Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/98

Cette quantité étant multipliée par $\cos(\varphi _{1}-\varphi _{2}),$ on aura

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

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

Enfin, multipliant la même quantité pa $-\sin(\varphi _{2}-\varphi _{1}),$ on aura

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

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