Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/83

${\begin{aligned}\mathrm {P} _{1}&={\mathfrak {S}}_{2}\left[{\frac {r_{1}p_{1}-r_{2}p_{2}}{\Delta (r_{1},r_{2})^{3}}}+{\frac {p_{2}}{r_{2}^{2}\left(1+p_{2}^{2}\right)^{\frac {3}{2}}}}\right]\\&+{\mathfrak {S}}_{3}\left[{\frac {r_{1}p_{1}-r_{3}p_{3}}{\Delta (r_{1},r_{3})^{3}}}-{\frac {p_{3}}{r_{3}^{2}\left(1+p_{3}^{2}\right)^{\frac {3}{2}}}}\right]\\&+{\mathfrak {S}}_{4}\left[{\frac {r_{1}p_{1}-r_{4}p_{4}}{\Delta (r_{1},r_{4})^{3}}}-{\frac {p_{4}}{r_{4}^{2}\left(1+p_{4}^{2}\right)^{\frac {3}{2}}}}\right]\\&+\circledast {\frac {r_{1}p_{1}}{\delta _{1}^{3}}}.\end{aligned}}$

XIII.

Telles sont les expressions des forces perturbatrices du satellite ${\mathfrak {S}}_{1}\,;$ d’où il est facile de déduire celles des trois autres satellites ${\mathfrak {S}}_{2},{\mathfrak {S}}_{3},{\mathfrak {S}}_{4}.$ En effet, un peu de réflexion suffit pour faire voir que les quantités $\mathrm {R_{1},Q_{1},P_{1}} ,$ deviendront $\mathrm {R_{2},Q_{2},P_{2}} ,$ en marquant seulement de deux traits les lettres qui sont marquées d’un trait, et réciproquement^[1] ; ainsi l’on aura pour les forces perturbatrices du second satellite les expressions suivantes

${\begin{aligned}\mathrm {R} _{2}&={\mathfrak {S}}_{1}\left[{\frac {r_{2}-r_{1}\cos(\varphi _{1}-\varphi _{2})}{\Delta (r_{2},r_{1})^{3}}}+{\frac {\cos(\varphi _{1}-\varphi _{2})}{r_{1}^{2}\left(1+p_{1}^{2}\right)^{\frac {3}{2}}}}\right]\\&+{\mathfrak {S}}_{3}\left[{\frac {r_{2}-r_{3}\cos(\varphi _{3}-\varphi _{2})}{\Delta (r_{2},r_{3})^{3}}}+{\frac {\cos(\varphi _{3}-\varphi _{2})}{r_{3}^{2}\left(1+p_{3}^{2}\right)^{\frac {3}{2}}}}\right]\\&+{\mathfrak {S}}_{4}\left[{\frac {r_{2}-r_{4}\cos(\varphi _{4}-\varphi _{2})}{\Delta (r_{2},r_{4})^{3}}}+{\frac {\cos(\varphi _{4}-\varphi _{2})}{r_{4}^{2}\left(1+p_{4}^{2}\right)^{\frac {3}{2}}}}\right]\\&+\circledast \ \ \left[{\frac {r_{2}-\rho _{1}\cos(\psi \ -\varphi _{2})}{\delta _{2}^{3}}}+{\frac {\cos(\ \psi \ -\varphi _{2})}{\rho _{1}^{2}}}\right],\\\\\mathrm {Q} _{2}&={\mathfrak {S}}_{1}\left[{\frac {r_{1}\sin(\varphi _{1}-\varphi _{2})}{\Delta (r_{2},r_{1})^{3}}}-{\frac {\sin(\varphi _{1}-\varphi _{2})}{r_{1}^{2}\left(1+p_{1}^{2}\right)^{\frac {3}{2}}}}\right]\\&+{\mathfrak {S}}_{3}\left[{\frac {r_{3}\sin(\varphi _{3}-\varphi _{2})}{\Delta (r_{2},r_{3})^{3}}}-{\frac {\sin(\varphi _{3}-\varphi _{2})}{r_{3}^{2}\left(1+p_{3}^{2}\right)^{\frac {3}{2}}}}\right]\\&+{\mathfrak {S}}_{4}\left[{\frac {r_{4}\sin(\varphi _{4}-\varphi _{2})}{\Delta (r_{2},r_{4})^{3}}}-{\frac {\sin(\varphi _{4}-\varphi _{2})}{r_{4}^{2}\left(1+p_{4}^{2}\right)^{\frac {3}{2}}}}\right]\\&+\circledast \ \ \left[{\frac {r_{4}\sin(\psi \ -\varphi _{2})}{\delta _{2}^{3}}}-{\frac {\sin(\ \psi \ -\varphi _{2})}{\rho _{1}^{2}}}\right],\end{aligned}}$

1. Voir la Note de la page 76.