The value of a function which satisfies Laplace's equation within a closed surface is determined by the values at the surface. If two surface distributions be superposed, the value at any internal point is the sum of those due to the two surface distributions considered separately. By means of this principle a very simple proof may be given of the known theorem that the value of the potential at the centre of a sphere is the mean of those distributed over the surface.

On account of the symmetry it is clear that the central value would not be affected by any rotation of the sphere, to which the surface values are supposed to be rigidly attached.

Thus, if we conceive the sphere to be turned into n different positions taken at random and the resulting surface distributions to be superposed, we obtain a new surface distribution, whose mean value is n times greater than before, determining a central value which is also n times greater than that due to the original distribution. When n is made infinite, the surface distribution becomes constant, in which case the central value is the same as the surface value. From this it follows that in the original state of things the central value was the mean of the surface values.