Page:Elektrische und Optische Erscheinungen (Lorentz) 054.jpg

dimensions of the molecule, we have committed an error of order $\textstyle {\frac {l}{V}}$, secondly, the inequality of the local times at the various locations of the molecule were not considered, and in this lies an error of order $\textstyle {\frac {l{\mathfrak {p}}}{V^{2}}}$ by (34). But even then, if we want to keep magnitudes of the order $\textstyle {\frac {\mathfrak {p}}{V}}$, we don't need to care about this second error, when already the first may be neglected. Now this is indeed the case when the dimensions of the molecule are much smaller than the wavelength of TV. Then also l/V is considerably smaller than T, and the state in the molecule will not noticeably change in the time l/V.

§ 35. The formulas for the propagation of oscillations is obtained, if goniometric functions of time are substituted into the equations (39) and (40) for ${\mathfrak {m}}_{x}$, ${\mathfrak {m}}_{y}$ ,${\mathfrak {m}}_{z}$. If, for example,

${\mathfrak {m}}_{y}=0,\ {\mathfrak {m}}_{z}=0{,}$

and, as a function of local time which is valid for the location of the molecule,

${\mathfrak {m}}_{x}=a\cos 2\pi {\frac {t'}{T}}{\text{, (a constant),}}$

thus at an external point in the distance r and for the local time $t'$ that belongs to it

${\mathfrak {H'}}_{x}=0,\ {\mathfrak {H'}}_{y}={\frac {\partial }{\partial t'}}\left({\frac {\partial \chi }{\partial z}}\right)',\ {\mathfrak {H'}}_{z}=-{\frac {\partial }{\partial t'}}\left({\frac {\partial \chi }{\partial y}}\right)'{,}$

${\mathfrak {F}}_{z}=-V^{2}\left({\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}\right)'\chi ,\ {\mathfrak {F}}_{y}=V^{2}\left({\frac {\partial ^{2}\chi }{\partial x\ \partial y}}\right)',\ {\mathfrak {F}}=V^{2}\left({\frac {\partial ^{2}\chi }{\partial x\ \partial z}}\right)'.$

$\chi ={\frac {a}{r}}\cos {\frac {2\pi }{T}}\left(t'-{\frac {r}{V}}\right).$

If we eventually want to consider a stationary light source once, so we simply have to omit all accents. The formulas then are in accordance with the expressions, by which Hertz^[1] represented the oscillations in the vicinity of his vibrator.

1. Hertz. Wied. Ann., Bd. 36, p. 1, 1889.