By using (47) equation (46) may be written, changing the constants,

where A is the wave-length in vacuo to which w applies, and Am is the wave-length, also measured in vacuo, which corresponds to the free vibration of the molecules.

150. Dispersion in transparent media. If the range of spectrum considered is far removed from any of the free periods of the molecule, the dispersion formula may conveniently be expressed differently. If the region of resonance lies in the ultra-violet, we may expand in terms of a series proceeding by and thus find

λ

m

where

A1 =ΣMm; A2 = ΣMmλm2; Ap=ΣMmλm2 2.

Equation (49) is known by the name of Cauchy's formula, but was deduced by Cauchy in quite a different manner.

If the region of resonance is in the infra-red, we may express the

In the case of many substances Cauchy's formula does not give a sufficient representation of the actual dispersion without the addition of a negative term proportional to λ2. This fact which has been clearly established by Ketteler suggests that though the dispersion in the visible part is mainly regulated by ultra-violet resonance, it is also to some extent influenced by free periods lying in the infra-red. Assuming for the sake of simplicity one infra-red and one ultra-violet free period, having wave-lengths A, and A, respectively, equation (48) becomes

This equation has been tested over a long range of wave-lengths for rock-salt, sylvin and fluorspar, and the agreement arrived at is sufficient to show that in its essential points, the present theory is correct, and that refraction is a consequence of the forced vibration of the molecules, which respond strongly to the periodic impulses of those waves which are in sympathy with its periods of free vibration. These experiments, which are fundamental in the theory

of refraction, have been made possible by the beautiful device of H. Rubens and E. F. Nichols*, which enabled them to obtain fairly homogeneous radiations of large wave-lengths by multiple reflexion. The success of the method itself is an excellent confirmation of the above theory.

As will appear in the next article, a substance will totally reflect the radiations having periods equal to that of the free vibrations of the molecules. By successive reflexion from a number of surfaces, all wave-lengths are eliminated except those for which there is approximately total reflexion. It was found in this manner that with quartz, wave-lengths of 20.75 and 8.25 mikroms, and with fluorspar a wave-length of 237, could be obtained.

The refractive index of quartz is represented with considerable accuracy by the formula

με K' +

Mr

Mr

Ms

λ - λ 2 λ - λε λ- λο

in which A, and λ, are directly determined by observation.

Rubens and Nichols also determined the refractive indices of rock-salt and sylvin for the wave-lengths 20'75μ and 8'85μ and hence could deduce an equation to represent the dispersion of these two substances through a wide range. In the following Table, I have collected the constants of the substances used by the authors, adding Paschen's† numbers for fluorspar.

As

All wave-lengths are given in mikroms, i.e. in 10-4 cms. has been stated, the resonance periods of quartz have been derived from observation, the others are calculated from the dispersion

*Wied. Ann. LX. p. 418 (1897).

+ Wied. Ann. LIII. p. 812.

formula. A good confirmation was subsequently obtained by Rubens and Aschkinass* in the direct determination of the resonance region in rock-salt and sylvin, though the observed free periods were not found to coincide as much as might have been wished with those derived from the dispersion formula. The wave-lengths for total reflexion were measured to be 512 and 611 instead of 56'1 and 67.2 as given in the table. There is still less agreement in the case of fluorspar, the wave-length best fitting the observation being 35:47, while the region of total reflexion lies at 237. The discrepancy may be due to the fact that as in quartz, fluorspar has a second region of total reflexion in the infra-red.

The next remark called for by the inspection of the table is connected with the relative small values of the constants M in the ultra-violet term. This must be due to the comparative smallness of the resonance for short periods. There is a gradual increase of the value of M for diminishing values of the resonance period. This increase is not very uniform, but is such that in general it is more rapid than the increase in the square of the wave-length at which resonance takes place. A closer investigation of this point seems called for, but it would be necessary for the purpose to take account of the 'molecular volumes of the different substances. If the refractive index for infinitely short waves is one, as required by (46), equation (48) shows that the constants should satisfy the condition

This relation is only approximately fulfilled by the numbers given in the Table, but its complete verification was not to be expected considering that there are probably unknown regions of resonance in the ultra-violet.

The constant K' should, according to theory, be equal to the specific inductive capacity of the medium; the last two rows of the table show that though the present agreement is not by any means perfect, its power to represent the facts is a considerable stage in advance of the older theories which gave Cauchy's formula. (Compare Art. 136.)

151. Extension of the theory. Our theory requires extension in two directions. Sellmeyer's equation

gives infinitely large values of μ whenever the period of the incident

* Wied. Ann. LXV. p. 241 (1898).

light coincides with one of the free periods of vibration. This is a consequence of the infinite amplitude of the forced vibration, as it appears e.g. in equation (22a).

These infinite amplitudes may be avoided by the introduction of a frictional term retarding the free vibrations as in the case treated in Art. 144. Real friction is not admissible in the treatment of molecular vibrations, but as there is loss of energy due to radiation, there must be some retarding force, which is in phase with the velocity. effect will therefore be the same as that of a frictional force. the constants determining the wave velocity in this case we apply equation (20). Confining ourselves for simplicity to a single variable, we may write for the displacement of the electrons in the molecule

Its To find

where RR, cos wt represents the electric force due to the incident light and p has the same meaning as in Art. 149.

Introducing R in place of R, the equation becomes

Hence (39) and (40) become, taking account of (12), and putting C=0,

If Ko

be small, so that 2 may be neglected, v, becomes equal to the refractive index of the substance, which then refracts according to the sine law. The introduction of k has got rid of the infinite value of v。, but the value of k will be shown in Chapter XIII. to be too small to be the cause of the observed absorption phenomena.

152. Finite range of Free Vibrations. In the case of solids and liquids, we may judge by their absorption effects that the free vibrations are not confined to definite wave-lengths, but extend over a finite range. In order to extend the theory so as to include this case we neglect k and we write ẞdn in (42) for 4πe NV2/p. We then find the total effect of the forced vibration by integrating over the absorption range, thus:

[n2 Bdn
n2 - w2'

Assuming it to be constant, we find

where the absolute value of the fraction, the logarithm of which occurs in the expression, is to be taken. The square of the refractive index is infinitely large on the positive side for wn, and infinitely large on the negative side for w = n2, n2 belonging to the higher frequency. The region of w for which μ2 is negative includes that range of waves which cannot enter the medium. It is bounded on the red side by the value of o for which

ய

and on the side of higher frequency by the value of for which

The infinity of the refractive index at the lower frequency edge of the absorption is avoided if the intensity of the absorption band is assumed to diminish gradually to zero on both sides instead of beginning and ending abruptly. As a simple example we may take the case that the absorption between n1 and n, is equal to ẞ(n − nı) (n − n). The expression for μ2-1 then becomes

In the second term the sign has to be chosen so as to give always