(66) |
$\left|{\frac {\partial L}{\partial x_{1}}},\ {\frac {\partial L}{\partial x_{2}}},\ {\frac {\partial L}{\partial x_{3}}},\ {\frac {\partial L}{\partial x_{4}}}\right|$ |

If $s=\left|s_{1},\ s_{2},\ s_{3},\ s_{4}\right|$ is a space-time vector of the 1st kind, then

(67) |
$lor\ {\bar {s}}={\frac {\partial s_{1}}{\partial x_{1}}}+{\frac {\partial s_{2}}{\partial x_{2}}}+{\frac {\partial s_{3}}{\partial x_{3}}}+{\frac {\partial s_{4}}{\partial x_{4}}}$ |

In case of a Lorentz transformation ${\mathsf {A}}$, we have

$lor'\ {\bar {s}}'=(lor\ {\mathsf {A}})({\mathsf {\bar {A}}}{\bar {s}})=lor\ {\bar {s}}$,
*i.e.*, *lor s* *is an invariant in a {sc|Lorentz}}-transformation.*

*In all these operations the operator* lor *plays the part of a space-time vector of the first kind.*

If *f* represents a space-time vector of the second kind, *-lor f* denotes a space-time vector of the first kind with the components

${\begin{array}{ccccccc}&&{\frac {\partial f_{12}}{\partial x_{2}}}&+&{\frac {\partial f_{13}}{\partial x_{3}}}&+&{\frac {\partial f_{14}}{\partial x_{4}}},\\\\{\frac {\partial f_{21}}{\partial x_{1}}}&&&+&{\frac {\partial _{23}}{\partial x_{3}}}&+&{\frac {\partial _{24}}{\partial x_{4}}},\\\\{\frac {\partial f_{31}}{\partial x_{1}}}&+&{\frac {\partial _{32}}{\partial x_{2}}}&&&+&{\frac {\partial _{34}}{\partial x_{4}}},\\\\{\frac {\partial f_{41}}{\partial x_{1}}}&+&{\frac {\partial _{42}}{\partial x_{2}}}&+&{\frac {\partial _{43}}{\partial x_{3}}},\end{array}}$
So the system o£ differential equations (A) can be expressed in the concise form

{A} |
$lor\ f=-s$ |

and the system (B) can be expressed in the form

{B} |
$lor\ F^{*}=0$ |

Referring back to the definition (67) for $lor\ {\bar {s}}$, we find that the combinations $lor({\overline {lor\ f}})$ and $lor({\overline {lor\ F^{*}}})$ vanish identically, when *f* and *F** are alternating matrices. Accordingly it follows out of (A), that

(68) |
${\frac {\partial s_{1}}{\partial x_{1}}}+{\frac {\partial s_{2}}{\partial x_{2}}}+{\frac {\partial s_{3}}{\partial x_{3}}}+{\frac {\partial s_{4}}{\partial x_{4}}}=0$, |