Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/152

celles-ci, dans lesquelles nous avons négligé les termes affectés de $n^{2},$

${\begin{aligned}(\mathrm {G} )&\ \ {\frac {d^{2}x_{1}}{dt^{2}}}+\mathrm {M} _{1}^{2}x_{1}+f_{1}\mathrm {L} _{1}-n\left(6\mu _{1}^{2}-3f_{1}\right)x_{1}^{2}-{\frac {1}{2}}nf_{1}z_{1}^{2}-nf_{1}^{2}\mathrm {Y} _{1}^{2}\\&-n\chi _{2}f_{1}x_{1}\left[{\breve {\Pi }}_{1}(a_{1},a_{2})+{\frac {2\mu _{1}}{\mu _{2}-\mu _{1}}}{\widehat {\Gamma }}_{1}(a_{1},a_{2})\right]\cos(\mu _{2}-\mu _{1})t\\&-n\chi _{2}f_{1}x_{1}\left[{\breve {\Pi }}_{2}(a_{1},a_{2})+{\frac {2\mu _{1}}{2(\mu _{2}-\mu _{1})}}{\widehat {\Gamma }}_{2}(a_{1},a_{2})\right]\cos 2(\mu _{2}-\mu _{1})t\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&-n\chi _{3}f_{1}x_{1}\left[{\breve {\Pi }}_{1}(a_{1},a_{3})+{\frac {2\mu _{1}}{\mu _{3}-\mu _{1}}}{\widehat {\Gamma }}_{1}(a_{1},a_{3})\right]\cos(\mu _{3}-\mu _{1})t\\&-n\chi _{3}f_{1}x_{1}\left[{\breve {\Pi }}_{2}(a_{1},a_{3})+{\frac {2\mu _{1}}{2(\mu _{3}-\mu _{1})}}{\widehat {\Gamma }}_{2}(a_{1},a_{3})\right]\cos 2(\mu _{3}-\mu _{1})t\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&-n\chi _{4}f_{1}x_{1}\left[{\breve {\Pi }}_{1}(a_{1},a_{4})+{\frac {2\mu _{1}}{\mu _{4}-\mu _{1}}}{\widehat {\Gamma }}_{1}(a_{1},a_{4})\right]\cos(\mu _{4}-\mu _{1})t\\&-n\chi _{4}f_{1}x_{1}\left[{\breve {\Pi }}_{2}(a_{1},a_{4})+{\frac {2\mu _{1}}{2(\mu _{4}-\mu _{1})}}{\widehat {\Gamma }}_{2}(a_{1},a_{4})\right]\cos 2(\mu _{4}-\mu _{1})t\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\\\&-n\mathrm {K} _{1}f_{1}x_{1}\left[{\frac {3}{2}}-{\frac {3\mu _{1}}{2(m-\mu _{1})}}\cos 2(m-\mu _{1})t\right]\\&-n\chi _{2}f_{1}x_{2}\\&\quad \times \left[{\breve {\Psi }}(a_{1},a_{2})+{\breve {\Psi }}_{1}(a_{1},a_{2})\cos(\mu _{2}-\mu _{1})t+{\breve {\Psi }}_{2}(a_{1},a_{2})\cos 2(\mu _{2}-\mu _{1})t+\ldots \right]\\&-n\chi _{3}f_{1}x_{3}\\&\quad \times \left[{\breve {\Psi }}(a_{1},a_{3})+{\breve {\Psi }}_{1}(a_{1},a_{3})\cos(\mu _{3}-\mu _{1})t+{\breve {\Psi }}_{2}(a_{1},a_{3})\cos 2(\mu _{3}-\mu _{1})t+\ldots \right]\\&-n\chi _{4}f_{1}x_{4}\\&\quad \times \left[{\breve {\Psi }}(a_{1},a_{4})+{\breve {\Psi }}_{1}(a_{1},a_{4})\cos(\mu _{4}-\mu _{1})t+{\breve {\Psi }}_{2}(a_{1},a_{4})\cos 2(\mu _{4}-\mu _{1})t+\ldots \right]\\&+n\mathrm {K} _{1}f_{1}\xi \left[{\frac {3}{2}}+{\frac {9}{2}}\cos 2(m-\mu _{1})t\right]\\&+n\chi _{2}f_{1}(y_{2}-y_{1})\\&\quad \times \left[{\breve {\Gamma }}_{1}(a_{1},a_{2})\sin(\mu _{2}-\mu _{1})t+2{\breve {\Gamma }}_{2}(a_{1},a_{2})\sin 2(\mu _{2}-\mu _{1})t-\ldots \right]\\&+n\chi _{3}f_{1}(y_{3}-y_{1})\\&\quad \times \left[{\breve {\Gamma }}_{1}(a_{1},a_{3})\sin(\mu _{3}-\mu _{1})t+2{\breve {\Gamma }}_{2}(a_{1},a_{3})\sin 2(\mu _{3}-\mu _{1})t-\ldots \right]\\&+n\chi _{4}f_{1}(y_{4}-y_{1})\\&\quad \times \left[{\breve {\Gamma }}_{1}(a_{1},a_{4})\sin(\mu _{4}-\mu _{1})t+2{\breve {\Gamma }}_{2}(a_{1},a_{4})\sin 2(\mu _{4}-\mu _{1})t-\ldots \right]\\\end{aligned}}$