Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/109

XXX.

Appliquons maintenant ces formules au premier satellite. Nous aurons d’abord en substituant la valeur de $\mathrm {R} _{1}$ (Article XXIX)

${\begin{aligned}&\mathrm {X} _{1}=g_{1}\\&-{\frac {\chi _{2}}{1+n\chi _{1}}}\left[{\breve {\Gamma }}(a_{1},a_{2})+{\breve {\Gamma }}_{1}(a_{1},a_{2})\cos(\mu _{2}-\mu _{1})t+{\breve {\Gamma }}_{2}(a_{1},a_{2})\cos 2(\mu _{2}-\mu _{1})t+\ldots \right]\\&-{\frac {\chi _{3}}{1+n\chi _{1}}}\left[{\breve {\Gamma }}(a_{1},a_{3})+{\breve {\Gamma }}_{1}(a_{1},a_{3})\cos(\mu _{3}-\mu _{1})t+{\breve {\Gamma }}_{2}(a_{1},a_{3})\cos 2(\mu _{3}-\mu _{1})t+\ldots \right]\\&-{\frac {\chi _{4}}{1+n\chi _{1}}}\left[{\breve {\Gamma }}(a_{1},a_{4})+{\breve {\Gamma }}_{1}(a_{1},a_{4})\cos(\mu _{4}-\mu _{1})t+{\breve {\Gamma }}_{2}(a_{1},a_{4})\cos 2(\mu _{4}-\mu _{1})t+\ldots \right]\\&-{\frac {\mathrm {K} _{1}}{1+n\chi _{1}}}\left[{\frac {1}{2}}+{\frac {3}{2}}\cos 2(m-\mu _{1})t\right]\\&-{\frac {n\chi _{2}}{1+n\chi _{1}}}x_{1}\left[{\breve {\Pi }}(a_{1},a_{2})+{\breve {\Pi }}_{1}(a_{1},a_{2})\cos(\mu _{2}-\mu _{1})t+{\breve {\Pi }}_{2}(a_{1},a_{2})\cos 2(\mu _{2}-\mu _{1})t+\ldots \right]\\&-{\frac {n\chi _{3}}{1+n\chi _{1}}}x_{1}\left[{\breve {\Pi }}(a_{1},a_{3})+{\breve {\Pi }}_{1}(a_{1},a_{3})\cos(\mu _{3}-\mu _{1})t+{\breve {\Pi }}_{2}(a_{1},a_{3})\cos 2(\mu _{3}-\mu _{1})t+\ldots \right]\\&-{\frac {n\chi _{4}}{1+n\chi _{1}}}x_{1}\left[{\breve {\Pi }}(a_{1},a_{4})+{\breve {\Pi }}_{1}(a_{1},a_{4})\cos(\mu _{4}-\mu _{1})t+{\breve {\Pi }}_{2}(a_{1},a_{4})\cos 2(\mu _{4}-\mu _{1})t+\ldots \right]\\&-{\frac {n\mathrm {K} _{1}}{1+n\chi _{1}}}x_{1}\left[{\frac {1}{2}}+{\frac {3}{2}}\cos 2(m-\mu _{1})\right]\\&-{\frac {n\chi _{2}}{1+n\chi _{1}}}x_{2}\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]\\&-{\frac {n\chi _{3}}{1+n\chi _{1}}}x_{3}\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]\\&-{\frac {n\chi _{4}}{1+n\chi _{1}}}x_{4}\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]\\&+{\frac {n\mathrm {K} _{1}}{1+n\chi _{1}}}\xi \left[{\frac {1}{2}}+{\frac {3}{2}}\cos 2(m-\mu _{1})t\right]\\&+{\frac {n\chi _{2}}{1+n\chi _{1}}}(y_{2}-y_{1})\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]\\&+{\frac {n\chi _{3}}{1+n\chi _{1}}}(y_{3}-y_{1})\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]\\&+{\frac {n\chi _{4}}{1+n\chi _{1}}}(y_{4}-y_{1})\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]\\&+{\frac {n\mathrm {K} _{1}}{1+n\chi _{1}}}(\mathrm {J} -y)3\sin 2(m-\mu _{1})t\\&+{\frac {\varkappa _{1}}{1+n\chi _{1}}}\left({\frac {1}{5}}-{\frac {4}{5}}nx_{1}\right).\end{aligned}}$