The theorem that, if four arbitrary lines be taken in a plane, the four circles about the triangles formed by threes of these lines, meet in a point, can be generalised to space of any even number of dimensions, as was recognised by Mr J. H. Grace in 1897. In the plane case the centres of the four circles lie on another circle passing through the point of concurrence of these four; it has been sought to prove that the centres of the n + 2 spheres, similarly arising in space of an even number, n, of dimensions, also lie on a sphere†.