Page:Elektrische und Optische Erscheinungen (Lorentz) 092.jpg

or

$\left.{\begin{array}{c}{\mathfrak {m}}_{x(1)}=f_{1}\left(t-{\frac {{\mathfrak {p}}_{x}}{V^{2}}}\xi _{1}-{\frac {{\mathfrak {p}}_{y}}{V^{2}}}\eta _{1}-{\frac {{\mathfrak {p}}_{z}}{V^{2}}}\zeta _{1}\right),\ \mathrm {etc.} ,\\{\mathfrak {m}}_{x(2)}=f_{2}\left(t-{\frac {{\mathfrak {p}}_{x}}{V^{2}}}\xi _{2}-{\frac {{\mathfrak {p}}_{y}}{V^{2}}}\eta _{2}-{\frac {{\mathfrak {p}}_{z}}{V^{2}}}\zeta _{2}\right),\ \mathrm {etc.} ,\\\mathrm {etc.} \end{array}}\right\}$

(73)

be the electric moments that occur within.

The condition that was caused by a single molecule in the point (x,y,z) of the aether, will be determined by equations (39) and (40). The latter one we additionally want to transform (to subsequently apply the theorem of § 59 more conveniently) by introducing the expressions ${\mathfrak {D}}$ and ${\mathfrak {D}}'$ for the aether. For this medium, as we know, ${\mathfrak {D}}$ is equal to ${\mathfrak {d}}$, and thus by (IX) (§ 56), $4\pi V^{2}{\mathfrak {D}}'$ is equal to

$4\pi V^{2}{\mathfrak {d}}+[{\mathfrak {p.H}}]$.

By means of equation $V_{b}$) we may replace ${\mathfrak {F}}$ by $4\pi V^{2}{\mathfrak {D}}'$ in (40).

Furthermore, if we denote by $\Sigma$ the sum of terms, any of them stemming from a luminous molecule, then we obtain from (39) and (40) the following formulas for the condition in the aether caused by ion oscillations (72):

$\left.{\begin{array}{c}{\mathfrak {H}}'_{x}={\frac {\partial }{\partial t'}}\left({\frac {\partial }{\partial y}}\right)^{'}\left\{\sum \left({\frac {m_{z}}{r}}\right)\right\}-{\frac {\partial }{\partial t'}}\left({\frac {\partial }{\partial z}}\right)^{'}\left\{\sum \left({\frac {{\mathfrak {m}}_{y}}{r}}\right)\right\},\ \mathrm {etc.} ,\\\\4\pi {\mathfrak {D}}'_{x}=\left({\frac {\partial S}{\partial x}}\right)^{'}-\Delta '\left\{\sum \left({\frac {{\mathfrak {m}}_{x}}{r}}\right)\right\},\ \mathrm {etc.} ,\\\\S=\left({\frac {\partial }{\partial x}}\right)^{'}\left\{\sum \left({\frac {{\mathfrak {m}}_{x}}{r}}\right)\right\}+\left({\frac {\partial }{\partial y}}\right)^{'}\left\{\sum \left({\frac {{\mathfrak {m}}_{y}}{r}}\right)\right\}+\left({\frac {\partial }{\partial z}}\right)^{'}\left\{\sum \left({\frac {{\mathfrak {m}}_{z}}{r}}\right)\right\}.\end{array}}\right\}$

(74)

Here, r denotes the distance of point (x, y, z) from the location ($\xi ,\eta ,\zeta$) of one of the luminous molecules, while ${\mathfrak {m}}_{x},{\mathfrak {m}}_{y},{\mathfrak {m}}_{z}$ represent the moments of this molecule at local time $t'-{\frac {r}{V}}$. The two first members of the sum

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

are for example