Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/105

${\begin{aligned}&-n{\frac {{\mathfrak {S}}_{4}}{\mathbb {Z} \!^{\upsilon }}}{\frac {\mathbb {Z} \!^{\upsilon }}{a_{1}^{2}}}\left[{\widehat {\Psi }}(a_{1},a_{4})+{\widehat {\Psi }}_{1}(a_{1},a_{4})\cos(\varphi _{4}-\varphi _{1})+\ldots \right]\\&-n{\frac {a_{1}^{3}}{\alpha ^{3}}}{\frac {\mathbb {Z} \!^{\upsilon }}{\circledast }}{\frac {\mathbb {Z} \!^{\upsilon }}{a_{1}^{2}}}\xi \left[{\frac {3}{2}}+{\frac {9}{2}}\cos 2(\psi -\varphi _{1})+\ldots \right],\\\mathrm {Q} _{1}=&-{\frac {{\mathfrak {S}}_{2}}{\mathbb {Z} \!^{\upsilon }}}{\frac {\mathbb {Z} \!^{\upsilon }}{a_{1}^{2}}}\left[{\widehat {\Gamma }}_{1}(a_{1},a_{2})\sin(\varphi _{2}-\varphi _{1})+\ldots \right]\\&-{\frac {{\mathfrak {S}}_{3}}{\mathbb {Z} \!^{\upsilon }}}{\frac {\mathbb {Z} \!^{\upsilon }}{a_{1}^{2}}}\left[{\widehat {\Gamma }}_{1}(a_{1},a_{3})\sin(\varphi _{3}-\varphi _{1})+\ldots \right]\\&-{\frac {{\mathfrak {S}}_{4}}{\mathbb {Z} \!^{\upsilon }}}{\frac {\mathbb {Z} \!^{\upsilon }}{a_{1}^{2}}}\left[{\widehat {\Gamma }}_{1}(a_{1},a_{4})\sin(\varphi _{4}-\varphi _{1})+\ldots \right]\\&-{\frac {a_{1}^{3}}{\alpha ^{3}}}{\frac {\circledast }{\mathbb {Z} \!^{\upsilon }}}{\frac {\mathbb {Z} \!^{\upsilon }}{\alpha _{1}^{2}}}\times {\frac {3}{2}}\sin 2(\psi -\varphi _{1})\\\\&-n{\frac {{\mathfrak {S}}_{2}}{\mathbb {Z} \!^{\upsilon }}}{\frac {\mathbb {Z} \!^{\upsilon }}{a_{1}^{2}}}x_{1}\left[{\widehat {\Pi }}_{1}(a_{1},a_{2})\sin(\varphi _{2}-\varphi _{1})+\ldots \right]\\&-n{\frac {{\mathfrak {S}}_{3}}{\mathbb {Z} \!^{\upsilon }}}{\frac {\mathbb {Z} \!^{\upsilon }}{a_{1}^{2}}}x_{1}\left[{\widehat {\Pi }}_{1}(a_{1},a_{3})\sin(\varphi _{3}-\varphi _{1})+\ldots \right]\\&-n{\frac {{\mathfrak {S}}_{4}}{\mathbb {Z} \!^{\upsilon }}}{\frac {\mathbb {Z} \!^{\upsilon }}{a_{1}^{2}}}x_{1}\left[{\widehat {\Pi }}_{1}(a_{1},a_{4})\sin(\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}\times {\frac {3}{2}}\sin 2(\psi -\varphi _{1})+\ldots \\&-n{\frac {{\mathfrak {S}}_{2}}{\mathbb {Z} \!^{\upsilon }}}{\frac {\mathbb {Z} \!^{\upsilon }}{a_{1}^{2}}}x_{2}\left[{\widehat {\Psi }}_{1}(a_{1},a_{2})\sin(\varphi _{2}-\varphi _{1})+\ldots \right]\\&-n{\frac {{\mathfrak {S}}_{3}}{\mathbb {Z} \!^{\upsilon }}}{\frac {\mathbb {Z} \!^{\upsilon }}{a_{1}^{2}}}x_{3}\left[{\widehat {\Psi }}_{1}(a_{1},a_{3})\sin(\varphi _{3}-\varphi _{1})+\ldots \right]\\&-n{\frac {{\mathfrak {S}}_{4}}{\mathbb {Z} \!^{\upsilon }}}{\frac {\mathbb {Z} \!^{\upsilon }}{a_{1}^{2}}}x_{4}\left[{\widehat {\Psi }}_{1}(a_{1},a_{4})\sin(\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}}}\xi \times {\frac {9}{2}}\sin 2(\psi -\varphi _{1}),\\\\\mathrm {P} _{1}=&n{\frac {{\mathfrak {S}}_{2}}{\mathbb {Z} \!^{\upsilon }}}{\frac {\mathbb {Z} \!^{\upsilon }}{a_{1}^{2}}}z_{1}\left[a_{1}^{3}\Gamma (a_{1},a_{2})+a_{1}^{3}\Gamma _{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}}}z_{1}\left[a_{1}^{3}\Gamma (a_{1},a_{2})+a_{1}^{3}\Gamma _{1}(a_{1},a_{3})\cos(\varphi _{3}-\varphi _{1})+\ldots \right]\end{aligned}}$