Page:Elektrische und Optische Erscheinungen (Lorentz) 093.jpg

${\frac {1}{r_{1}}}f_{1}\left(t'-{\frac {r_{1}}{V}}\right)$ und ${\frac {1}{r_{2}}}f_{2}\left(t'-{\frac {r_{2}}{V}}\right)$,

when $r_{1}$ and $r_{2}$ are the distances between (x, y, z) and the two first molecules.

§ 65. From the preceding formulas, others immediately arise, which apply to a stationary light source when we simply erase all accents. If in this case in the luminous molecules the moments exist

$\left.{\begin{array}{c}{\mathfrak {m}}_{x(1)}=f_{1}(t),\ {\mathfrak {m}}_{y(1)}=g_{1}(t),\ {\mathfrak {m}}_{z(1)}=h_{1}(t),\ \\{\mathfrak {m}}_{x(2)}=f_{2}(t),\ {\mathfrak {m}}_{y(2)}=g_{2}(t),\ {\mathfrak {m}}_{z(2)}=h_{2}(t),\ \\\mathrm {etc.} ,\end{array}}\right\}$

(75)

then we have in the aether

$\left.{\begin{array}{c}{\mathfrak {H}}{}_{x}={\frac {\partial }{\partial t}}{\frac {\partial }{\partial y}}\left\{\sum \left({\frac {{\mathfrak {m}}_{z}}{r}}\right)\right\}-{\frac {\partial }{\partial t}}{\frac {\partial }{\partial z}}\left\{\sum \left({\frac {{\mathfrak {m}}_{y}}{r}}\right)\right\},\ \mathrm {etc.} ,\\4\pi {\mathfrak {D}}{}_{x}={\frac {\partial S}{\partial x}}-\Delta \left\{\sum \left({\frac {{\mathfrak {m}}_{x}}{r}}\right)\right\},\ \mathrm {etc.} ,\\S={\frac {\partial }{\partial x}}\left\{\sum \left({\frac {{\mathfrak {m}}_{x}}{r}}\right)\right\}+{\frac {\partial }{\partial y}}\left\{\sum \left({\frac {{\mathfrak {m}}_{y}}{r}}\right)\right\}+{\frac {\partial }{\partial z}}\left\{\sum \left({\frac {{\mathfrak {m}}_{z}}{r}}\right)\right\}.\end{array}}\right\}$

(76)

where ${\mathfrak {m}}_{x},{\mathfrak {m}}_{y},{\mathfrak {m}}_{z}$ are now the moments of a molecule at time $t-{\frac {r}{V}}$, so that e.g. the two first members of the sum

$\sum \left({\frac {{\mathfrak {m}}_{x}}{r}}\right)$

have the values

${\frac {1}{r_{1}}}f_{1}\left(t-{\frac {r_{1}}{V}}\right)$ and ${\frac {1}{r_{2}}}f_{2}\left(t-{\frac {r_{2}}{V}}\right)$

Of course, $\xi ,\eta ,\zeta$, x, y, z are now the coordinates related to stationary axes.

§ 66. The two cases considered in §§ 64 and 65 (with or without translation) shall be compared to one another. Here, we imagine that the spatial arrangement of the luminous molecules is the same in the two cases, i.e. that all $\xi ,\eta ,\zeta$ have the same value; the latter we also assume for x, y, z, with the result that we consider the state of the aether in a point