Page:Elektrische und Optische Erscheinungen (Lorentz) 058.jpg

If the law, as it is usually applied, should be given, it must of course still be assumed, that the translation does not change the actual period of oscillation of the luminous particles. I must abstain from giving an account of this hypothesis, since we know nothing about nature of the molecular forces that determine the oscillation period.

§ 38. The case that the light source is at rest and the observer progresses, allows of a similar treatment. If namely, as above, $\mathbf {x}$, $\mathbf {y}$, $\mathbf {z}$ are the coordinates based on stationary axes, then in a distant point P, any of the magnitudes ${\mathfrak {d}}_{x},...{\mathfrak {H}}_{x},...$ shall now be represented by

$A\cos {\frac {2\pi }{T}}\left\{t-{\frac {b_{x}\mathbf {x} +b_{y}\mathbf {y} +b_{z}\mathbf {z} }{V}}+C\right\}.$

(46)

We most conveniently describe the perception of motion by means of a co-ordinate system, which shares the translation ${\mathfrak {p}}$ of the observer. Here, again the relations (44) are applicable, and (46) transforms into

$A\cos {\frac {2\pi }{T}}\left\{t-{\frac {{\mathfrak {p}}_{r}}{V}}t-{\frac {b_{x}x+b_{y}y+b_{z}z}{V}}+C\right\}{,}$

from which it is given for the "observed" period of oscillation

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