The dispersion of light within the limits of the visible spectrum is for most substances controlled by a natural frcquency p which corresponds to a vibration beyond the violet end of the visible spectrum: so that, n being smaller than p, we may expand the fraction in the formula of dispersion, and obtain the equation
,
which resembles the formula of dispersion in Cauchy's theory[1]; indeed, we may say that Cauchy's formula is the expansion of Maxwell's formula in a series which, as it converges only when a has values within a limited range, fails to represent the phenomena outside that range.
The theory as given above is defective in that it becomes meaningless when the frequency n of the incident light is equal to the frequency p of the free vibrations of the atoms. This defect may be remedied by supposing that the motion of an atomic particle relative to the shell in which it is contained is opposed by a dissipative force varying as the relative velocity: such a force suffices to prevent the forced vibration from becoming indefinitely great as the period of the incident light approaches the period of free vibration of the atoms; its introduction is justified by the fact that vibrations in this part of the spectrum suffer absorption in passing through the medium. When the incident vibration is not the same region of the spectrum as the free vibration, the absorption is not of much importance, and may be neglected.
It is shown by the spectroscope that the atomic systems which emit and absorb radiation in actual bodies possess more than one distinct free period. The theory already given may, however, readily be extended[2] to the case in which the atoms have several natural frequencies of vibration; we have only to suppose that the external massless rigid shell is connected by springs to an interior massive rigid shell, and that this again
