Joseph Boussinesq

The attempt to represent the properties of the aether by those of an elastic solid lost some of its interest after the rise of the electromagnetic theory of light.

But in 1867, 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[81] by Joseph Boussinesq (b. 1842).

Until this time, investigators had been divided into 2 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 took up a position apart from both these schools.

He 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

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,

where the constant A depends on the nature of the ponderable body—our equation becomes

which is essentially the same equation as is obtained in those. older theories which suppose the inertia of the luminiferous. medium to vary from one medium to another. So far there would seem to be nothing very new in Boussinesq’s work. But when we proceed to consider crystal-optics, dispersion, and rotatory polarization, the advantage of his method becomes evident: he retains equation (1) as a formula universally true—at any rate for bodies at rest—while equation (2) is varied to suit the circumstances of the case. Thus dispersion can be explained if, instead of equation (2), we take the relation

where D is a constant which measures the dispersive power of the substance: the rotation of the plane of polarization of sugar solutions can be explained if we suppose that in these bodies equation (2) is replaced by

where B is a constant which measures the rotatory power, and the optical properties of crystals can be explained if we suppose that for them equation (2) is to be replaced by the equations

When these values for the components of e′ are substituted in equation (1), we evidently obtain the same formulae as were derived from the Stokes-Rankine-Rayleigh hypothesis of inertia different in different directions in a crystal; to which Boussinesq’s theory of crystal-optics is practically equivalent.

The optical properties of bodies in motion may be accounted for by modifying equation (1), so that it takes the form

where w denotes the velocity of the ponderable body. If the body is an ordinary isotropic one, and if we consider light propagated parallel to the axis of z, in a medium moving in that direction; the light-vector being parallel to the axis of x, the equation reduces to

substituting

where V denotes the velocity of propagation of light in the medium estimated with reference to the fixed aether, we obtain for V the value

The absolute velocity of light is therefore increased by the amount ρ1Aw/(ρ + ρ1A) owing to the motion of the medium; and this may be written (μ2 - 1) w/μ2, where μ denotes the refractive index; so that Boussinesq’s theory leads to the same formula as had been given half a century previously by Fresnel.[82]

It is Boussinesq’s merit to have clearly asserted that all space, both within and without ponderable bodies, is occupied by one identical aether, the same everywhere both in inertia and elasticity; and that all aethereal processes are to be represented by two kinds of equations, of which one kind expresses the invariable equations of motion of the aether, while the other kind expresses the interaction between aether and matter. Many years afterwards these ideas were revived in connexion with the electromagnetic theory, in the modern forms of which they are indeed of fundamental importance.