Page:AbrahamMinkowski1.djvu/7

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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.