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1917]                       THE THEORY  OF  ANOMALOUS DISPERSION                           489
On the assumption that
i},£ = (C,D')e'iğ*erM+into*l   . ....................... (3)
we get Maxwell's results*
1 _   1   _ p + a-     crn*         p- — n2                                    ,..
~                     '
E       H!!'( n2           Rn
,„. ( *
Here v is the velocity of propagation of phase, and I is the distance the waves must run in order that the amplitude of vibration may be reduced in the ratio & : 1.
When we suppose that It = 0, and consequently that I = oo , (4) simplifies. If VQ be the velocity in aether (a- = 0), and v be the refractive index,
For comparison with experiment, results are often conveniently expressed in terms of the wave-lengths in free aether corresponding with the frequencies in question. Thus, if X correspond with n and A with p, (6) may be written
— the dispersion formula commonly named after Sellmeier. It will be observed that p, A refer to the vibrations which the atoms might freely execute when the aether is maintained at rest (77 = 0).
If we suppose that n is infinitely small, or X infinitely great,
^2 = 1+^ ................................. (8)
thus remaining finite.
Helmholtz in his investigation also introduces a dissipative force, as is necessary to avoid infinities when n =p, but one differing from Maxwell's, in that it is dependent upon the absolute velocity of the atoms instead of upon the relative velocity of aether and matter. A more important difference is the introduction of an additional force of restitution (a2#), proportional to the absolute displacement of the atoms. His equations are
* Thus in Maxwell's original statement.   lu my quotation of 1899 the sign of the second term in (4) was erroneously given as plus.
t What was doubtless meant to be d-%/dyz appears as.^/da;2, bringing in x in two senses.. 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.