In this paper we consider the effect of the passage of shallow water waves over vertical walled objects on a flat seabed. The effect of reflections at the successive walls is taken into account when determining the place of breaking of the wave. The results are obtained by the introduction of special reflection and transmission coefficients at each wall and recurrence relations are formulated connecting their values at successive walls. The effect of this is to reduce the calculation of the place of breaking of the wave over a stepped seabed to an equivalent one for a wave over a flat seabed, the result of which is well known.

The first part of this paper is concerned with the elastic response of an isotropie infinite plane containing a central circular inhomogeneity to a deformation such that in the homogeneous infinite plane the associated elastic displacement is the gradient of a harmonic function. Based upon the Papkovich stress-function approach, it is shown that it is possible to represent all solutions to such problems by means of only one family of equations. The second part of the paper is concerned with the elastokinetics of two bonded dissimilar half-planes subjected to any uniformly moving body force. It is again shown that there exist simple relations between components of the elastic field in the composite plane and those of a particular field in the homogeneous infinite plane.

This paper deals with the nature of movable singularities of solutions of Emden's equation

at which the solution becomes infinite. If m = 1 + 2/p with p > 1 an integer, then the solution becomes infinite at a given point x = c as

By the general theory of P. Painlevé on movable poles of solutions of non-linear second order differential equations this ‘pseudo-pole’ cannot actually be a pole of order p. Instead of a bona fide Laurent series at x = c we obtain a series expansion of the form

where Pn(t) is a polynomial in t of degree at most [n/(2p + 2)]. The object of this paper is to derive these series and to prove convergence for p = 2. In this case deg [P6m] is strictly equal to m. For other values of p, see Section 8, Addenda.

A way is suggested of modifying the kernel of an integral equation of the second kind so that truncation of the algebraic system of equations corresponding to the new kernel is subject to less error than that for the original kernel. A computable bound for the error is derived.

The measure of non-compactness of linear integral operators on the half-line [0, ∞) of a special type is studied. In particular, a necessary and sufficient condition is established for an operator of this type to define a compact operator from L2(0, ∞) into itself. These results are then used to discuss the spectrum of second-order differential operators. A necessary and sufficient condition for the spectrum to be discrete is established together with estimates for the distance of a point in the resolvent set from the essential spectrum.

This paper presents a generalisation of earlier results on the dimension of the space of integrable-square solutions of the ordinary linear differential equation Su = λTu, where S and T are formally symmetric ordinary differential operators and λ is a spectral parameter.

1-Nitro-, 1-bromo-, and 1-acetamidofluoranthene undergo bromination to yield 4,9-dibrominated products. 1-Acetamidofluoranthene also yields 1-acetamido-4,8,9-tribrofluoranthene and as a side product 1-acetamido-4,8,9,x-tetrabromofluoranthene, from the former of which 1,4,8,9-tetrabromofluoranthene is obtained and found to be identical with the product resulting from the interaction of 1-bromofluoranthene with three molecules of bromine.

The propagation of sound across a vortex sheet separating two fluids in relative subsonic motion is examined for both harmonic and impulsive sources, the sound source being in the moving fluid. It is found that instability waves arise in a certain region depending on M, the Mach number.

The propagation of sound across a vortex sheet separating two fluids, with the same density and sound speed, which are in supersonic relative motion is examined for both harmonic and impulsive sources. It is found that instability waves are present when M < 2√2, M being the Mach number, and it is suggested that these may account for the phenomena observed in certain experiments. For M > 2 there are also neutral stability waves and these lie in regions in the supersonic stream which are otherwise silent when the excitation is an impulsive source.

The differential expression Mf = −f″+qf, on a half-line [a, ∞), is said to be ‘separated’ in L2(a, ∞) if the collection of all functions f ∈ L2(a, ∞) such that Mf is defined and also in L2(a, ∞), has the property that both the terms f″ and qf are separately in L2(a, ∞). When q is positive and differentiable on [a, ∞) this paper obtains sufficient conditions on the coefficient q for M to be separated; these take the form of bounds for q′q−3/2 on [a, ∞).

In this note it will be proved that any small relatively bounded perturbation of a self-adjoint operator with discrete spectrum also has discrete spectrum.