In the Proceedings of 1905–6 Mr Pinkerton gave an extension of the nine point circle to a nine point conic. This raises the question of the extension of the geometry of the circle and triangle to that of the conic and triangle. If a triangle with its associated system of lines and circles be orthogonally projected on a second plane we have a triangle with an associated system of lines and homothetic ellipses. Pairs of perpendicular lines are projected into lines parallel to pairs of conjugate diameters. Such lines will be called, for shortness, in the sequel, conjugate lines. In any relation between lengths of lines, these lengths will be replaced by their ratios to the lengths of the parallel radii of one of the homothetic ellipses.

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.

Ellipsoidal harmonics are defined to be those solutions of Laplace's equation

(where x, y, z are rectangular coordinates) which are useful in problems relating to ellipsoids. If the equation

represents a family of confocal quadrics, it is known that the ellipsoidal harmonics belonging to the family are products of the form

where l1, l2… are constants: one term is to be picked out of the square brackets as a multiplier of the other factors. Now if we consider the case in which two of the principal axes of the ellipsoids are equal, the latter become spheroids. If then we put b = 0 in (1) the family of confocal spheroids has the equation

and belonging to this family there will be spheroidal harmonics of the form given by (2) with b zero.

The purpose of this note is to provide a simplified proof of the following result due to Crawley-Boevey, Kropholler and Linnell.

It can be shown by means of relative motion that if two bodies A and B move with velocities u and v in the same straight line, and a third body C move with velocity u + v also in the same straight line, the space passed over by C is equal to the sum of the spaces passed over by A and by B in the same time.

Let G be a p-solvable group with a p-Sylow subgroup P of order pa and let t(G) be the nilpotency index of the radical of a group algebra of G over a field of characteristic p. The purpose of this paper is to give an elementary proof of the following result of Koshitani [1, Theorem].

I propose to prove the following theorem.

With n > 2, the (n + 2) equations derived from the matrix

where

by equating to zero all (n + 1)-rowed determinants from the matrix ‖Δ‖ are equivalent to only two, one of which is linear in li (i = 1, 2, …, n) and the other is homogeneous and quadratic in a certain n – 1 of li (i = 1, 2, …, n); the elements of the matrix are real; (r, s = 1, 2, …, n) and d is arbitrary.

A semigroup is said to be completely regular if and only if each of its elements lies in a subgroup. It is shown that the algebra of a completely regular monoid (semigroup with identity) over a field of characteristic zero is directly finite.

The term “flat” is used to indicate that the minimum modulus of a function in a region is (in some sense) of the same order as the maximum modulus. Some properties concerned with this notion are described below. They came to light during an attempt to answer a question put to me by Professor Littlewood.

If ABC, A′B′C′ are any two equilateral triangles in a plane, their vertices being taken in the same sense of rotation, of the three lines AA′, BB′, CC′, the sum of any two is not less than the third.