The effect of a twofold bond alternation on the bond orders and and on the π-electron contribution to the diamagnetic susceptibility of a cyclic even hydrocarbon polyene is investigated. Power series expansions and asymptotic expressions are given for and χπ in terms of the parameter t2 (0 ≤ t2 < 1) which characterizes the amount of bond alternation. It is found that bond alternation reduces both bond order and χπ strongly, the effect being more drastic the greater the bond alternation (t2→0).

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?

Various authors, e.g. Sommerfeld (8), Bouwkamp(i), Groschwitz and Hönl(2), Müller and Westpfahl(7), have considered approximate solutions to the problem of the diffraction of a plane wave by a narrow perfectly conducting strip. If the electric vector is polarized parallel to the strip they find that the scattering coefficient, for normal incidence, is given by, where k is the wave number, 2b the width of the strip and γ(= 0·5572…) being Euler's constant. Any attempt to discuss the convergence of such a pseudo-power series requires a knowledge of the higher order terms and naturally leads to the question of what is the proper expansion parameter.

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.

This paper is concerned with dual series equations of the form where is the associated Legendre function of degree n + m and order −m of the first kind, Hn is a known coefficient of n, f(θ) and g(θ) are given functions of the variable θ and the equations are to be solved for the unknown coefficients An. In hydrodynamics, elasticity and electromagnetism such equations occur in many problems involving a spherical or spheroidal boundary when different conditions are prescribed on different parts of the boundary. These dual series equations are analogous to the dual integral equations which arise in the application of Hankel transforms to problems involving plane or paraboloidal boundaries, the theory and applications of these latter equations having been discussed by several writers amongst whom are Sneddon, Noble and Lebedev(1), (2), (3). The main result of this paper is that the solution of the dual series equations (1·1) and (1·2) for m a positive integer or zero can be reduced to the solution of a Fredholm integral equation of the second kind in one independent variable. The restriction that m be a positive integer or zero covers those dual series equations which arise in three-dimensional boundary-value problems but excludes those which arise in two-dimensional ones where m is . These latter equations have, however, been discussed in a recent paper by Tranter (13).

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 3-dimensional potential due to a charged circle can be derived in various ways which lead to the same expression in terms of the toroidal coordinate σ through Legendre functions of s (= cosh σ). Applying the same methods to the magnetic potential of a circular current we find various results: (a) Legendre's equation yields no solution (§4·1). (b) The appropriate transformation from cylindrical to toroidal coordinates gives a Fourier series (equation (48)) the coefficients of which, though not Legendre functions, are related to them by the identity (§4·2) By this method we also derive expressions in elliptic integrals (equations (56), (58)). (c) Direct integration of the solid angle gives an alternative expression in elliptic integrals (equation (59)).

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.

A study is made of the waves generated by a magnetic line source placed at the vertex of a wedge of high conductivity and arbitrary angle. The boundary-value problem is reduced to the solution of a difference equation and an exact solution obtained. The method is also applied to the case of dielectric coated wedges where the surface reactance and resistance are arbitrary, and the propagation of surface waves along such surfaces is considered briefly. The forms of the solution for large and small values of the surface impedance are obtained and show complete agreement with the known results available for a right-angled wedge and a plane.

The flow pattern past a thin wedge-like profile set at a small angle of attack in a gas flowing with high subsonic or sonic velocity is discussed within the order of the transonic approximation. In the model considered the flow has a stagnation point at the nose of the wedge and breaks away, with velocity equal to that of sound, from the shoulders. The velocity is subsonic throughout the whole field of flow. The solution of the boundary-value problem for the wedge in a channel is formulated as a pair of dual integral equations. The complete solution is given for the wedge in a free stream and the dimensions of the profile, together with the lift coefficient, are computed as functions of the transonic similarity parameter.

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 effect of surface tension on the stability of two superposed fluids can be described in a universal way by a non-dimensional ‘surface tension number’ S which provides a measure of the relative importance of surface tension and viscosity. When both fluids extend to infinity, the problem can be reduced to the finding of the roots of a quartic equation. The character of these roots is first analysed so as to obtain all possible modes of stability or instability. Two illustrative cases are then considered in further detail: an unstable case for which the density of the lower fluid is zero and a stable case for which the density of the upper fluid is zero, the latter case corresponding to gravity waves. Finally, the variational principle derived by Chandrasekhar for problems of this type is critically discussed and it is shown to be of less usefulness than had been thought, especially in those cases where periodic modes exist.