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THE THEORY OF ANOMALOUS DISPERSION.
[Philosophical Magazine, Vol. xxxin. pp. 496 — 499, 1917.]
IN a short note* with the above title I pointed out that Maxwell as early as 1869 in a published examination paper had given the appropriate formulse, thus anticipating the work of Sellmeierf and HelmholtzJ. It will easily be understood that the German writers were unacquainted with Maxwell's formulse, which indeed seem to have been little known even in England. I have thought that it would be of more, than historical interest to examine the relation between Maxwell's and Helmholtz's work. It appears that the generalization attempted by the latter is nugatory, unless we are prepared to accept a refractive index in the dispersive medium becoming infinite with the wave-length in vacuo.
In the aether the equation of plane waves propagated in the direction of x is in Maxwell's notation
pdhijdV = Ed yd#2, .............................. (1)
where 17 is the transverse displacement at any point cc and time t, p is the density and E the coefficient of elasticity. Maxwell supposes "that every part of this medium is connected with an atom of other matter by an attractive force varying as distance, and that there is also a force of resistance between the medium and the atoms varying as their relative velocity, the atoms being independent of each other"; and he shows that the equations of propagation in this compound medium are
where p and a- are the quantities of the medium and of the atoms respectively in unit of volume, tj is the displacement of the medium/ and 17 4- f that of the atoms, o-jD2^ is the attraction, and aRdQdt is the resistance to the relative. motion per unit of volume.
* Phil Mag. Vol. XLVIII. p. 151 (1899); Scientific Papers, Vol. iv. p. 413. •' A misprint is now corrected, see (4) below. . . -
f Pogg. Ann. CXLIII. p. 272 (1871). t Pogg. Ann. CLIV. p. 582 (1874) ; Wissenschaftliche Abhandlungen, Band rt. p. 213.ion of F (experimentally or otherwise) does not require the variation of both a and v. There is advantage in keeping a constant; for if a be varied, geometrical similarity demands that any roughnesses shall be in proportion.