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Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/101
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XXV.
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{\displaystyle {\begin{aligned}{\widehat {\Gamma }}_{1}(a_{1},a_{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 {a_{1}^{2}}{a_{2}^{2}}},\\{\widehat {\Gamma }}_{2}(a_{1},a_{2})=&{\frac {a_{1}^{2}a_{2}\Gamma _{3}(a_{1},a_{2})+a_{1}^{2}a_{2}\Gamma _{1}(a_{1},a_{2})}{2}},\\{\widehat {\Gamma }}_{3}(a_{1},a_{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 \,;\\{\widehat {\Pi }}_{1}(a_{1},a_{2})=&3{\frac {a_{1}^{2}a_{2}\Pi _{2}(a_{1},a_{2})-2a_{1}^{2}a_{2}\Pi (a_{1},a_{2})}{2}},\\{\widehat {\Pi }}_{2}(a_{1},a_{2})=&3{\frac {a_{1}^{2}a_{2}\Pi _{3}(a_{1},a_{2})+a_{1}^{2}a_{2}\Pi _{1}(a_{1},a_{2})}{2}}\\{\widehat {\Pi }}_{3}(a_{1},a_{2})=&3{\frac {a_{1}^{2}a_{2}\Pi _{4}(a_{1},a_{2})+a_{1}^{2}a_{2}\Pi _{2}(a_{1},a_{2})}{2}}\\\ldots \ldots \ldots \ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\\{\widehat {\Psi }}_{1}(a_{1},a_{2})=&3{\frac {a_{1}^{2}a_{2}\Psi _{2}(a_{1},a_{2})-2a_{1}^{2}a_{2}\Psi (a_{1},a_{2})}{2}}+{\widehat {\Gamma }}(a_{1},a_{2})-3{\frac {2a_{1}^{2}}{a_{2}^{2}}},\\{\widehat {\Psi }}_{2}(a_{1},a_{2})=&3{\frac {a_{1}^{2}a_{2}\Psi _{3}(a_{1},a_{2})-a_{1}^{2}a_{2}\Psi _{1}(a_{1},a_{2})}{2}}+{\widehat {\Gamma }}(a_{1},a_{2}),\\{\widehat {\Psi }}_{3}(a_{1},a_{2})=&3{\frac {a_{1}^{2}a_{2}\Psi _{4}(a_{1},a_{2})-a_{1}^{2}a_{2}\Psi _{2}(a_{1},a_{2})}{2}}+{\widehat {\Gamma }}_{3}(a_{1},a_{2}),\\\ldots \ldots \ldots \ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\end{aligned}}}
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{\displaystyle {\begin{aligned}&{\mathfrak {S}}_{2}\left[{\frac {r_{2}\sin(\varphi _{2}-\varphi _{1})}{\Delta (r_{1},r_{2})^{3}}}-{\frac {\sin(\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[{\widehat {\Gamma }}_{1}(a_{1},a_{2})\sin(\varphi _{2}-\varphi _{1})+{\widehat {\Gamma }}_{2}(a_{1},a_{2})\sin(\varphi _{2}-\varphi _{1})+\ldots \right]\\&-n{\frac {{\mathfrak {S}}_{2}}{a_{1}^{2}}}x_{1}\left[{\widehat {\Pi }}_{1}(a_{1},a_{2})\sin(\varphi _{2}-\varphi _{1})+{\widehat {\Pi }}_{2}(a_{1},a_{2})\sin(\varphi _{2}-\varphi _{1})+\ldots \right]\\&-n{\frac {{\mathfrak {S}}_{2}}{a_{1}^{2}}}x_{2}\left[{\widehat {\Psi }}_{1}(a_{1},a_{2})\sin(\varphi _{2}-\varphi _{1})+{\widehat {\Psi }}_{2}(a_{1},a_{2})\sin(\varphi _{2}-\varphi _{1})+\ldots \right].\end{aligned}}}