Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/100

${\begin{aligned}{\breve {\Psi }}\ \,(a_{1},a_{2})=&3{\frac {a_{1}^{2}a_{2}\Psi _{1}(a_{1},a_{2})-2a_{1}^{3}\Psi (a_{1},a_{2})}{2}}+{\frac {a_{1}^{2}a_{2}\Gamma _{1}(a_{1},a_{2})}{2}},\\{\breve {\Psi }}_{1}(a_{1},a_{2})=&3{\frac {a_{1}^{2}a_{2}\Psi _{2}(a_{1},a_{2})-2a_{1}^{3}\Psi _{1}(a_{1},a_{2})+2a_{1}^{2}a_{2}\Psi (a_{1},a_{2})}{2}}\\&+{\frac {a_{1}^{2}a_{2}\Gamma _{2}(a_{1},a_{2})+2a_{1}^{2}a_{2}\Gamma (a_{1},a_{2})}{2}}+{\frac {2a_{1}^{2}}{a_{2}^{2}}},\\{\breve {\Psi }}_{2}(a_{1},a_{2})=&3{\frac {a_{1}^{2}a_{2}\Psi _{3}(a_{1},a_{2})-2a_{1}^{3}\Psi _{2}(a_{1},a_{2})+a_{1}^{2}a_{2}\Psi _{1}(a_{1},a_{2})}{2}}\\&+{\frac {a_{1}^{3}a_{2}\Gamma _{3}(a_{1},a_{2})+a_{1}^{2}a_{2}\Gamma _{1}(a_{1},a_{2})}{2}},\\{\breve {\Psi }}_{3}(a_{1},a_{2})=&3{\frac {a_{1}^{2}a_{2}\Psi _{4}(a_{1},a_{2})-2a_{1}^{3}\Psi _{3}(a_{1},a_{2})+a_{1}^{2}a_{2}\Psi _{2}(a_{1},a_{2})}{2}}\\&+{\frac {a_{1}^{2}a_{2}\Gamma _{4}(a_{1},a_{2})+a_{1}^{2}a_{2}\Gamma _{2}(a_{1},a_{2})}{2}},\\\ldots \ldots \ldots \ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\end{aligned}}$

On aura

${\begin{aligned}&{\mathfrak {S}}_{2}\left[{\frac {r_{1}-r_{2}\cos(\varphi _{2}-\varphi _{1})}{\Delta (r_{1},r_{2})^{3}}}+{\frac {\cos(\varphi _{2}-\varphi _{1})}{r_{2}^{2}\left(1+p_{2}^{2}\right)^{\frac {3}{2}}}}\right]=\\&-{\frac {{\mathfrak {S}}_{2}}{a_{1}^{2}}}\left[{\breve {\Gamma }}(a_{1},a_{2})+{\breve {\Gamma }}_{1}(a_{1},a_{2})\cos(\varphi _{2}-\varphi _{1})+{\breve {\Gamma }}_{2}(a_{1},a_{2})\cos(\varphi _{2}-\varphi _{1})\ldots \right]\\&-n{\frac {{\mathfrak {S}}_{2}}{a_{1}^{2}}}x_{1}\left[{\breve {\Pi }}(a_{1},a_{2})+{\breve {\Pi }}_{1}(a_{1},a_{2})\cos(\varphi _{2}-\varphi _{1})+{\breve {\Pi }}_{2}(a_{1},a_{2})\cos(\varphi _{2}-\varphi _{1})\ldots \right]\\&-n{\frac {{\mathfrak {S}}_{2}}{a_{1}^{2}}}x_{2}\left[{\breve {\Psi }}(a_{1},a_{2})+{\breve {\Psi }}_{1}(a_{1},a_{2})\cos(\varphi _{2}-\varphi _{1})+{\breve {\Psi }}_{2}(a_{1},a_{2})\cos(\varphi _{2}-\varphi _{1})\ldots \right].\end{aligned}}$

C’est la partie de la force $\mathrm {R} ,$ qui résulte de l’action du satellite ${\mathfrak {S}}_{2}$ (Article X).