Page:Elektrische und Optische Erscheinungen (Lorentz) 099.jpg

The magnitude $U$ is, what we usually call the velocity of the light ray.

If we now pass to the corresponding motion in the progressing body, then the considered line remains (§ 60, b) a light ray, and we obtain for the determination of the deviations of equilibrium, in the different points of it, expressions as

$A\ \cos {\frac {2\pi }{T}}\left(t'-{\frac {s}{U}}+B'\right)$.

or, by (34) and (85),

$A\ \cos {\frac {2\pi }{T}}\left(t-{\frac {{\mathfrak {p}}_{s}s}{V^{2}}}-{\frac {s}{U}}+B''\right)$,

(86)

where ${\mathfrak {p}}_{s}$ is the component of ${\mathfrak {p}}$ in the direction of the light ray, while the new constant $B''$ has the value

$B'-{\frac {{\mathfrak {p}}_{x}x_{0}+{\mathfrak {p}}_{y}y_{0}+{\mathfrak {p}}_{z}z_{0}}{V^{2}}}$

The expression (86) goes over into

$A\ \cos {\frac {2\pi }{T}}\left(t'-{\frac {s}{U'}}+B''\right)$.

and here, $U'$ is the velocity of the light ray in the moving body, when we put

${\frac {1}{U'}}={\frac {1}{U}}+{\frac {{\mathfrak {p}}_{s}}{V^{2}}}$

From that we conclude

$U'=U-{\mathfrak {p}}_{s}{\frac {U^{2}}{V^{2}}}$,

(87)

a formula, whose shape agrees with (82), in which $U$ and $U'$ are now related to light rays of the same direction.

§ 71. Formula (84) has found a nice confirmation by the experiment, that were first executed by Fizeau and later