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.

Professor Hemraj has given a proof of a part of a theorem of Gauss without using the theory of quadratic residues. Proceeding on similar lines, I have obtained a complete proof which is rather simpler and certainly more concise.

Let K be a complete field with respect to a discrete valuation and let L be a finite Galois extension of K. If the residue field extension is separable then the different of L/K can be expressed in terms of the ramification groups by a well-known formula of Hilbert. We will identify the necessary correction term in the general case, and we give inequalities for ramification groups of subextensions L′/K in terms of those of L/K. A question of Krasner in this context is settled with a counterexample. These ramification phenomena can be related to the structure of the module of differentials of the extension of valuation rings. For the case that [L: K] = p2, where p is the residue characteristic, this module is shown to determine the correction term in Hilbert's formula.

The object of this paper is to show how the leading properties of Spherical Harmonic Functions may be readily deduced by employing as a typical harmonic a certain simple algebraical expression, which obviously satisfies Laplace's equation; and to extend a similar method to the case of any number of variables.