A mathematical proof of a theorem concerning the flux density of parti
cles at any point interior to a simple closed convex surface in which
the surface emits such particles uniformly and isotropically is presen
ted. The relationship of this theorem, which is of rather general vali
dity, to related observations of other workers is discussed. The ramif
ications of the theorem are considered with particular attention devot
ed to conditions commonly found at particle accelerators. It is conclu
ded that results of this theorem are directly applicable to efforts to
apply the principle of maintaining dose equivalents received by perso
nnel as low as reasonably achievable.