Page:Grundgleichungen (Minkowski).djvu/45

(92)

$N_{h}=-{\frac {1}{2}}\Phi {\overline {\Phi }}{\frac {\partial \epsilon }{\partial x_{h}}}-{\frac {1}{2}}\Psi {\overline {\Psi }}{\frac {\partial \mu }{\partial x_{h}}}$ $+(\epsilon \mu -1)\left(\Omega _{1}{\frac {\partial w_{1}}{\partial x_{h}}}+\Omega _{2}{\frac {\partial w_{2}}{\partial x_{h}}}+\Omega _{3}{\frac {\partial w_{3}}{\partial x_{h}}}+\Omega _{4}{\frac {\partial w_{4}}{\partial x_{h}}}\right)$

for $h=1,2,3,4$.

Now if we make use of (59), and denote the space-vector which has $\Omega _{1},\ \Omega _{2},\ \Omega _{3}$ as the x-, y-, z-components by the symbol ${\mathfrak {W}}$, then the third component of 92) can be expressed in the form

(93)	${\frac {\epsilon \mu -1}{\sqrt {1-{\mathfrak {w}}^{2}}}}\left({\mathfrak {W}}{\frac {d{\mathfrak {w}}}{dx_{h}}}\right)$

The round bracket denoting the scalar product of the vectors within it.

§ 14. The Ponderomotive Force.

Let us now write out the relation $K=lor\ S=-sF+N$ in a more practical form; we have the four equations

(94)

$K_{1}={\frac {\partial X_{x}}{\partial x}}+{\frac {\partial X_{y}}{\partial y}}+{\frac {\partial X_{z}}{\partial z}}-{\frac {\partial X_{t}}{\partial t}}=\varrho {\mathfrak {E}}_{x}+{\mathfrak {s}}_{y}{\mathfrak {M}}_{z}-{\mathfrak {s}}_{z}{\mathfrak {M}}_{y}$ $-{\frac {1}{2}}\Phi {\overline {\Phi }}{\frac {\partial \epsilon }{\partial x}}-{\frac {1}{2}}\Psi {\overline {\Psi }}{\frac {\partial \mu }{\partial x}}+{\frac {\epsilon \mu -1}{\sqrt {1-{\mathfrak {w}}^{2}}}}\left({\mathfrak {W}}{\frac {\partial {\mathfrak {w}}}{\partial x}}\right)$,

(95)

$K_{2}={\frac {\partial Y_{x}}{\partial x}}+{\frac {\partial Y_{y}}{\partial y}}+{\frac {\partial Y_{z}}{\partial z}}-{\frac {\partial Y_{t}}{\partial t}}=\varrho {\mathfrak {E}}_{y}+{\mathfrak {s}}_{z}{\mathfrak {M}}_{x}-{\mathfrak {s}}_{x}{\mathfrak {M}}_{z}$ $-{\frac {1}{2}}\Phi {\overline {\Phi }}{\frac {\partial \epsilon }{\partial y}}-{\frac {1}{2}}\Psi {\overline {\Psi }}{\frac {\partial \mu }{\partial y}}+{\frac {\epsilon \mu -1}{\sqrt {1-{\mathfrak {w}}^{2}}}}\left({\mathfrak {W}}{\frac {\partial {\mathfrak {w}}}{\partial y}}\right)$,

(96)

$K_{3}={\frac {\partial Z_{x}}{\partial x}}+{\frac {\partial Z_{y}}{\partial y}}+{\frac {\partial Z_{z}}{\partial z}}-{\frac {\partial Z_{t}}{\partial t}}=\varrho {\mathfrak {E}}_{z}+{\mathfrak {s}}_{x}{\mathfrak {M}}_{y}-{\mathfrak {s}}_{y}{\mathfrak {M}}_{x}$ $-{\frac {1}{2}}\Phi {\overline {\Phi }}{\frac {\partial \epsilon }{\partial z}}-{\frac {1}{2}}\Psi {\overline {\Psi }}{\frac {\partial \mu }{\partial z}}+{\frac {\epsilon \mu -1}{\sqrt {1-{\mathfrak {w}}^{2}}}}\left({\mathfrak {W}}{\frac {\partial {\mathfrak {w}}}{\partial z}}\right)$,

(97)

${\frac {1}{i}}K_{4}=-{\frac {\partial T_{x}}{\partial x}}-{\frac {\partial T_{y}}{\partial y}}-{\frac {\partial T_{z}}{\partial z}}-{\frac {\partial T_{t}}{\partial t}}={\mathfrak {s}}_{x}{\mathfrak {E}}_{x}+{\mathfrak {s}}_{y}{\mathfrak {E}}_{y}+{\mathfrak {s}}_{z}{\mathfrak {E}}_{z}$ $+{\frac {1}{2}}\Phi {\overline {\Phi }}{\frac {\partial \epsilon }{\partial t}}-{\frac {1}{2}}\Psi {\overline {\Psi }}{\frac {\partial \mu }{\partial t}}+{\frac {\epsilon \mu -1}{\sqrt {1-{\mathfrak {w}}^{2}}}}\left({\mathfrak {W}}{\frac {\partial {\mathfrak {w}}}{\partial t}}\right)$.

It is my opinion that when we calculate the ponderomotive force which acts upon a unit volume at the space-time point x,y,z,t, it has got x-, y-, z-