Page:AbrahamMinkowski1.djvu/7

From the momentum equations (8), we find the expression for the performance of work of the ponderomotive force

${\begin{array}{ll}{\mathfrak {mK}}=-{\mathfrak {w}}{\frac {\delta {\mathfrak {g}}}{\delta t}}&+{\frac {\partial }{\partial x}}\left({\mathfrak {w}}_{x}X'_{x}+{\mathfrak {w}}_{y}Y'_{x}+{\mathfrak {w}}_{z}Z'_{x}\right)\\\\&+{\frac {\partial }{\partial y}}\left({\mathfrak {w}}_{x}X'_{y}+{\mathfrak {w}}_{y}Y'_{y}+{\mathfrak {w}}_{z}Z'_{y}\right)\\\\&+{\frac {\partial }{\partial z}}\left({\mathfrak {w}}_{x}X'_{z}+{\mathfrak {w}}_{y}Y'_{z}+{\mathfrak {w}}_{z}Z'_{z}\right)\end{array}}$ ${\begin{array}{r}-\left\{X'_{x}{\frac {\partial {\mathfrak {w}}_{x}}{\partial x}}+Y'_{x}{\frac {\partial {\mathfrak {w}}_{y}}{\partial x}}+Z'_{x}{\frac {\partial {\mathfrak {w}}_{z}}{\partial x}}+X'_{y}{\frac {\partial {\mathfrak {w}}_{x}}{\partial y}}+Y'_{y}{\frac {\partial {\mathfrak {w}}_{y}}{\partial y}}+Z'_{y}{\frac {\partial {\mathfrak {w}}_{z}}{\partial y}}\right.\\\\\left.+X'_{z}{\frac {\partial {\mathfrak {w}}_{x}}{\partial z}}+Y'_{z}{\frac {\partial {\mathfrak {w}}_{y}}{\partial z}}+Z'_{z}{\frac {\partial {\mathfrak {w}}_{z}}{\partial z}}\right\}\end{array}}$

If we set here for abbreviation

(13)

${\begin{cases}P'&=X'_{x}{\frac {\partial {\mathfrak {w}}_{x}}{\partial x}}+X'_{y}{\frac {\partial {\mathfrak {w}}_{x}}{\partial y}}+X'_{z}{\frac {\partial {\mathfrak {w}}_{x}}{\partial z}}\\\\&+Y'_{x}{\frac {\partial {\mathfrak {w}}_{y}}{\partial x}}+Y'_{y}{\frac {\partial {\mathfrak {w}}_{y}}{\partial y}}+Y'_{y}{\frac {\partial {\mathfrak {w}}_{y}}{\partial z}}\\\\&+Z'_{x}{\frac {\partial {\mathfrak {w}}_{z}}{\partial x}}+Z'_{y}{\frac {\partial {\mathfrak {w}}_{z}}{\partial y}}+Z'_{z}{\frac {\partial {\mathfrak {w}}_{z}}{\partial z}}\end{cases}}$

then the energy equation (9) gives with respect to (12)

(14)	$Q+\mathrm {div} {\mathfrak {S}}'=-{\frac {\delta \psi }{\delta t}}+{\mathfrak {w}}{\frac {\delta {\mathfrak {g}}}{\delta t}}+P'$

This relation gained from the momentum and energy theorem, will prove itself to be still important.

§ 4. The main equations.

Common to all theories of electrodynamics of moving bodies, is the form of the two first main-equations

(I)	$c\ \mathrm {curl{\mathfrak {H}}'={\frac {\partial '{\mathfrak {D}}}{\partial t}}+{\mathfrak {J}},}$

(II)	$c\ \mathrm {curl{\mathfrak {E}}'=-{\frac {\partial '{\mathfrak {B}}}{\partial t}}.}$

They are nothing else than a general scheme, which obtains a physical sense only by addition of two relations between the four arising vectors; and two such relations are necessary, to reduce the number of unknown vectors to two; the temporal change of the field of these two vectors, is then described by the two first main-equations.