Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/96

Soit aussi

${\begin{aligned}&\Psi \ (a_{1},a_{2})={\frac {a_{1}a_{2}\Lambda _{1}(a_{1},a_{2})-2a_{1}^{2}\Lambda (a_{1},a_{2})}{2}},\\&\Psi _{1}(a_{1},a_{2})={\frac {a_{1}a_{2}\Lambda _{2}(a_{1},a_{2})-2a_{1}^{2}\Lambda _{1}(a_{1},a_{2})+2a_{1}a_{2}\Lambda (a_{1},a_{2})}{2}},\\&\Psi _{2}(a_{1},a_{2})={\frac {a_{1}a_{2}\Lambda _{3}(a_{1},a_{2})-2a_{1}^{2}\Lambda _{2}(a_{1},a_{2})+a_{1}a_{2}\Lambda _{1}(a_{1},a_{2})}{2}},\\&\Psi _{3}(a_{1},a_{2})={\frac {a_{1}a_{2}\Lambda _{4}(a_{1},a_{2})-2a_{1}^{2}\Lambda _{3}(a_{1},a_{2})+a_{1}a_{2}\Lambda _{2}(a_{1},a_{2})}{2}},\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\end{aligned}}$

On aura la quantité ${\frac {1}{\Delta (r_{1},r_{2})^{3}}},$ exprimée de la manière suivante

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

On trouvera de même, en changeant simplement $r_{2}$ en $r_{3},$ $a_{2}$ en $a_{3},$ $\varphi _{2}$ en $\varphi _{3}$ et $x_{2}$ en $x_{3},$

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

et pareillement

${\begin{aligned}&{\frac {1}{\Delta (r_{1},r_{4})^{3}}}\\&=\Gamma (a_{1},a_{4})+\Gamma _{1}(a_{1},a_{4})\cos(\varphi _{4}-\varphi _{1})+\Gamma _{2}(a_{1},a_{4})\cos 2(\varphi _{4}-\varphi _{1})+\ldots \\&\quad +3nx_{1}\left[\Pi (a_{1},a_{4})+\Pi _{1}(a_{1},a_{4})\cos(\varphi _{4}-\varphi _{1})+\Pi _{2}(a_{1},a_{4})\cos 2(\varphi _{4}-\varphi _{1})+\ldots \right]\\&\quad +3nx_{4}\left[\Psi (a_{1},a_{4})+\Psi _{1}(a_{1},a_{4})\cos(\varphi _{4}-\varphi _{1})+\Psi _{2}(a_{1},a_{4})\cos 2(\varphi _{4}-\varphi _{1})+\ldots \right].\end{aligned}}$