The present paper contains solutions of the tensor generalisation of Laplace's Equation. The results obtained are summarised in the two theorems enunciated in § 1. They apply only to the case when the Riemannian space forming the background of the theory is flat. In the concluding paragraph a special case is considered, and it is shown that the present theory is closely connected with Whittaker's well known general solution of the ordinary Laplace's Equation.

In a recent paper Dr J. M. Whittaker has shown that the Fourier series

of a function f(θ) which has a Lebesgue integral in (— π, π), is absolutely summable (A) to sum l, if

exists, where

.

In the following pages we give a complete system of projective concomitants for any number of linear complexes and one quadric, in a quaternary field. The investigation follows the earlier1 paper on mixed quaternary forms, in which the corresponding system, omitting the quadric, was given. Such a system, for one quadric and one linear complex was given by Weitzenböck who used complex symbols. A detailed investigation of the case of two linear complexes and a quadric, together with their geometry, has been given elsewhere.

An elementary proposition states that an absolutely convergent series is convergent, i.e. that if

then

All points in an n-space equidistant from a fixed point (the centre) constitute what may be called a spherical continuum of the nth order,—the continuum being of n — 1 dimensions ((n — l)-dimensional spread) and of the 2nd degree. Any region of this spherical continuum bounded by n (n — l)-dimensional linear continua or, primes (spaces of n — 1 dimensions), passing through the centre shall be called a spherical simplex of the nth order. This spherical simplex is bounded by n faces, spherical simplexes of the (n — l)th order, each of which in turn is bounded by n — 1 spherical simplexes of the (n — 2)th order, and so on till we reach spherical triangles, arcs and lastly points, the vertices. The total number of spherical simplexes of different orders connected with one of the nth order is 2n — 2. The n spherical continua of the (n — l)th order which contain the faces of the spherical simplex of the nth order determine a set of 2n spherical simplexes of the same order, 2n–1 pairs, the two spherical simplexes of a pair being symmetrically situated with respect to the centre and therefore congruent.

It is well known that the polynomial in x,

has the following properties:—

(A) it is the coefficient of tn in the expansion of (1–2xt+t2)–½;

(B) it satisfies the three-term recurrence relation

(C) it is the solution of the second order differential equation

(D) the sequence Pn(x) is orthogonal for the interval (— 1, 1),

i.e. when

Several other familiar polynomials, e.g., those of Laguerre Hermite, Tschebyscheff, have properties similar to some or all of the above. The aim of the present paper is to examine whether, given a sequence of functions (polynomials or not) which has one of these properties, the others follow from it : in other words we propose to examine the inter-relation of the four properties. Actually we relate each property to the generating function.

Hadamard defines the “elementary solution” of the general linear partial differential equation of the second order, namely

(Aik, BiC being functions of the n variables x1, x2, .., xn, which may be regarded as coordinates in a space of n dimensions), to be one of those solutions which are infinite to as low an order as possible at a given point and on every bicharacteristic through that point.

It is well known that in many cases the solutions of a linear differential equation can be expressed as definite integrals, different solutions of the same equation being represented by integrals which have the same integrand, but different paths of integration. Thus, the various solutions of the hypergeometric differential equation

can be represented by integrals of the type

the path of integration being (for one particular solution) a closed circuit encircling the point t = 0 in the positive direction, then the point t = 1 in the positive direction, then the point t = 0 in the negative direction, and lastly the point t = 1 in the negative direction; or (for another particular solution) an arc in the t-plane joining the points t = 1 and t = ∞.

Einstein has recently adopted a new set of field-equations in his Unified Field-Theory of Gravitation and Electricity, the so-called theory of parallelism at a distance or Teleparallelism, and has given a solution of these equations with spherical symmetry, corresponding to the field of a charged mass-particle. In the present paper we discuss the solution of these equations with axial symmetry, which corresponds to a statical field whose field-variables depend on a single coordinate only, viz. the coordinate which is measured along the axis of symmetry.

In the preceding paper Professor Whittaker has given a general method for the solution of differential equations by means of definite integrals. It depends on finding a solution χ (q, Q) of an auxiliary pair of simultaneous partial differential equations to be derived from an arbitrary contact transformation by changing the momentum variables into differential operators. The first object of the present paper is to arrive at a method for passing from the contact transformation in its algebraic form to these partial differential equations, in a manner which is unambiguous and which makes them compatible. We show too how to obtain any number of such, pairs of equations from any given contact transformation. Successive transformations are also discussed.

