Abstract
The straightforward problem is considered of finding the mean distance from the origin of points randomly distributed inside the unit circle, unit sphere and the generalized n‐dimensional spherical region Bn. This basic investigation is then extended to the problems of finding the mean distance between two randomly distributed points on the surface of Bn and inside Bn. These mean distances are denoted by E(ls,n ) and E(lv,n ), respectively. By introducing generalized n‐dimensional spherical coordinates, the appropriate multiple integrals can be evaluated, and a simple relationship is shown to exist between E(lv,n ) and E(ls,n + 2 ). It is also noted that as n becomes large the volume distribution of points inside Bn approximates to a surface distribution on Bn. It is further shown that the sequence {E(ls,n )} is increasing and has the limit ?2. This property of the sequence {E(ls,n )} leads directly to a generalization of Wallis’ formula for π.
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