Page:Grundgleichungen (Minkowski).djvu/53

and therefore $\xi _{1},\ \xi _{2},\ \xi _{3},\ \xi _{4}$ are nil. Then by partial integration, the integral is transformed into the form

$\int \int \int \int {\underset {h}{\sum }}\xi _{h}\left({\frac {\partial v\ w_{h}w_{1}}{\partial x_{1}}}+{\frac {\partial \nu \ w_{h}w_{2}}{\partial x_{2}}}+{\frac {\partial \nu \ w_{h}w_{3}}{\partial x_{3}}}+{\frac {\partial \nu \ w_{h}w_{4}}{\partial x_{4}}}\right)dx\ dy\ dz\ dt$.

the expression within the bracket may be written as

$=w_{h}{\underset {k}{\sum }}{\frac {\partial \nu \ w_{k}}{\partial x_{k}}}+\nu {\underset {k}{\sum }}w_{k}{\frac {\partial w_{h}}{\partial x_{k}}}$.

The first sum vanishes in consequence of the continuity equation (6). The second may be written as

${\frac {\partial w_{h}}{\partial x_{1}}}{\frac {dx_{1}}{d\tau }}+{\frac {\partial w_{h}}{\partial x_{2}}}{\frac {dx_{2}}{d\tau }}+{\frac {\partial w_{h}}{\partial x_{3}}}{\frac {dx_{3}}{d\tau }}+{\frac {\partial w_{h}}{\partial x_{4}}}{\frac {dx_{4}}{d\tau }}={\frac {dw_{h}}{d\tau }}={\frac {d}{d\tau }}\left({\frac {dx_{h}}{d\tau }}\right)$

whereby ${\frac {d}{d\tau }}$ is meant the differential quotient in the direction of the space-time line at any position. For the differential quotient (12), we obtain the final expression

(14)	$\int \int \int \int \nu \left({\frac {dw_{1}}{d\tau }}\xi _{1}+{\frac {dw_{2}}{d\tau }}\xi _{2}+{\frac {dw_{3}}{d\tau }}\xi _{3}+{\frac {dw_{4}}{d\tau }}\xi _{4}\right)dx\ dy\ dz\ dt$.

For a virtual displacement in the sickle we have postulated the condition that the points supposed to be substantial shall advance normally to the curves giving their actual motion, which is $\vartheta =0$, this condition denotes that the $\xi _{h}$ is to satisfy the condition

(15)	$w_{1}\xi _{1}+w_{2}\xi _{2}+w_{3}\xi _{3}+w_{4}\xi _{4}=0$

Let us now turn our attention to the Maxwellian tensions in the electrodynamics of stationary bodies, and let us consider the results in §§ 12 and 13; then we find that Hamilton's Principle can be reconciled to the relativity postulate for continuously extended elastic media.

At every space-time point (as in § 18), let a space time matrix of the 2nd kind be known

(16)

$S=\left|{\begin{array}{cccc}S_{11},&S_{12},&S_{13},&S_{14}\\S_{21},&S_{22},&S_{23},&S_{24}\\S_{31},&S_{32},&S_{33},&S_{34}\\S_{41},&S_{42},&S_{43},&S_{44}\end{array}}\right|=\left|{\begin{array}{cccc}X_{x},&Y_{x},&Z_{x},&-iT_{x}\\X_{y},&Y_{y},&Z_{y},&-iT_{y}\\X_{z},&Y_{z},&Z_{z},&-iT_{z}\\-iX_{t},&-iY_{t},&-iZ_{t},&T_{t}\end{array}}\right|$

where $X_{x},Y_{x},\dots Z_{z},\dots T_{z},\dots X_{t},\dots T_{t}$ are real magnitudes.