It is well known that any map of n regions on a sphere may be coloured in five or fewer colours. The purpose of the present note is to prove the following

Theorem. If Pn(λ)denotes the number of ways of colouring any ma: of n regions on the sphere in λ (or fewer) colours, then

(1)

This inequality obviously holds for λ = 1, 2, 3 so that we may confine attention to the case λ > 4. Furthermore it holds for n = 3, 4 since the first region may be coloured in λ ways, the second in at least λ — 1 ways, the third in at least λ — 2 ways, and the fourth, if there be one, in at least λ — 3 ways.

These notes are intended to be read in connexion with Dr A. C. Aitken's paper, Proc. Edinburgh Math. Soc. (2) 1 (1929), 199-203. It is proposed to show (by a simple line of direct algebraic demonstration which is also applicable to the original formula) that Aitken's Theorem can be extended to the Everett types, i.e. the types which include two sets of terms—one set involving u (0) and the resultant of generalised operations on u (0), and the other set involving u (1) and the resultant of similar operations on u (1).

The various formulae for the Legendre Functions, and the relations between these formulae, have been studied by Kummer,Riemann, Olbricht, Hobson, Barnes, Whipple, and others. Hobson obtained some of the relations directly, by expres ing the functions as Pochhammer integrals, and expanding in a number of series each with its own region of convergence. To obtain some of the other formulae, such as (i) below, he transformed the differential equation, and then expressed the functions in terms of the solutions of the transformed equation. Barnes succeeded, by means of his wellknown integrals involving Gamma Functions, in deducing all the formulae directly from the formulae which define the functions.Notes on the history of the subject and references to previous work will be found in the papers by Hobson and Barnes.

The earliest proof that every rational number (R) can be expressed as a sum of cubes of three rational numbers (x, y, z), not necessarily positive, was published in 1825 by S. Ryley, a schoolmaster of Leeds: formulae were given for x, y, z in terms of a parameter, such that every value of the parameter led to a system of values of x, y, z satisfying the above relation, and every rational value of the parameter led to a system of rational values of x, y, z. The later solutions referred to by Dickson are found to give the same results as Ryley's formula, as does another method, quoted in a modified form by Landau from a paper by the present writer. Thus it might almost be believed that Ryley's century-old result embodies all that is known with regard to the resolution of a number into three cubes, and that his formula is unique. I propose to examine the rationale of his method and the causes of its success; it will then appear that an infinity of similar formulae exist, and that one of them is at least as simple as his. It is convenient to state Ryley's formula, and the modification made by Landau, in section 2; and to generalize the method in section 3.

Mr D. E. Littlewood has recently discussed the properties of the quadratic equation over the real quaternions and shown that the solutions correspond to the common intersections of four quadrics in four-space. Although complex quaternion solutions may arise, the system of real quaternions to which the coefficients belong is a division algebra. It is of interest, therefore, to discuss the solution of the quadratic when the coefficients are drawn from a system containing divisors of zero.

In Part I it has been shown that, given a contact transformation, two equations

can be derived which lead to the compatible differential equations

It will be shown in the present communication that the necessary and sufficient condition that (1.3), (1.4) should be compatible is that

regarded as an equation in the non-commutative variables q, p which themselves satisfy the condition

We shall call functions satisfying this condition conjugate functions. From this point of view the method employed by Professor Whittaker in his original paper, involving the use of a contact transformation,, was really a particular method of generating conjugate functions. This powerful method may be supplemented and extended by the other methods developed in the following pages.

In a paper entitled “On differentiating a Matrix” H. W. Turnbull deduced some interesting and elegant results by the use of a matrix operator Ω, a matrix whose elements were the partial differential operators with respect to the elements of a square matrix x. Throughout the present paper the differential operator Ω. is used, or rather a matrix operator, which is the product of Ω and another square matrix Y.

By means of this operator in §§ 1 and 2 Bazin's matrix and Reiss's matrix are considered from the standpoint of matrices as distinct from that of determinants. Reiss's matrix is shown to be a constant times a compound of Bazin's matrix; and the latent roots of Reiss's matrix are immediately determined in terms of the latent roots of Bazin's matrix. From this result a theorem, discovered by Deruyts, is deduced as well as a more general theorem.

An algebraic equation

determines, in general, s as a many valued function of z. If s and z can be expressed as one valued functions of a third variable t, then t is called the uniformising variable. As Poincaré showed, s and z are automorphic functions of t.

Einstein has studied a universe in which the time coordinate t is uncurved and the spatial section is the surface of a sphere

in four dimensions. Some interest attaches to the case where this surface is replaced by

where n is a positive integer.

By applying Fourier's Integral Theorem to a well-known formula, due to Cauchy, expressed in the form

where R (μ + v) > 1, Ramanujan has shown that

where t is any real number.