Fano's theorem states that the fluence of particles, emitted uniformly per unit mass, is constant throughout an infinite medium of uniform composition but varying density. O'Connor's scaling theorem says that the ratio of the fluence of secondary particles to that of primary particles, caused by an external source irradiating a medium in a collimated beam, is the same in two uniform media of the same composition but different density, provided geometrical distances are scaled inversely to density. These two theorems are proved by one line of reasoning. The scaling theorem is given a more general formulation.