Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/449

denotes the refractive index of the dielectric; and therefore the refractive index is determined in terms of the frequency by the equation

$\mu ^{2}=1+4\pi e^{2}c^{2}N/(k^{2}-mn^{2})$.

This formula is equivalent to that which Maxwell and Sellmeier^[1] had derived from the elastic-solid theory. Though superficially different, the derivations are alike in their essential feature, which is the assumption that the molecules of the dielectric contain systems which possess free periods of vibration, and which respond to the oscillations of the incident light. The formula may be derived on electromagnetic principles without any explicit reference to electrons; all that is necessary is to assume that the dielectric polarization has a free period of vibration.^[2]

When the luminous vibrations are very slow, so that n is small, μ² reduces to the dielectric constant ε^[3]; so that the theory of Lorentz leads to the expression

$\epsilon =1+4\pi Ne^{2}c^{2}/k^{2}$

1. Ct. p. 293.
2. A theory of dispersion, which, so far as its physical assumptions and results. are concerned, resembles that described above, was published in the same year (1892) by Helmholtz, Berl. Ber., 1892, p. 1093, Ann d. Phys. xlviii (1893), pp. 389, 723. Io this, as in Lorentz' theory, the incident light is supposed to excite sympathetic vibrations in the electric doublets which exist in the molecules of transparent bodies. Helmholtz' equations were, however, derived in a different way from those of Lorentz, being deduced from the Principle of Least Action. The final result is, as in Lorentz' theory, represented (when the effect of damping is neglected) by the Maxwell-Sellmeier formula. Helmholtz' theory was developed further by Reiff, Ann. d. Phys. lv (1895), p. 82.
   In a theory of dispersion given by Planck, Berl. Ber., 1902, p. 470, the damping of the oscillations is assumed to be due to the loss of energy by radiation: so that no new constant is required in order to express it.
   Lorentz, in his lectures on the Theory of Electrons (Leipzig, 1909), p. 141, suggested that the dissipative term in the equations of motion of dielectric electrons might be ascribed to the destruction of the regular vibrations of the electrons within a molecule by the collisions of the molecule with other molecules.
   Some interesting references to the ideas of Hertz on the electromagnetic explanation of dispersion will be found in a memoir by Drude, Ann. d. Phys. (6) i (1900), p. 437.
3. Cf. p. 283.