Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/205

before the electromagnetic hypothesis had attracted much attention, an elastic-solid theory in many respects preferable to its predecessors was presented to the French Academy^[1] by Joseph Boussinesq (b. 1842). Until this time, as we have seen, investigators had been divided into two parties, according as they attributed the optical properties of different bodies to variations in the inertia of the luminiferous medium, or to variations in its elastic properties. Boussinesq, taking up a position apart from both these schools, assumed that the aether is exactly the same in all material bodies as in interplanetary space, in regard both to inertia and to rigidity, and that the optical properties of matter are due to interaction between the aether and the material particles, as had been imagined more or less by Neumann and O'Brien. These material particles he supposed to be disseminated in the aether, in much the same way as dust-particles floating in the air.

If e denote the displacement at the point (x, y, z) in the aether, and e′ the displacement of the ponderable particles at the same place, the equation of motion of the aether is

$\rho {\frac {\partial ^{2}\mathbf {e} }{\partial t^{2}}}=-(k+{\tfrac {1}{3}}n){\text{ grad div }}\mathbf {e} +n\nabla ^{2}\mathbf {e} -\rho _{1}{\frac {\partial ^{2}\mathbf {e} ^{\prime }}{\partial t^{2}}},\qquad$(1)

where ρ and ρ₁ denote the densities of the aether and matter respectively, and k and n denote as usual the elastic constants of the aether. This differs from the ordinary Cauchy-Green equation only in presence of the term ρ₁&part²e′/∂t², which represents the effect of the inertia of the matter. To this equation we must adjoin another expressing the connexion between the displacements of the matter and of the aether: if we assume that these are simply proportional to each other—say,

$\mathbf {e} ^{\prime }=A\mathbf {e} ,\qquad \qquad$(2)

1. Journal de Math. (2) xiii (1868), pp. 313, 425: cf. also Comptes Rendus, cxvii (1893), pp. 80, 139, 193. Equations kindred to some of those of Boussinesq were afterwards deduced by Karl Pearson, Proc. Lond. Math. Soc, xx (1889), p. 297, from the hypotbesis that the strain-energy involves the velocities.