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In this paper we consider a new class of Riemannian spaces which arise in the theory of the solution of the tensor generalisation of Laplace's equation ∇2V = o. To obtain this generalisation Beltrami's second differential parameter is defined in terms of the metric
of the associated n-dimensional Riemannian space by the usual formulæ
where denotes the Christoffel symbol . The generalised Laplace's equation is then Δ2V = o. For simplicity the quadratic differential form (1.1) is taken to be positive definite, which involves no essential loss of generality.
A partial differential equation of physics may be defined as a linear second-order equation which is derivable from a Hamiltonian Principle by means of the methods of the Calculus of Variations. This principle states that the actual course of events in a physical problem is such that it gives to a certain integral a stationary value.
is convergent for any non-integral value of z, it is uniformly convergent in any finite region of the z-plane and represents an integral function C(z), say, such that C(n) = an for n = 0, ± 1, ± 2, … It is called the Cardinal Function of the table of values, and is identical with Gauss's Interpolation Formula (suitably bracketed).
The function C(z) defined by the series (1) has been given two different definite integral representations, due to Ferrar and Ogura respectively.
§ 1. Notation.—Suppose that a0, a1a2,… are real constants, such that, for some fixed p (≥1), the series is convergent. It is a consequence of the generalised Riesz-Fischer Theorem that there exists a function f(t) L1+p over ( – 1 , 1), of which
In 1932, Hardy and Littlewood  proved the inequality
The constant 4 is best possible; equality occurs when f(x) = A Y(Bx), where
y(x) = e−½x sin (x sin y−y) (y = ⅓π), (x ≧ o)
and A and B (>0) are constants. In , three proofs are given. The inequality has also been discussed in [3, 4]. A very elementary proof in which the function Y(x) emerges naturally is given in this paper.
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.
In 1902, Professor E. T. Whittaker gave a general solution of Laplace's equation in the form
where f is an arbitrary function of the two variables. It appears that this is not the most general solution, since there are harmonic functions, such as r−1Q0(cos θ), which cannot be expressed in this form near the origin. The difficulty is naturally connected with the location of the singular points of the harmonic function. It seems therefore to be worth while considering afresh the conditions under which Whittaker's solution is valid.
Professor E. T. Whittaker has recently discovered a Third Quantum-Mechanical Principal Function R(q, Q, t - T) and has worked out the theory of this function in detail when the Hamiltonian is
By using the Sturm-Liouville theory of linear differential equations and the properties of Green's function, it is shown that the function is an elementary solution of the adjoint of the Schrodinger wave equation associated with the Hamiltonian H.
It is pointed out that the modified Planck constant ħ arises solely from the commutation relation and may, from the analytical view-point, be any constant, real or complex. In particular, if ħ = i, the use of an algebra with the commutation relation leads to an elementary solution of the real equation of parabolic type
An account is given of Professor Marcel Riesz's generalisation of the Riemann-Liouville integral of fractional order. It is shown that the new ideas introduced by Riesz may prove valuable in the theory of partial differential equations and in the theory of the wave-equation in momentum space.
In 1932, Hardy and Littlewood proved the inequalities
The best possible values of the constants K1 and K2 being 1 and 4 respectively. The object of this paper is to prove analogous results for infinite series in which the derivative of the real function f is replaced by the finite difference
In this paper, the electrostatic potential of a point charge in a Reisser-Nordström gravitational field is found in closed form by using the theory of Hadamard's elementary solution of a partial differential equation of elliptic type.