Page:Henri Poincaré - Théorie mathématique de la lumière, Tome 1, 1889.djvu/71

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PROPAGATION D’UNE ONDE PLANE — INTERFÉRENCES

(1)

${\begin{aligned}{\frac {d^{2}\xi }{dt^{2}}}&=\mathrm {V} ^{2}\left(\Delta \xi -{\frac {d\Theta }{dx}}\right)\\[1.5ex]{\frac {d^{2}\eta }{dt^{2}}}&=\mathrm {V} ^{2}\left(\Delta \eta -{\frac {d\Theta }{dy}}\right)\\[1.5ex]{\frac {d^{2}\zeta }{dt^{2}}}&=\mathrm {V} ^{2}\left(\Delta \zeta -{\frac {d\Theta }{dz}}\right)\cdot \\\end{aligned}}$

Ces équations peuvent se mettre sous une autre forme en posant

$u={\frac {d\eta }{dz}}-{\frac {d\zeta }{dy}},\quad \qquad v={\frac {d\zeta }{dx}}-{\frac {d\xi }{dz}},\quad \qquad w={\frac {d\xi }{dy}}-{\frac {d\eta }{dx}}\cdot$

On a en effet :

${\begin{aligned}\Delta \xi -{\frac {d\Theta }{dx}}&={\frac {d^{2}\xi }{dx^{2}}}+{\frac {d^{2}\xi }{dy^{2}}}+{\frac {d^{2}\xi }{dz^{2}}}-{\frac {d^{2}\xi }{dx^{2}}}-{\frac {d^{2}\eta }{dx\,dy}}-{\frac {d^{2}\zeta }{dx\,dz}}\\[1.5ex]&={\frac {d^{2}\xi }{dy^{2}}}-{\frac {d^{2}\eta }{dx\,dy}}+{\frac {d^{2}\xi }{dz^{2}}}-{\frac {d^{2}\zeta }{dx\,dz}},\\\end{aligned}}$

ou

$\Delta \xi -{\frac {d\Theta }{dx}}={\frac {dw}{dy}}-{\frac {dv}{dz}}\cdot$

En transformant de la même manière les quantités

$\Delta \eta -{\frac {d\Theta }{dy}},\qquad \qquad \Delta \zeta -{\frac {d\Theta }{dz}}$

qui entrent dans les deux dernières équations du mouvement, on obtient :

${\begin{aligned}{\frac {d^{2}\xi }{dt^{2}}}&=\mathrm {V} ^{2}\left({\frac {dw}{dy}}-{\frac {dv}{dz}}\right),\\[1.5ex]{\frac {d^{2}\eta }{dt^{2}}}&=\mathrm {V} ^{2}\left({\frac {du}{dz}}-{\frac {dw}{dx}}\right),\\[1.5ex]{\frac {d^{2}\zeta }{dt^{2}}}&=\mathrm {V} ^{2}\left({\frac {dv}{dx}}-{\frac {du}{dy}}\right)\cdot \\\end{aligned}}$