Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/104

${\begin{aligned}\circledast &\left[{\frac {\rho _{1}\sin(\psi -\varphi _{1})}{\delta _{1}^{3}}}+{\frac {\sin(\psi -\varphi _{1})}{\rho _{1}^{2}}}\right]\\&={\frac {3a_{1}\circledast }{2\alpha ^{3}}}\sin 2(\psi -\varphi _{1})+n{\frac {3a_{1}\circledast }{2\alpha ^{3}}}(x_{1}-3\xi )\sin 2(\psi -\varphi _{1}),\\\circledast &{\frac {r_{1}p_{1}}{\delta _{1}^{3}}}=n{\frac {a_{1}\circledast }{\alpha _{1}^{3}}}z_{1}.\end{aligned}}$

Ce sont les valeurs des forces $\mathrm {R_{1},Q_{1},P_{1}} ,$ qui viennent de l’action du Soleil (Article XI).

XXVIII.

En joignant ensemble toutes ces différentes valeurs, on aura les valeurs complètes des forces $\mathrm {R_{1},Q_{1},P_{1}}$ exprimées de la manière suivante

${\begin{aligned}\mathrm {R} _{1}=&-{\frac {{\mathfrak {S}}_{2}}{\mathbb {Z} \!^{\upsilon }}}{\frac {\mathbb {Z} \!^{\upsilon }}{a_{1}^{2}}}\left[{\breve {\Gamma }}(a_{1},a_{2})+{\breve {\Gamma }}_{1}(a_{1},a_{2})\cos(\varphi _{2}-\varphi _{1})+\ldots \right]\\&-{\frac {{\mathfrak {S}}_{3}}{\mathbb {Z} \!^{\upsilon }}}{\frac {\mathbb {Z} \!^{\upsilon }}{a_{1}^{2}}}\left[{\breve {\Gamma }}(a_{1},a_{3})+{\breve {\Gamma }}_{1}(a_{1},a_{3})\cos(\varphi _{3}-\varphi _{1})+\ldots \right]\\&-{\frac {{\mathfrak {S}}_{4}}{\mathbb {Z} \!^{\upsilon }}}{\frac {\mathbb {Z} \!^{\upsilon }}{a_{1}^{2}}}\left[{\breve {\Gamma }}(a_{1},a_{4})+{\breve {\Gamma }}_{1}(a_{1},a_{4})\cos(\varphi _{4}-\varphi _{1})+\ldots \right]\\&-{\frac {a_{1}^{3}}{\alpha ^{3}}}{\frac {\circledast }{\mathbb {Z} \!^{\upsilon }}}{\frac {\mathbb {Z} \!^{\upsilon }}{a_{1}^{2}}}\left[{\frac {1}{2}}+{\frac {3}{2}}\cos 2(\psi -\varphi _{1})+\ldots \right]\\&-n{\frac {{\mathfrak {S}}_{2}}{\mathbb {Z} \!^{\upsilon }}}{\frac {\mathbb {Z} \!^{\upsilon }}{a_{1}^{2}}}x_{1}\left[{\breve {\Pi }}(a_{1},a_{2})+{\breve {\Pi }}_{1}(a_{1},a_{2})\cos(\varphi _{2}-\varphi _{1})+\ldots \right]\\&-n{\frac {{\mathfrak {S}}_{3}}{\mathbb {Z} \!^{\upsilon }}}{\frac {\mathbb {Z} \!^{\upsilon }}{a_{1}^{2}}}x_{1}\left[{\breve {\Pi }}(a_{1},a_{3})+{\breve {\Pi }}_{1}(a_{1},a_{3})\cos(\varphi _{3}-\varphi _{1})+\ldots \right]\\&-n{\frac {{\mathfrak {S}}_{4}}{\mathbb {Z} \!^{\upsilon }}}{\frac {\mathbb {Z} \!^{\upsilon }}{a_{1}^{2}}}x_{1}\left[{\breve {\Pi }}(a_{1},a_{4})+{\breve {\Pi }}_{1}(a_{1},a_{4})\cos(\varphi _{4}-\varphi _{1})+\ldots \right]\\&-n{\frac {a_{1}^{3}}{\alpha ^{3}}}{\frac {\circledast }{\mathbb {Z} \!^{\upsilon }}}{\frac {\mathbb {Z} \!^{\upsilon }}{a_{1}^{2}}}x_{1}\left[{\frac {1}{2}}+{\frac {3}{2}}\cos 2(\psi -\varphi _{1})+\ldots \right]\\&-n{\frac {{\mathfrak {S}}_{2}}{\mathbb {Z} \!^{\upsilon }}}{\frac {\mathbb {Z} \!^{\upsilon }}{a_{1}^{2}}}x_{2}\left[{\breve {\Psi }}(a_{1},a_{2})+{\breve {\Psi }}_{1}(a_{1},a_{2})\cos(\varphi _{2}-\varphi _{1})+\ldots \right]\\&-n{\frac {{\mathfrak {S}}_{3}}{\mathbb {Z} \!^{\upsilon }}}{\frac {\mathbb {Z} \!^{\upsilon }}{a_{1}^{2}}}x_{3}\left[{\breve {\Psi }}(a_{1},a_{3})+{\breve {\Psi }}_{1}(a_{1},a_{3})\cos(\varphi _{3}-\varphi _{1})+\ldots \right]\\\end{aligned}}$