Sets of Equally Inclined Spheres

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Abstract
In n-dimensional Euclidean space En, where we shall throughout assume that n ≥ 2, the maximum number of (n — 1)-dimensional spheres which can be mutually orthogonal is n + 2, and it is well known that the sum of the squares of the reciprocals of their radii is zero, so that the spheres cannot be all real. The maximum number of such spheres which can be mutually tangent is also n + 2, and in 1936 Soddy (7) indicated the beautiful relation connecting their radii. These two formulae are the particular cases γ = 0, γ - 1 of Theorem 1 below, which gives the relation connecting the radii of a set of n + 2 such spheres when every pair is inclined at a given non-zero angle 0, where γ is written for cos θ.

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