A theorem on multiple integrals

Abstract

Suppose thatdυ anddυ′ are two volume elements situated at pointsPand P′ respectively in a three-dimensional right circular cylinder, thatyis the distancePP′, thatz(y) is a given function ofy, and that we wish to evaluate the sixfold integraltaken over all pairs of pointsP, P′within the cylinder. We observe thatz(y) is a function ofyonly; so that the sixfold integral can be expressed as a single integralthat is to say a weighted mean ofz(y) over the relevant values ofy,where the weight function is evidently given by