In the normal procedure for establishing linearized stability criteria a mean steady state which we denote by U is perturbed by a small additional disturbance u in the Eulerian sense. In the linearization it is usual to regard the disturbance as resolved into Fourier components which are additive so far as the first order goes. The linearized problem will then determine a time amplification factor eiσr so that the linearized disturbance is of the form uexp{i(αx + σt)} and the whole motion U + uexp{i(αx + σt)}. As usual it is implied that the real part of complex functions such as uexp{i(αx + σt)} be taken. In cases of instability (imaginary part of σ negative), where the magnitude of the disturbance increases with the time, we are prompted to ask how energy is supplied to the disturbance. This question is especially relevant to many of the classical problems of fluid motion instability in which the systems are self-contained in the sense that energy is neither supplied nor withdrawn by an outside agency. If such a system is non-dissipative we expect that the total energy of the organized motion and the electromagnetic field, if present, shall be conserved, whilst such energy in a dissipative system will be gradually reduced by heat losses. In either case energy which is taken into a growing disturbance must be provided from the mean energy stored in the system in its initial state. One of the simplest cases of instability to examine from this point of view is the Helmholtz instability of a plane in viscid vortex sheet. This system is self-contained and conservative in the energy of the organized motion, and moreover the only form of energy with which we have to deal is kinetic energy. In the first part of this note we examine how disturbance kinetic energy is provided in this case by the break up of the organized motion of the mean stream. In the second part we consider what happens to the energy balance of a current-vortex sheet in the non-dissipative case. There are here two forms of energy, kinetic and magnetic, and it is shown that when the interface is unstable the overall magnetic energy increases at the expense of the kinetic energy.

A train of surface waves (water waves under gravity) is normally incident on a cylinder with horizontal generators fixed near the free surface, and is partially transmitted and partially reflected. At a great distance behind the cylinder the wave motion tends to a regular wave train travelling towards infinity; the ratio of its amplitude to the amplitude of the incident wave is the transmission coefficient . The transmission coefficient is studied when the wavelength is short compared to the dimensions of the body; physically (though not for engineering applications) this is the most interesting range of wavelengths, which corresponds to the range of shadow formation and ray propagation in optics and acoustics. The waves are then confined to a thin layer near the free surface, and the transmission under a partially immersed obstacle is then small. In the calculation the boundary condition at the free surface is linearized, viscosity is neglected, and the motion is assumed to be irrotational.

At present the transmission coefficient is known only for a few configurations, all of them relating to infinitely thin plane barriers. A method is now given which is applicable to cylinders of finite cross-section and which is worked out in detail for a half-immersed cylinder of circular cross-section. The solution of the problem is made to depend on the solution of an integral equation which is solved by iteration. Only the first two terms can be obtained with any accuracy, and it appears at first that this is not sufficient to give the leading term in the transmission coefficient at short wavelengths; this difficulty is characteristic of transmission problems. By various mathematical devices which throw light on the mechanism of wave transmission, it is, nevertheless, found possible to prove that the transmission coefficient for waves of short wavelength λ and period 2π/ω incident on a half-immersed circular cylinder of radius a is asymptotically given by

when N = 2πα/λ = ω2α/g is large. Earlier evidence had pointed towards an exponential law. It is suggested that transmission coefficients of order N−4 are typical for obstacles having vertical tangents and finite non-zero radius of curvature at the points where they meet the horizontal mean free surface. For obstacles having both front and rear face plane vertical to a depth a, is probably of order e−2N approximately; if only one of the two faces is plane vertical, is probably of order e−N approximately. Thus is seen to depend critically on the details of the cross-section.

An account is given of Eshelby's point-force method for solving elastic inclusion problems, and of his equations relating an in homogeneity to its equivalent inclusion. The introduction of complex variable formalism enables explicit solutions to be found in various two-dimensional cases. Strain energies are calculated. The equilibrium shape of an elliptic inclusion exhibits an interesting feature not previously expected. A fresh analysis of stress magnification effects is developed.

A doubly-stochastic matrix is an n × n matrix with non-negative elements such that each row and each column sums to 1. A permutation matrix is the result of permuting the rows of the unit n × n matrix. Birkhoff's theorem states that the doubly-stochastic matrices constitute the convex hull of the permutation matrices. Using Birkhoff's theorem, Farahat and Mirsky (1) showed that each doubly-stochastic matrix could be expressed as a convex combination of n2 − 2n + 2 permutation matrices, though not in general of fewer. Given Birkhoff's theorem, the Farahat-Mirsky refinement can also be proved quite shortly as follows.

In a recent paper (3) Mordell proved the following theorem.

Theorem 1. Let xn + i = xi ≥ 0 for all i and x1 + … + xn = 1. Then

The problem is no. 226, Colloq. Math. 5 (1958), a problem of Steinhaus. If T is a closed set of tangent lines to a sphere S, are conditions (1) and (2) compatible:

(1) Every great circle of S has at least one tangent belonging to T.

(2) No great circle of S has two parallel tangents belonging to T?

We solve a problem of Steinhaus (New Scottish Book, Problem 328; also Colloq. Math. 5 (1958), Problem 225); if p is a prime, does there exist a regular p-gon such that three of its diagonals cut in a point? The answer is in the negative, except in the case where the said diagonals are all parallel, which can clearly occur for any p ≥ 7.

Let m ≥ 0 and p, p0, …, pm be integers such that p, pj are relatively prime and 1 ≤ Pj < p for 0 ≤ j ≤ m. In the (2m + 2)-dimensional Euclidean space R2m+2 with points (x0, …, x2m+1), the (2m + l)-sphere S2m+1

The aim of this note is to find out some self-reciprocal functions and kernels for Fourier-Bessel integral transforms. Following Hardy and Titchmarsh(i), we shall denote by Rp the class of functions which satisfy the homogeneous integral equation

where Jp(x) is a Bessel function of order p ≥ − ½. For particular values of p = ½, − ½, we write Rs and Rc irrespectively.

The following theorem will be established:

Provided the roots of the associated characteristic equation are all real, any solution of a linear homogeneous differential equation with constant coefficients has at most n − 1 zeros for real finite values of the independent variable, where n is the order of the equation.

The theorem applies to equations with a complex independent variable, but since the conclusion concerns only real values of the variable there is no loss of generality in considering an equation of order n in the form

with y = y(x), where x is real. The constant coefficients ar may be complex.

In order to complete fully the proof of the fixed momentum-transfer dispersion relations for elastic scattering in quantum field theory given by Bremermann, Oehme and Taylor (1) it is necessary (2) to discuss more fully the analyticity of the absorptive part A(t, γ, Δ) in the mass variable γ. We are using the notation of (1), with 2Δ the momentum transfer and 4t2 the total energy of the initial system. For simplicity of discussion we will only treat the case of the elastic scattering of equal mass particles of mass m, the other cases discussed in (1) can be treated similarly. The crucial point of the proof, as first pointed out in (3), is to prove that

is regular in γ in a strip . It was only proved explicitly in (1) that A(t, γ, Δ) is regular for each t, in a strip , where δ may depend on t. In order to prove the analyticity in S of the integral in equation (1) it is necessary to show that δ(t) may be chosen to be independent of t, at least above some finite value of t (possibly large).

The differential equation

which is of importance in boundary-layer theory, has recently been discussed in detail for all non-negative values of the parameter λ(1). I wish to point out here that when λ = − 1 the equation can be solved completely in terms of known functions.