Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/97

XXIII.

Cela posé, on aura

${\begin{aligned}&{\frac {r_{1}}{\Delta (r_{1},r_{2})^{3}}}\\&=a_{1}\left[\Gamma (a_{1},a_{2})+\Gamma _{1}(a_{1},a_{2})\cos(\varphi _{2}-\varphi _{1})+\Gamma _{2}(a_{1},a_{2})\cos 2(\varphi _{2}-\varphi _{1})+\ldots \right]\\&\quad +nx_{1}a_{1}\left[3\Pi (a_{1},a_{2})+\Gamma (a_{1},a_{2})+\left[3\Pi _{1}(a_{1},a_{2})+\Gamma _{1}(a_{1},a_{2})\right]\cos(\varphi _{2}-\varphi _{1})\right.\\&\qquad \qquad \qquad \qquad \left.+\left[3\Pi _{2}(a_{1},a_{2})+\Gamma _{2}(a_{1},a_{2})\right]\cos 2(\varphi _{2}-\varphi _{1})+\ldots \right]\\&\quad +3nx_{2}a_{1}\left[\Psi (a_{1},a_{2})+\Psi _{1}(a_{1},a_{2})\cos(\varphi _{2}-\varphi _{1})+\Psi _{2}(a_{1},a_{2})\cos 2(\varphi _{2}-\varphi _{1})+\ldots \right].\end{aligned}}$

On trouvera de la même manière

${\begin{aligned}&{\frac {r_{2}}{\Delta (r_{1},r_{2})^{3}}}\\&=a_{2}\left[\Gamma (a_{1},a_{2})+\Gamma _{1}(a_{1},a_{2})\cos(\varphi _{2}-\varphi _{1})+\Gamma _{2}(a_{1},a_{2})\cos 2(\varphi _{2}-\varphi _{1})+\ldots \right]\\&\quad +3nx_{1}a_{2}\left[\Pi (a_{1},a_{2})+\Pi _{1}(a_{1},a_{2})\cos(\varphi _{2}-\varphi _{1})+\Pi _{2}(a_{1},a_{2})\cos 2(\varphi _{2}-\varphi _{1})+\ldots \right]\\&\quad +nx_{2}a_{2}\left[3\Psi (a_{1},a_{2})+\Gamma (a_{1},a_{2})+\left[3\Psi _{1}(a_{1},a_{2})+\Gamma _{1}(a_{1},a_{2})\right]\cos(\varphi _{2}-\varphi _{1})\right.\\&\qquad \qquad \qquad \qquad \left.+\left[3\Psi _{2}(a_{1},a_{2})+\Gamma _{2}(a_{1},a_{2})\right]\cos 2(\varphi _{2}-\varphi _{1})+\ldots \right].\end{aligned}}$

On aura ensuite

${\frac {1}{r_{2}^{2}\left(1+p_{2}^{2}\right)^{\frac {3}{2}}}}={\frac {1}{a_{2}^{2}}}(1-2nx_{2}+\ldots ).$

Donc

$-{\frac {r_{2}}{\Delta (r_{1},r_{2})^{3}}}+{\frac {1}{r_{2}^{2}\left(1+p_{2}^{2}\right)^{\frac {3}{2}}}}$

${\begin{aligned}&={\frac {1}{a_{2}^{2}}}-a_{2}\Gamma (a_{1},a_{2})-a_{2}\Gamma _{1}(a_{1},a_{2})\cos(\varphi _{2}-\varphi _{1})-a_{2}\Gamma _{2}(a_{1},a_{2})\cos 2(\varphi _{2}-\varphi _{1})-\ldots \\&\quad -3nx_{1}a_{2}\left[\Pi (a_{1},a_{2})+\Pi _{1}(a_{1},a_{2})\cos(\varphi _{2}-\varphi _{1})+\Pi _{2}(a_{1},a_{2})\cos 2(\varphi _{2}-\varphi _{1})+\ldots \right]\\&\quad -nx_{2}\left[{\frac {2}{a_{2}^{2}}}+3a_{2}\Psi (a_{1},a_{2})+a_{2}\Gamma (a_{1},a_{2})\right.\\&\qquad \qquad \qquad \qquad +\left[3a_{2}\Psi _{1}(a_{1},a_{2})+a_{2}\Gamma _{1}(a_{1},a_{2})\right]\cos(\varphi _{2}-\varphi _{1})\\&\qquad \qquad \qquad \qquad +{\biggl .}\left[3a_{2}\Psi _{2}(a_{1},a_{2})+a_{2}\Gamma _{2}(a_{1},a_{2})\right]\cos 2(\varphi _{2}-\varphi _{1})+\ldots {\biggr ]}.\end{aligned}}$