Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/203

the point-centres within the sphere of influence of m, we obtain an equation of the form ${\frac {\partial ^{2}\xi }{\partial t^{2}}}=\alpha {\frac {\partial ^{2}\xi }{\partial z^{2}}}+\beta {\frac {\partial ^{4}\xi }{\partial z^{4}}}+\gamma {\frac {\partial ^{6}\xi }{\partial z^{6}}}$…, where α, β, γ denote constants. Each successive term on the right-hand side of this equation involves an additional factor (Δz)/λ² as compared with the preceding term, where λ denotes the wave-length of the light: so if the radii of influence of the point-centres were indefinitely small in comparison with the wave-length of the light, the equation would reduce to ${\frac {\partial ^{2}\xi }{\partial t^{2}}}=\alpha {\frac {\partial ^{2}\xi }{\partial z^{2}}}$ which is the ordinary equation of wave-propagation in one dimension in non-dispersive media. But if the medium is so coarse-grained that λ is not large compared with the radii of influence, we must retain the higher derivates of ξ. Substituting $\xi =e^{{\frac {2\pi i}{\lambda }}(z-c_{1}t)}$ in the differential equation with these higher derivates retained, we have $c_{1}^{2}=\alpha -\beta \left({\frac {2\pi }{\lambda }}\right)^{2}+\gamma \left({\frac {2\pi }{\lambda }}\right)^{4}$…, which shows that c₁, the velocity of the light in the medium, depends on the wave-length λ; as it should do in order to explain dispersion.

Dispersion is, then, according to the view of Fresnel and Cauchy, a consequence of the coarse-grainedness of the medium. Since the luminiferous medium was found to be dispersive only within material bodies, it seemed natural to suppose that in these bodies the aether is loaded by the molecules of matter, and that dispersion depends essentially on the ratio of the wave-length to the distance between adjacent material molecules.