![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d024aa0e55a68b9353584393921bcfcb9565a5d)
XIII.
Telles sont les expressions des forces perturbatrices du satellite
d’où il est facile de déduire celles des trois autres satellites
En effet, un peu de réflexion suffit pour faire voir que les quantités
deviendront
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
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b0a9a6d3678b482189730bb55294d134161c3a1)
- ↑ Voir la Note de la page 76.