Page:Elektrische und Optische Erscheinungen (Lorentz) 137.jpg

§ 102. Let, with respect to the previously introduced axes,

$\cos \alpha ,\ 0,\ \sin \alpha$

be the direction constants of the rays incident on the plate. Neglecting magnitudes of second order, we consequently obtain the direction of the wave normal by application of the fundamental law of aberration; namely we have to compose a velocity V in the direction of the rays with a translational velocity ${\mathfrak {p}}$. Now, if the latter is parallel to the z-axis, then the direction constants of the wave normal become,

$\cos \alpha ',\ 0,\ \sin \alpha '$

where

$\alpha '=\alpha +{\frac {{\mathfrak {p}}_{z}}{V}}\cos \alpha$

The absolute velocity of the waves is V; however, the relative velocity $V'$ will be found, when we diminish V by the component of ${\mathfrak {p}}$ with respect to the wave normal. If we understand by x, y, z relative coordinates, then for the incident light, expressions of the form

$A\ \cos {\frac {2\pi }{T}}\left(t-{\frac {x\ \cos \alpha '+z\ \sin \alpha '}{V'}}+B\right)$

apply, or

$A\ \cos {\frac {2\pi }{T}}\left(t-{\frac {x\ \cos \alpha +z\ \sin \alpha }{V}}-{\frac {{\mathfrak {p}}_{z}z}{V^{2}}}+B\right)$

(126)

On the other hand, for glass we have to assume Fresnel's dragging coefficient. Consequently, when we denote the propagation velocity in stationary glass by W, and the directions constants of the wave normal in the plate by

$\cos \beta ,\ 0,\ \sin \beta$

we have to put for the relative velocity of the waves with respect to glass, by (82),

$W'=W-{\mathfrak {p}}_{x}\sin \beta {\frac {W^{2}}{V^{2}}}$

(127)

To light in the plate an expression applies now, which has the form: