Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/147

it is well known that v₁ and v₂ are the roots of the equation in v

${\frac {l^{2}}{{\frac {1}{\epsilon _{1}}}-v^{2}}}+{\frac {m^{2}}{{\frac {1}{\epsilon _{2}}}-v^{2}}}+{\frac {n^{2}}{{\frac {1}{\epsilon _{3}}}-v^{2}}}=0$;

and accordingly Fresnel conjectured that the roots of this equation represent the velocities, in a biaxal crystal, of the two plane-polarized waves whose normals are in the direction (l, m, n)

Having thus arrived at his result by reasoning of a purely geometrical character, he now devised a dynamical scheme to suit it.

The vibrating medium within a crystal he supposed to be ultimately constituted of particles subjected to mutual forces; and on this assumption he showed that the elastic force of restitution when the system is disturbed must depend linearly on the displacement. In this first proposition a difference is apparent between Fresnel's and a true elastic-solid theory; for in actual elastic solids the forces of restitution depend not on the absolute displacement, but on the strains, i.e., the relative displacements.

In any crystal there will exist three directions at right angles to each other, for which the force of restitution acts in the same line as the displacement: the directions which possess this property are named axes of elasticity. Let these be taken as axes, and suppose that the elastic forces of restitution for unit displacements in these three directions are 1/ε₁, 1/ε₂, 1/ε₃ respectively. That the elasticity should vary with the direction of the molecular displacement seemed to Fresnel to suggest that the molecules of the material body either take part in the luminous vibration, or at any rate influence in some way the elasticity of the aether.

A unit displacement in any arbitrary direction (α, β, γ) can be resolved into component displacements (cos α, cos β, cos γ) parallel to the axes, and each of these produces its own effect