§1. Whittaker has shewn that a general solution of Laplace's Equation

may be put in the form

where f (v, u) denotes an arbitrary function of the two variables u and v; such a representation is valid only in the neighbourhood of a regular point.

The Partial Differential Equations of Physics may be defined as those equations which can be derived from a “least action principle,” that is, as those which are obtained by making a certain integral stationary by the methods of the Calculus of Variations. But, generally speaking, such equations belong to conservative physical systems, and not to those which involve dissipation of energy. In this note it is shewn that a certain class of dissipative equation, of which the best known example is the equation of telegraphy, can be derived from such a calculus of variations problem.

§1. The Lamé Functions of degree n (where n is a positive integer) may be defined as those solutions of the equation

which are polynomials in the elliptic functions sn x, cn x, dn x of real modulus K. Such solutions only exist for certain particular values of the constant a; there are 2n + 1 such values and 2n + 1 corresponding Lamé functions.

§1. The Cardinal Function of Interpolation Theory is the function

which takes the values an at the points x = n. Ferrar has recently proved

Theorem1. If are convergent, C(x)is an m-function3for

This means that C(x) is a solution of the intergral equation

Ferrar's proof deals with functions of a real variable and involves some rather difficult double limit considerations.

One of the many interesting problems discussed by Ramanujan is concerned with the effect of truncating at its maximum term nn/n! the exponential series for en, where n is a positive integer. When n is large, the sum of the first n terms is, roughly speaking, half the sum of the whole series.

A bounded monotonic sequence is convergent. Dr J. M. Whittaker recently suggested to me a generalisation of this result, that, if a bounded sequence {an} of real numbers satisfies the inequality

then it is convergent. This I was able to prove by considering the corresponding difference equation

§ 1. It is well known that, if , the convergence of sn to a limit implies the convergence of tn to the same limit. The converse theorem, that the convergence of tn implies the convergence of sn, is false. Mercer1 proved, however, that if , then both sn and tn tend to l. This theorem has recently been extended in various directions.2 In the present note the case of Abel limits is considered.

Riemann's method of solution of a linear second order partial differential equation of hyperbolic type was introduced in his memoir on sound waves. It has been used by Darboux in discussing the equation

where α, β, γ are functions of x and y.

Binet shewed that the function

can be expanded as an inverse factorial series. This note furnishes a new and much simpler proof of his result, based on a formula which is an analogue of the Binomial Theorem for factorials.

This formula is that, if we denote by [x]n the ratio

then

where denotes the coefficient of xr in the expansion of (1 + x)m.

The function

is, as is well-known, a general solution of Laplace's equation of degree −1 in (x, y, z). In 1926* I proved that the particular solution r−1Q0 (z/r) cannot be represented in this form whereas the solution r−1Q0 (y/r) can. In the present note I find a very simple expression for the latter solution in the form (1.1), and I deduce from it an apparently new integral formula for Qn (cos θ).

When a perfectly conducting uniform thin circular disc is kept at a potential V0 in an external electrostatic field of potential Φ, electric charge is induced on the surface of the disc; the problem is to find the surface-density σ of this induced charge and its potential V so that the total potential V + Φ has the constant value V0 on the surface of the disc. This problem was first discussed by Green in 1832, and the solution in the case when there is no external field was deduced by Lord Kelvin from the known formula for the gravitational potential of an elliptic homoeoid. The problem is still of interest since similar ideas occur in the theory of diffraction by a circular disc and in the theory of the generation of sound waves by a vibrating disc when the wave-length is large compared with the radius of the disc.

Let be a linear differential expression involving n independent variables xi the coefficients Aik Bi, and C being functions of the independent variables but not involving the dependent variable u. Associated with F(u) is the adjoint expression

In a lecture at the Oslo Congress in 1936, Marcel Riesz introduced an important generalisation of the Riemann-Liouville integral of fractional order. Riesz's integral Iaf of order α is a multiple integral in m variables which converges uniformly when the real part of αexceeds m —2 and so represents an analytic function of the complex variable α. This integral is important in the theory of the generalised wave equation, for it provides a direct method of solving Cauchy's initial-value problem. The most recent developments show that it is likely to be also of great importance in quantum electrodynamics.