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.

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.

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.

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 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 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, ∞).

The differential expression M[f] = −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 M[f] is defined and also in L2(a, ∞), has the property that both terms f″ and qf ar separately in L2(a, ∞). When q is positive and differentiable on [a, ∞) it is known that separation holds for M[·] if q satisfies the condition |q| ≦ on [a, ∞)(*) provided the constant c satisfies 0 < c < 2. This paper constructs a class of examples of the coefficient q to show that (*) does not necessarily yield separation if c > 4/√3>2. The precise upper bound of c for which (*) yields separation is not known.

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.