The Distribution of Distance in a Hypersphere

Abstract

Deltheil ([1], pp. 114-120) has considered the distribution of distance in an $n$-dimensional hypersphere. In this paper I put his results (17) in a more compact form (16); and I investigate in greater detail the asymptotic form of the distribution for large $n$, for which the rather surprising result emerges that this distance is almost always nearly equal to the distance between the extremities of two orthogonal radii. I came to study this distribution by the need to compute a doubly-threefold integral, which measures the damage caused to plants by the presence of radioactive tracers in their fertilizers; for the distribution affords a method of evaluating numerically certain multiple integrals. I hope to describe elsewhere this application of the theory.