Page:Elektrische und Optische Erscheinungen (Lorentz) 105.jpg

with the condition

$lf+mg+nh=0$.

(95)

We can easily see, that these values satisfy all equations of motion. The vectors ${\mathfrak {d}}$ and ${\mathfrak {H}}$ are perpendicular to one another and to the wave normal; the direction of the light rays (§ 60, b) falls into the latter.

§ 75. If the Earth is moving, then by the theorem of § 59 a condition is possible, which (related to a moving coordinate system), will be represented by

${\mathfrak {d}}'_{x}=af\ \cos \psi ',\ {\mathfrak {d}}'_{y}=ag\ \cos \psi ',\ {\mathfrak {d}}'_{z}=ah\ \cos \psi '$,

(96)

${\mathfrak {H}}'_{x}=4\pi aV(mh-ng)\cos \psi ',\ {\mathfrak {H}}'_{y}=4\pi aV(nf-lh)\cos \psi '$, ${\mathfrak {H}}'_{z}=4\pi aV(lg-mf)\cos \psi '$,

(97)

$\psi '={\frac {2\pi }{T}}\left(t-{\frac {{\mathfrak {p}}_{x}x+{\mathfrak {p}}_{y}y+{\mathfrak {p}}_{z}z}{V^{2}}}-{\frac {lx+my-nz}{V}}+q\right)$,

(98)

By ${\mathfrak {d}}'$ we have to understand a vector ${\mathfrak {D}}'$ for the pure aether, which is defined by (IX) (§ 56).

While the light rays, which determine the lateral limitation of the bundle, have still the direction (l, m, n), the wave normal deviates from it. Its direction constants l', m', n' satisfy, as it can be seen from (98), the conditions

$l':m':n'=\left(l+{\frac {{\mathfrak {p}}_{x}}{V}}\right):\left(m+{\frac {{\mathfrak {p}}_{y}}{V}}\right):\left(n+{\frac {{\mathfrak {p}}_{z}}{V}}\right)$.

We will neglect all magnitudes of second order again. Then, by denoting the components of ${\mathfrak {p}}$ in the direction of the rays by ${\mathfrak {p}}_{s}$, we have

$l'\left(1+{\frac {{\mathfrak {p}}_{s}}{V}}\right)=l+{\frac {{\mathfrak {p}}_{x}}{V}}$, etc.

(99)

by which (98) is transformed into

$\psi '={\frac {2\pi }{T}}\left\{t-\left(1+{\frac {{\mathfrak {p}}_{s}}{V}}\right){\frac {l'x+m'y-n'z}{V}}+q\right\}$.

While T is now the relative oscillation period, we find for the absolute one (§§ 60 and 37)

$T'=T\left(1-{\frac {{\mathfrak {p}}_{s}}{V}}\right)$.