The mutual action of two electrified bodies was regarded by Maxwell as transmitted by a medium. According to him the stress in the medium consists of a “tension like a rope” along the lines of electrical force whose intensity per unit of area is R2/8π, where R is the resultant electric intensity, and of a pressure numerically equal to this in all orthogonal directions. Maxwell's remarks are somewhat vague but his notation is strongly suggestive of an elastic solid medium. It has, however, been pointed out by Minchin that Maxwell's stress system would not in an ordinary elastic solid give origin to strains consistent with the “equations of compatibility” which the theory of elastic solids supplies. Considerable interest still attaches to the theory of an elastic solid medium propagating stresses equivalent to the action between distant bodies of forces varying inversely as the square of the distance. For in the first place, it has been pointed out that the stress system given by Maxwell does not constitute a unique solution of his equations; and, in the second place, it has been suggested that some medium must exist for the transmission of gravitational forces. The statical problem of the propagation of gravitational forces by an isotropic elastic medium has been treated by Minchin. His treatment how ever neglects a certain surface condition. I have thus thought it worth while to consider the problem independently, employing the ordinary surface conditions. The first part of the paper is devoted more especially to the electrostatic problem, but the elastic solid problem is essentially the same throughout.