It is singular to observe the comparative ease with which elementary propositions in attraction can be proved by one of the obvious methods, while the proof by the other is tedious.

Thus nothing can be simpler than Newton's proof that a uniform spherical shell exerts no gravitating force on an internal particle. But, so far as I know, there is no such simple proof (of a direct character) that the potential is constant throughout the interior.

On the other hand the direct proof that a spherical shell, whose surface-density is inversely as the cube of the distance from an internal point, is centrobaric is neither short nor simple. (See, for instance, Thomson and Tait's Elements of Natural Philosophy, § 491.) But we may prove at once that its potential at external points is the same as if its mass were condensed at the internal point.