In an attempt to evaluate the integral (5) below, using a decomposition of an orthogonal matrix (Jack 1968), the author is led to define a set of polynomials, one for each partition of an integer k, which are invariant under the orthogonal group and which depend on a real parameter α. An explicit representation of these polynomials is given in an operational form. When α = − 1, these polynomials coincide with the augmented monomial symmetric functions. When α = 1, a systematic way of taking linear combinations of these polynomials is explained and it is shown that the resulting polynomials coincide with the Schur functions from the representation theory of the symmetric group. A similar procedure in the case α = 2 then appears to give the zonal polynomials as defined by James (1964, p. 478).

Hadamard elementary solutions are found for the tri-axially symmetric potential equation in space of three dimensions and for the bi-axially symmetric potential equation in space of two dimensions. The elementary solutions involve hypergeometric functions of several variables.

When the Helmholtz equation ∇2V + k2V = o is separated in the general paraboloidal coordinate system, the three ordinary differential equations obtained each take, after a suitable change of variable, the form of the Whittaker Hill equation. For the case k2 < o, a considerable amount is known about the periodic solutions of this equation. The theory for k2 < o does not carry over immediately to the case k2 < o, and so far only perturbation solutions have been obtained. This paper gives, in sections 1–5, explicit solutions for the case k2 < o, in the form of trigonometric series determined by three term recurrence relations. In sections 6, 7 some important relations and orthogonality properties are discussed, which are of particular significance in respect of application to boundary value problems. Section 8 discusses some degenerate cases, and in the Appendix an important property is established of continuity of the solutions with respect to the parameters of the equation.

The solution of certain integral equations, which are related to the Hilbert transform, is discussed from the viewpoint of generalized functions. The behaviour of the corresponding transforms in the complex plane is examined for generalized functions which effectively vanish at infinity. A space of generalized functions which need not be small at infinity is introduced and the integral equations solved in this space. Various possible extensions of the theory are also mentioned.

In a recent paper, Jyoti Chaudhuri and W. N. Everitt linked the spectral properties of certain second order ordinary differential operators with the analytic properties of the solutions of the corresponding differential equations. This paper considers similar properties of the spectrum of the corresponding partial differential operators.