Page:Becquerel - Le Principe de relativité et la théorie de la gravitation, 1922.djvu/266

246

deuxième partie. — la relativité généralisée.

Les équations (1-15) s’écrivent, avec cette notation,

(3-15)

$\left\{{\begin{aligned}\mathrm {X} &={\frac {\partial \varphi _{1}}{\partial {x_{4}}}}-{\frac {\partial \varphi _{4}}{\partial {x_{1}}}},&\mathrm {L} &={\frac {\partial \varphi _{2}}{\partial {x_{3}}}}-{\frac {\partial \varphi _{3}}{\partial {x_{2}}}},\\\mathrm {Y} &={\frac {\partial \varphi _{2}}{\partial {x_{4}}}}-{\frac {\partial \varphi _{4}}{\partial {x_{2}}}},&\mathrm {M} &={\frac {\partial \varphi _{3}}{\partial {x_{1}}}}-{\frac {\partial \varphi _{1}}{\partial {x_{3}}}},\\\mathrm {Z} &={\frac {\partial \varphi _{3}}{\partial {x_{4}}}}-{\frac {\partial \varphi _{4}}{\partial {x_{3}}}},&\mathrm {N} &={\frac {\partial \varphi _{1}}{\partial {x_{2}}}}-{\frac {\partial \varphi _{2}}{\partial {x_{1}}}}\cdot \end{aligned}}\right.$

Écrivons maintenant les équations de Maxwell-Lorentz. L’unité de charge étant choisie de manière que le facteur $4\pi$ disparaisse (système d’Heaviside-Lorentz), soient $u,$ $v,$ $w$ les composantes de la densité de courant (unités électromagnétiques) et $\mathrm {P}$ la densité de charge (unités électrostatiques). Les équations bien connues sont les suivantes :

$-{\frac {1}{c}}{\frac {\partial \mathrm {L} }{\partial t}}={\frac {\partial \mathrm {Z} }{\partial y}}-{\frac {\partial \mathrm {Y} }{\partial z}}\,;$		${\frac {1}{c}}{\frac {\partial \mathrm {X} }{\partial t}}+u={\frac {\partial \mathrm {N} }{\partial y}}-{\frac {\partial \mathrm {M} }{\partial z}},$
...............,		.................,
$\operatorname {Div} \,(\mathrm {L} ,\mathrm {M} ,\mathrm {N} )=0,$		$\operatorname {Div} \,(\mathrm {X} ,\mathrm {Y} ,\mathrm {Z} )=\mathrm {P} ,$

c’est-à-dire, avec notre notation^[1],

(4-15)

$\left\{{\begin{aligned}{\frac {\partial {\mathrm {L} }}{\partial {x_{4}}}}+{\frac {\partial {\mathrm {Z} }}{\partial {x_{2}}}}-{\frac {\partial {\mathrm {Y} }}{\partial {x_{3}}}}&=0,\\{\frac {\partial {\mathrm {M} }}{\partial {x_{4}}}}+{\frac {\partial {\mathrm {X} }}{\partial {x_{3}}}}-{\frac {\partial {\mathrm {Z} }}{\partial {x_{1}}}}&=0,\\{\frac {\partial {\mathrm {N} }}{\partial {x_{4}}}}+{\frac {\partial {\mathrm {Y} }}{\partial {x_{1}}}}-{\frac {\partial {\mathrm {X} }}{\partial {x_{2}}}}&=0,\\{\frac {\partial {\mathrm {L} }}{\partial {x_{1}}}}+{\frac {\partial {\mathrm {M} }}{\partial {x_{2}}}}+{\frac {\partial {\mathrm {N} }}{\partial {x_{3}}}}&=0,\end{aligned}}\right.$

(5-15)

$\left\{{\begin{aligned}-{\frac {\partial {\mathrm {X} }}{\partial {x_{4}}}}+{\frac {\partial {\mathrm {N} }}{\partial {x_{2}}}}-{\frac {\partial {\mathrm {M} }}{\partial {x_{3}}}}&=u,\\-{\frac {\partial {\mathrm {Y} }}{\partial {x_{4}}}}+{\frac {\partial {\mathrm {L} }}{\partial {x_{3}}}}-{\frac {\partial {\mathrm {N} }}{\partial {x_{1}}}}&=v,\\-{\frac {\partial {\mathrm {Z} }}{\partial {x_{4}}}}+{\frac {\partial {\mathrm {M} }}{\partial {x_{1}}}}-{\frac {\partial {\mathrm {L} }}{\partial {x_{2}}}}&=w,\\{\frac {\partial {\mathrm {X} }}{\partial {x_{1}}}}+{\frac {\partial {\mathrm {Y} }}{\partial {x_{2}}}}+{\frac {\partial {\mathrm {Z} }}{\partial {x_{3}}}}&=\mathrm {P} .\end{aligned}}\right.$

1. Dans un champ statique la quatrième des équations (5-15) s’écrit
   $\Delta \varphi _{4}=-\mathrm {P} :$

   c’est l’expression analytique de la loi de Coulomb.