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

.

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.

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.

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.

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.

In an earlier paper, which this note is intended to supplement and in some respects improve, the writer gave a general theorem of duality relating to isobaric determinants with elements Cr and Hr, the elementary and the complete homogeneous symmetric functions of a set of variables. The result was shewn to include as special cases the dual forms of “bi-alternant” symmetric functions given by Jacobi and Naegelsbach, as well as two equivalent forms of isobaric determinant used by MacMahon as a generating function in an important problem of permutations.

There are two planes, called “parabolic planes,” through eachline on the cubic surface cutting the surface in residual conics touching the line in question at two points called the “parabolic points” of that line.