Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/174

et de plus

${\begin{aligned}{\overset {\circ }{\Pi }}_{1}=&{\frac {-2\mu _{2}}{\mathrm {M} _{2}^{2}}}{\widehat {\Gamma }}_{1}(a_{1},a_{2})\\&\quad +\left[2{\widehat {\Psi }}_{1}(a_{1},a_{2})+4{\frac {\mu _{2}(\mu _{2}-\mu _{1})}{\mathrm {M} _{2}^{2}}}{\widehat {\Gamma }}_{1}(a_{1},a_{2})\right]{\frac {\mu _{1}}{\mathrm {M} _{2}^{2}-(\mu _{2}-\mu _{1})^{2}}},\\{\overset {\circ }{\Pi }}_{1}=&{\frac {-2\mu _{3}}{\mathrm {M} _{3}^{2}}}{\widehat {\Gamma }}_{1}(a_{1},a_{3})\\&\quad +\left[2{\widehat {\Psi }}_{1}(a_{1},a_{3})+4{\frac {\mu _{3}(\mu _{3}-\mu _{1})}{\mathrm {M} _{3}^{2}}}{\widehat {\Gamma }}_{1}(a_{1},a_{3})\right]{\frac {\mu _{1}}{\mathrm {M} _{3}^{2}-(\mu _{3}-\mu _{1})^{2}}},\\{\overset {\circ }{\Pi }}_{1}=&{\frac {-2\mu _{4}}{\mathrm {M} _{4}^{2}}}{\widehat {\Gamma }}_{1}(a_{1},a_{4})\\&\quad +\left[2{\widehat {\Psi }}_{1}(a_{1},a_{4})+4{\frac {\mu _{4}(\mu _{4}-\mu _{1})}{\mathrm {M} _{4}^{2}}}{\widehat {\Gamma }}_{1}(a_{1},a_{4})\right]{\frac {\mu _{1}}{\mathrm {M} _{4}^{2}-(\mu _{4}-\mu _{1})^{2}}},\\\end{aligned}}$

on aura, pour le premier satellite, l’équation

${\begin{aligned}(&\mathrm {M} _{1})\ {\frac {d^{2}\mathrm {x} _{1}}{dt^{2}}}+\mathrm {M} _{1}^{2}\mathrm {x} _{1}\\&-nf_{1}\chi _{2}{\overset {\circ }{\Psi }}_{1}(a_{1},a_{2})\mathrm {x} _{2}\cos(\mu _{2}-\mu _{1})t-nf_{1}\chi _{2}{\overset {\circ }{\Pi }}_{1}(a_{1},a_{2}){\frac {d\mathrm {x} _{2}}{dt}}\sin(\mu _{2}-\mu _{1})t\\&-nf_{1}\chi _{3}{\overset {\circ }{\Psi }}_{1}(a_{1},a_{3})\mathrm {x} _{3}\cos(\mu _{3}-\mu _{1})t-nf_{1}\chi _{3}{\overset {\circ }{\Pi }}_{1}(a_{1},a_{3}){\frac {d\mathrm {x} _{3}}{dt}}\sin(\mu _{3}-\mu _{1})t\\&-nf_{1}\chi _{4}{\overset {\circ }{\Psi }}_{1}(a_{1},a_{4})\mathrm {x} _{4}\cos(\mu _{4}-\mu _{1})t-nf_{1}\chi _{4}{\overset {\circ }{\Pi }}_{1}(a_{1},a_{4}){\frac {d\mathrm {x} _{4}}{dt}}\sin(\mu _{4}-\mu _{1})t=0\,;\end{aligned}}$

et de même pour les trois autres satellites

${\begin{aligned}(&\mathrm {M} _{2})\ {\frac {d^{2}\mathrm {x} _{2}}{dt^{2}}}+\mathrm {M} _{2}^{2}\mathrm {x} _{2}\\&-nf_{2}\chi _{1}{\overset {\circ }{\Psi }}_{1}(a_{2},a_{1})\mathrm {x} _{1}\cos(\mu _{1}-\mu _{2})t-nf_{2}\chi _{1}{\overset {\circ }{\Pi }}_{1}(a_{2},a_{1}){\frac {d\mathrm {x} _{1}}{dt}}\sin(\mu _{1}-\mu _{2})t\\&-nf_{2}\chi _{3}{\overset {\circ }{\Psi }}_{1}(a_{2},a_{3})\mathrm {x} _{3}\cos(\mu _{3}-\mu _{2})t-nf_{2}\chi _{3}{\overset {\circ }{\Pi }}_{1}(a_{2},a_{3}){\frac {d\mathrm {x} _{3}}{dt}}\sin(\mu _{3}-\mu _{2})t\\&-nf_{2}\chi _{4}{\overset {\circ }{\Psi }}_{1}(a_{2},a_{4})\mathrm {x} _{4}\cos(\mu _{4}-\mu _{2})t-nf_{2}\chi _{4}{\overset {\circ }{\Pi }}_{1}(a_{2},a_{4}){\frac {d\mathrm {x} _{4}}{dt}}\sin(\mu _{4}-\mu _{2})t=0,\end{aligned}}$