Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/202

to become imaginary for certain kinds of light, in order to explain satisfactorily both the surface colours of the aniline dyes and the strong reflecting powers of the metals.

Dispersion was the subject of several memoirs by the founders of the elastic-solid theory. So early as 1830 Cauchy's attention was directed^[1] to the possibility of constructing a mathematical theory of this phenomenon on the basis of Fresnel's "Hypothesis of Finite Impacts"^[2]—i.e. the assumption that the radius of action of one particle of the luminiferous medium on its neighbours is so large as to be comparable with the wave-length of light. Cauchy supposed the medium to be formed, as in Navier's theory of elastic solids, of a system of point-centres of force: the force between two of these point-centres, m at (x, y, z), and μ at (x + Δx, y + Δy,z + Δz), may be denoted by mμf(r), where r denotes the distance between m and μ. When this medium is disturbed by light-waves propagated parallel to the z-axis, the displacement being parallel to the x-axis, the equation of motion of m is evidently

${\frac {\partial ^{2}\xi }{\partial t^{2}}}=\sum _{\mu }\mu f(r+\rho ){\frac {\Delta x+\Delta \xi }{r+\rho }}$,

where ξ denotes the displacement of m, (ξ + Δξ) the displacement of μ, and (r + ρ) the new value of r. Substituting for ρ its value, and retaining only terms of the first degree in Δξ, this equation becomes

${\frac {\partial ^{2}\xi }{\partial t^{2}}}=\textstyle {\sum }\mu {\frac {f(r)}{r}}\Delta \xi +\textstyle {\sum }\mu {\frac {d}{dr}}\left\{{\frac {f(r)}{r}}\right\}{\frac {(\Delta x)^{2}}{r}}\Delta \xi$.

Now, by Taylor's theorem, since ξ depends only on z, wo have

$\Delta \xi ={\frac {\partial \xi }{\partial z}}\Delta z+{\frac {1}{2!}}{\frac {\partial ^{2}\xi }{\partial z^{2}}}(\Delta z)^{2}+{\frac {1}{3!}}{\frac {\partial ^{3}\xi }{\partial z^{3}}}(\Delta z)^{3}$…

Substituting, and remembering that summations which involve odd powers of Δz: must vanish when taken over all

1. Bull. des Sc. Math. xiv (1830), p. 9: "Sur la dispersion de la lumière," Nouv. Erercices de Math., 1836.
2. Cf. p. 132.