Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/223

observer, who looks in the direction opposite to that of the propagation of the light,^[1] we have

$\theta ={\frac {\pi }{p}}\left({\frac {1}{{\text{V}}_{\text{L}}}}-{\frac {1}{{\text{V}}_{\text{R}}}}\right)\cdot$

(35)

By the preceding equation, this reduces to

$\theta ={\frac {\pi \phi }{p{\text{V}}_{\text{R}}^{2}{\text{V}}_{\text{L}}^{2}}}\cdot$

(36)

Without any appreciable error, we may substitute ${\text{U}}^{4}$ for ${\text{V}}_{\text{R}}^{2}{\text{V}}_{\text{L}}^{2},$ which will give^[2]

$\theta ={\frac {\pi \phi }{p{\text{U}}^{4}}}\cdot$

(37)

19. Since these equations involve unknown functions of the period they will not serve for an exact determination of the relation between and the period. For a rough approximation, however, we may assume that the manner in which the general displacement in any small part of the medium distributes itself among the molecules and intermolecular spaces is independent of the period, being determined entirely by the values of $\xi ,\eta ,\zeta ,$ and their differential coefficients with respect to the coordinates.^[3] For a fixed direction of the wave-normal, $\Phi$ and $\Phi '$ will then be constant. Now equations (15) and (36) give

$\theta ={\frac {\Phi }{2p^{2}{\text{V}}_{\text{R}}^{2}{\text{V}}_{\text{L}}^{2}}}-{\frac {2\pi ^{2}\Phi '}{p^{4}{\text{V}}_{\text{R}}^{2}{\text{V}}_{\text{L}}^{2}}}\cdot$

(38)

To express this result in terms of the quantities directly observed, we may use the equations

$p={\frac {\lambda }{k}},$${\text{V}}_{\text{R}}={\frac {k}{n_{\text{R}}}},$${\text{V}}_{\text{L}}={\frac {k}{n_{\text{L}}}},$${\text{U}}={\frac {k}{n}},$

where $k$ denotes the velocity of light in vacuo, $\lambda$ the wave-length in vacuo of the light employed, $n_{\text{R}},n_{\text{L}}$ the absolute indices of refraction of the two rays, and $n$ the index for the optic axis as derived from the ellipsoid (24) by Fresnel's law. We thus obtain

$\theta ={\frac {\Phi n_{\text{R}}^{2}n_{\text{L}}^{2}}{2k^{2}\lambda ^{2}}}-{\frac {2\pi ^{2}\Phi 'n_{\text{R}}^{2}n_{\text{L}}^{2}}{\lambda ^{4}}}\cdot$

(39)

1. When the rotation of the plane of polarization appears clockwise to the observer, it has the character of a left-handed screw. But the circularly polarized ray to which ${\text{V}}_{\text{R}}$ relates, the rotation of which also appears clockwise to the observer, has the character of a right-handed screw.
2. The degree of accuracy of this substitution may be shown as follows. By (33)

   ${\text{V}}_{\text{R}}({\text{V}}_{\text{R}}^{2}-{\text{U}}^{2})={\text{V}}_{\text{L}}({\text{U}}^{2}-{\text{V}}_{\text{L}}^{2}),$

   whence

   ${\text{V}}_{\text{R}}^{3}+{\text{V}}_{\text{L}}^{3}=({\text{V}}_{\text{R}}+{\text{V}}_{\text{L}}){\text{U}}^{3},$ ${\text{V}}_{\text{R}}^{2}-{\text{V}}_{\text{R}}{\text{V}}_{\text{L}}+{\text{V}}_{\text{L}}^{2}={\text{U}}^{2},$ ${\text{V}}_{\text{R}}{\text{V}}_{\text{L}}={\text{U}}^{2}({\text{V}}_{\text{R}}-{\text{V}}_{\text{L}})^{2}.$

3. Compare § 12 of the former paper, on page 189 of this volume.