The modern algebraic treatment of geometry in projective spaces focuses attention on the properties of homogeneous ideals in polynomial and power-series rings. This inevitably raises questions concerning how far ordinary ideal theory needs to be modified if only homogeneous ideals are to be regarded as significant. In practice, one can usually answer any particular question of this type without undue difficulty when it arises but, it seems to the author, the topic has enough intrinsic interest to merit a connected discussion by itself.

The object of this note is to construct a set of real three-dimensional Lie groups such that every real three-dimensional Lie group is locally isomorphic with some group in the set. The construction is effected by first finding canonical forms for the constants of structure of real three-dimensional Lie algebras; these canonical forms give rise to certain bilinear forms, and the Lie groups are obtained as linear groups isomorphic with groups of automorphisms which leave these bilinear forms invariant.

Groups that can be represented as the product of two proper subgroups have been studied extensively; one of the latest contributions is a paper by Wielandt (8), in which references to previous work can be found. In the case where the two proper subgroups have only the unit element in common, we adopt the term ‘general product’introduced by Neumann (1).

The formula to be proved is

In proving (1) the following formulae are required:

where p≥q + 1, (1).

The formulae to be proved are as follows.

If p ≧ q + 1, R(l + m) > 0, R(αr − l − m + n)> −1, R(αr − l − m − n)> 0, r = 1,2,…p,

If p ≧ q + 1, R(l >0, R(l+m >0, R(αr − l − m + n)> −1,

The formula to be proved is

Recently R. S. Varma [11] gave the generalisation

for the Laplace transform

Since

(1) gives (2) when k = − m + ½.

We shall represent (1) by

and as usual, (2) will be denoted by

Let Sn denote the “surface” of an n-dimensional unit sphere in Euclidean space of n dimensions. We may suppose that the sphere is centred at the origin of coordinates O, so that the points P(x1, x2, …, xn) of Sn satisfy

We suppose that n≥2.

A point x in real Hilbert space is represented by an infinite sequence (x1, x2, x3, …) of real numbers such that

is convergent. The unit “sphere“ S consists of all points × for which ‖x‖ ≤ 1. The sphere of radius a and centre y is denoted by Sa(y) and consists of all points × for which ‖x−y‖ ≤ a.

In § 2 a product of two modified Bessel Functions of the Second Kind is expressed as an integral with a function of the same type as a factor of the integrand. In § 3 an integral involving a product of these functions, regarded as functions of their orders, is evaluated in terms of another function of this kind. These results were suggested by a study of Mellin's inversion formula.