Page:Elektrische und Optische Erscheinungen (Lorentz) 107.jpg

Here, we only consider such ones, that are again propagating in the aether, and we represent (for the case that the earth is at rest) one of the bundles, which emerge from the incident motion considered in § 74, by the following formula

${\mathfrak {d}}_{x(1)}=a_{1}f_{1}\ \cos \psi _{1},\ {\mathfrak {d}}_{y(1)}=a_{1}g_{1}\ \cos \psi _{1},\ {\mathfrak {d}}_{z(1)}=a_{1}h_{1}\ \cos \psi _{1}$, ${\mathfrak {H}}_{x(1)}=4\pi a_{1}V(m_{1}h_{1}-n_{1}g_{1})\cos \psi _{1},\ etc.,$ $\psi _{1}={\frac {2\pi }{T}}\left(t-{\frac {l_{1}x+m_{1}y-n_{1}z}{V}}+q_{1}\right)$.

§ 77. With this motion only that corresponds, which (in case Earth is moving together with the reflecting or refracting body) emerges from the light represented by (96)-(98). For this new state of motion we can thus write

${\mathfrak {d}}'_{x(1)}=a_{1}f_{1}\ \cos \psi '_{1},\ {\mathfrak {d}}'_{y(1)}=a_{1}g_{1}\ \cos \psi '_{1},\ {\mathfrak {d}}'_{z(1)}=a_{1}h_{1}\ \cos \psi '_{1}$, ${\mathfrak {H}}'_{x(1)}=4\pi a_{1}V(m_{1}h_{1}-n_{1}g_{1})\cos \psi '_{1}$, etc., $\psi '_{1}={\frac {2\pi }{T}}\left(t-{\frac {{\mathfrak {p}}_{x}x+{\mathfrak {p}}_{y}y+{\mathfrak {p}}_{z}z}{V^{2}}}-{\frac {l_{1}x+m_{1}y-n_{1}z}{V}}+q_{1}\right)$,

from which it again follows — see (100) and (101) —

${\mathfrak {d}}_{x(1)}=a_{1}\left\{f_{1}-{\frac {{\mathfrak {p}}_{y}}{V}}(l_{1}g_{1}-m_{1}f_{1})+{\frac {{\mathfrak {p}}_{z}}{V}}(n_{1}f_{1}-l_{1}h_{1})\right\}\cos \psi '_{1}$, etc., ${\mathfrak {H}}_{x(1)}=4\pi a_{1}\left\{V(m_{1}h_{1}-n_{1}g_{1})+({\mathfrak {p}}_{y}h_{1}-{\mathfrak {p}}_{z}g_{1})\right\}\cos \psi '_{1}$, etc.

In these equations $l_{1},m_{1},n_{1}$ determine the direction of the rays, which we also want to denote by $s_{1}$.

§ 78. In the course of mirroring or refraction, the absolute period will be changed in general, while, as it nearly goes without saying and as it is also expressed by our formulas, the relative period is the same for all relevant light bundles. The absolute period of the incident motion is (§ 75)

$T\left(1-{\frac {{\mathfrak {p}}_{s}}{V}}\right)$.

Also, as regards the bundle considered in the previous paragraph it becomes

$T\left(1-{\frac {p_{s1}}{V}}\right)$.