be reversed in direction. Since the resultant force at the latter point vanishes, we must have S=s; so the resultant force at the exterior point is 2s. But s is proportional to the charge per unit area of the surface, as is seen by considering the case of an infinite plate; which establishes the theorem.
When several conductors are in presence of each other, the distribution of electricity on their surfaces may be determined by the principle, which Poisson took as the basis of his work, that at any point in the interior of any one of the conductors, the resultant force due to all the surface-layers must be zero. He discussed, in particular, one of the classical problems of electrostatics— namely, that of determining the surface-density on two charged conducting spheres placed at any distance from each other. The solution depends on Double Gamma Functions in the general case; when the two spheres are in contact, it depends on ordinary Gamma Functions. Poisson gave a solution in terms of definite integrals, which is equivalent to that in terms of Gamma Functions; and after reducing his results to numbers, compared them with Coulomb's experiments.
The rapidity with which in a single memoir Poisson passed from the barest elements of the subject to such recondite problems as those just mentioned may well excite admiration. His success is, no doubt, partly explained by the high state of development to which analysis had been advanced by the great mathematicians of the eighteenth century; but even after allowance has been made for what is due to his predecessors, Poisson's investigation must be accounted a splendid memorial of his genius.
Some years later Poisson turned his attention to magnetism; and, in a masterly paper[1] presented to the French Academy in 1824, gave a remarkably complete theory of the subject.
His starting-point is Coulomb's doctrine of two imponderable magnetic fluids, arising from the decomposition of a neutral fluid, and confined in their movements to the individual elements
- ↑ Mem, de l'Acad., v, p. 247.
