constant, is slightly disturbed by the application of a constant pressure ρPU2 to a band of finite width 2a of the surface of the stream; ρ denotes the density of the water and P a non-dimensional constant. In the solution Lamb, using an artifice due originally to Rayleigh and widely used since, assumes that ‘the deviation of any particle of the fluid from the state of uniform flow is resisted by a force proportional to the relative velocity’, and states that this ‘law of friction does not profess to be altogether a natural one, but it serves to represent in a rough way the effect of small dissipative forces; and it has the great mathematical convenience that it does not interfere with the irrotational character of the motion’.

In this paper we are concerned with a material which can support a finite stress elastically without flow and which flows with constant mobility(1) (or plastic fluidity) when the stresses are sufficiently great. Following Bingham(1) and Houwink(2), such a material is called a Bingham solid and the type of flow (purely) plastic. The transition from elastic to plastic behaviour takes place at the yield point.

A classical theory of a spinning particle with charge and dipole moment in an electromagnetic field is obtained by working symmetrically with respect to retarded and advanced fields, and with respect to the ingoing and outgoing fields. The equations are in a simpler form than those of Bhabha and Corben or those of Bhabha, and involve fewer constants. On the assumption that the spin angular momentum tensor θμν satisfies the equation θ2 ≡ θμν θμν = constant, the value of the dipole moment Zμν is chosen to be Cθμν, where C is a constant. The theory is generalized to the case of several particles with charge and dipole moment. By using a suitable Hamiltonian equation, the classical equations of motion, obtained on the assumption that θ is a constant, are put into Hamiltonian form by means of the ‘Wentzel field’ and the λ-limiting process. The passage to the quantum theory is effected by the usual rules of quantization. The theory is extended to the case of particles with charge and dipole moment in the generalized wave field by defining the Wentzel potential in terms of the generalized relativistic δ-function.

Some data obtained at Leiden on relaxation effects in a diluted potassium chrome alum are discussed in the light of a paper by Temperley (10), and excellent agreement is found with the conclusions of that paper. The primary effect of diluting the salt is to reduce the magnetic interaction, and the observed effects are a big increase in the relaxation time in zero field, and a decrease of the relaxation time with increasing field instead of an increase. It is predicted that a decrease should set in with an ordinary salt at liquid helium temperatures at fields somewhat higher than have so far been used. Van Vleck's criticisms of Temperley's paper are discussed, and it is concluded that they are not valid in their present form.

It is shown that In113* can be obtained by nuclear excitation of indium by X-rays. The cross-section per nucleus for the production of In113* with 2 MeV. X-rays is of the same order as the cross-section for the production of In115*. The decay period of In115*, for which some discrepancy was apparent in the literature, was definitely proved to be 4·5 hr.

A simple formula is obtained for the visibility of Talbot's bands. This formula shows that the bands occur only if the glass plate is inserted at one particular side of the telescope objective, and that when they occur, their visibility is proportional to the plate thickness up to a certain optimum thickness, and then decreases linearly, vanishing at and beyond twice the optimum thickness.

It is shown that the white light fringe obtained with Brewster's fringes or with ‘scatter’ fringes has a breadth which decreases with the reflexion coefficient R of the film surfaces. The decrease of intensity from the centre of the fringe is given by the usual Fabry-Perot formula except that R is replaced by R2.