Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 10.djvu/123

en substituant ces valeurs dans les trois équations différentielles précédentes, elles se changeront dans celles-ci

(1)

$\left\{{\begin{aligned}0=&{\frac {d^{2}x'}{dt^{2}}}+{\frac {x'}{r^{3}}}\\&+{\frac {d^{2}\rho }{dt^{2}}}\cos \theta \cos \alpha -2{\frac {d\rho }{dt}}{\frac {d\theta }{dt}}\sin \theta \cos \alpha \\&-2{\frac {d\rho }{dt}}{\frac {d\alpha }{dt}}\cos \theta \sin \alpha \\&-\rho {\frac {d^{2}\theta }{dt^{2}}}\sin \theta \cos \alpha -\rho \left({\frac {d\theta }{dt}}\right)^{2}\cos \theta \cos \alpha \\&+2\rho {\frac {d\theta }{dt}}{\frac {d\alpha }{dt}}\sin \theta \sin \alpha \\&-\rho {\frac {d^{2}\alpha }{dt^{2}}}\cos \theta \sin \alpha -\rho \left({\frac {d\alpha }{dt}}\right)^{2}\cos \theta \cos \alpha \\&+\rho {\frac {\cos \theta \cos \alpha }{r^{3}}},\end{aligned}}\right.$

(2)

$\left\{{\begin{aligned}0=&{\frac {d^{2}y'}{dt^{2}}}+{\frac {y'}{r^{3}}}\\&+{\frac {d^{2}\rho }{dt^{2}}}\cos \theta \sin \alpha -2{\frac {d\rho }{dt}}{\frac {d\theta }{dt}}\sin \theta \sin \alpha \\&+2{\frac {d\rho }{dt}}{\frac {d\alpha }{dt}}\cos \theta \cos \alpha \\&-\rho {\frac {d^{2}\theta }{dt^{2}}}\sin \theta \sin \alpha -\rho \left({\frac {d\theta }{dt}}\right)^{2}\cos \theta \sin \alpha \\&-2\rho {\frac {d\theta }{dt}}{\frac {d\alpha }{dt}}\sin \theta \cos \alpha \\&+\rho {\frac {d^{2}\alpha }{dt^{2}}}\cos \theta \cos \alpha -\rho \left({\frac {d\alpha }{dt}}\right)^{2}\cos \theta \sin \alpha \\&+\rho {\frac {\cos \theta \sin \alpha }{r^{3}}},\end{aligned}}\right.$

(3)

$\left\{{\begin{aligned}0=&{\frac {d^{2}\rho }{dt^{2}}}\sin \theta +2{\frac {d\rho }{dt}}{\frac {d\theta }{dt}}\cos \theta \\&+\rho {\frac {d^{2}\theta }{dt^{2}}}\cos \theta -\rho \left({\frac {d\theta }{dt}}\right)^{2}\sin \theta +{\frac {\rho \sin \theta }{r^{3}}}.\end{aligned}}\right.$

Il ne s’agit plus maintenant que de tirer de ces équations les valeurs