Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/99

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

XXIV.

Soit maintenant

${\begin{aligned}{\breve {\Gamma }}\ \,(a_{1},a_{2})=&{\frac {a_{1}^{2}a_{2}\Gamma _{1}(a_{1},a_{2})-2a_{1}^{3}\Gamma (a_{1},a_{2})}{2}},\\{\breve {\Gamma }}_{1}(a_{1},a_{2})=&{\frac {a_{1}^{2}a_{2}\Gamma _{2}(a_{1},a_{2})-2a_{1}^{3}\Gamma _{1}(a_{1},a_{2})+2a_{1}^{2}a_{2}\Gamma (a_{1},a_{2})}{2}}-{\frac {a_{1}^{2}}{a_{2}^{2}}},\\{\breve {\Gamma }}_{2}(a_{1},a_{2})=&{\frac {a_{1}^{2}a_{2}\Gamma _{3}(a_{1},a_{2})-2a_{1}^{3}\Gamma _{2}(a_{1},a_{2})+a_{1}^{2}a_{2}\Gamma _{1}(a_{1},a_{2})}{2}},\\{\breve {\Gamma }}_{3}(a_{1},a_{2})=&{\frac {a_{1}^{3}a_{2}\Gamma _{4}(a_{1},a_{2})-2a_{1}^{3}\Gamma _{3}(a_{1},a_{2})+a_{1}^{2}a_{2}\Gamma _{2}(a_{1},a_{2})}{2}},\\\ldots \ldots \ldots \ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\\{\breve {\Pi }}\ \,(a_{1},a_{2})=&3{\frac {a_{1}^{2}a_{2}\Pi _{1}(a_{1},a_{2})-2a_{1}^{3}\Pi (a_{1},a_{2})}{2}}-a_{1}^{3}\Gamma (a_{1},a_{2}),\\{\breve {\Pi }}_{1}(a_{1},a_{2})=&3{\frac {a_{1}^{2}a_{2}\Pi _{2}(a_{1},a_{2})-2a_{1}^{3}\Pi _{1}(a_{1},a_{2})+2a_{1}^{2}a_{2}\Pi (a_{1},a_{2})}{2}}\\&-a_{1}^{3}\Gamma _{1}(a_{1},a_{2}),\\\\{\breve {\Pi }}_{2}(a_{1},a_{2})=&3{\frac {a_{1}^{2}a_{2}\Pi _{3}(a_{1},a_{2})-2a_{1}^{3}\Pi _{2}(a_{1},a_{2})+a_{1}^{2}a_{2}\Pi _{1}(a_{1},a_{2})}{2}}\\&-a_{1}^{3}\Gamma _{2}(a_{1},a_{2}),\\\\{\breve {\Pi }}_{3}(a_{1},a_{2})=&3{\frac {a_{1}^{2}a_{2}\Pi _{4}(a_{1},a_{2})-2a_{1}^{3}\Pi _{3}(a_{1},a_{2})+a_{1}^{2}a_{2}\Pi _{2}(a_{1},a_{2})}{2}}\\&-a_{1}^{3}\Gamma _{3}(a_{1},a_{2}),\\\ldots \ldots \ldots \ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\end{aligned}}$