Page:Joseph Louis de Lagrange - Œuvres, Tome 5.djvu/361

Employant donc les séries du n^o 5 ci-dessus, on aura

${\begin{aligned}\int (d\Omega )=&-{\frac {\mathrm {T} '}{n'}}\left[\left[{\overline {r}},{\overline {r}}'\ \right]+\left(\left[{\overline {r}},{\overline {r}}'\ \right]_{1}-{\frac {\overline {r}}{{\overline {r}}'^{2}}}\right)\cos \left({\overline {p}}-{\overline {p}}'\ \right)\right.\\&\quad \qquad \qquad \qquad \qquad \qquad {\Bigl .}+\left[{\overline {r}},{\overline {r}}'\ \right]_{2}\cos 2\left({\overline {p}}-{\overline {p}}'\ \right)+\ldots {\Bigr ]}\\&-{\frac {\mathrm {T} ''}{n''}}\left[\left[{\overline {r}},{\overline {r}}''\right]+\left(\left[{\overline {r}},{\overline {r}}''\right]_{1}-{\frac {\overline {r}}{{\overline {r}}''^{2}}}\right)\cos \left({\overline {p}}-{\overline {p}}''\ \right)\right.\\&\quad \qquad \qquad \qquad \qquad \qquad {\Bigl .}+\left[{\overline {r}},{\overline {r}}''\right]_{2}\cos 2\left({\overline {p}}-{\overline {p}}''\ \right)+\ldots {\Bigr ]}\\&-\ldots +\chi .\end{aligned}}$

et de là

${\begin{aligned}\int d\,{\overline {p}}\int (d\Omega )=&-{\frac {\mathrm {T} '}{n'^{2}}}\left[\left(\left[{\overline {r}},{\overline {r}}'\ \right]_{1}-{\frac {\overline {r}}{{\overline {r}}'^{2}\ }}\right)\sin \left({\overline {p}}-{\overline {p}}'\ \right)\right.\\&\qquad \qquad \qquad \qquad \quad \left.+{\frac {1}{2}}\left[{\overline {r}},{\overline {r}}'\ \right]_{2}\sin 2\left({\overline {p}}-{\overline {p}}'\ \right)+\ldots \right]\\&-{\frac {\mathrm {T} ''}{n''^{2}}}\left[\left(\left[{\overline {r}},{\overline {r}}''\right]_{1}-{\frac {\overline {r}}{{\overline {r}}''^{2}}}\right)\sin \left({\overline {p}}-{\overline {p}}''\ \right)\right.\\&\qquad \qquad \qquad \qquad \quad \left.+{\frac {1}{2}}\left[{\overline {r}},{\overline {r}}''\right]_{2}\sin 2\left({\overline {p}}-{\overline {p}}''\ \right)+\ldots \right]\\&-\ldots +\left(\chi -{\frac {\mathrm {T} '}{n'}}\left[{\overline {r}},{\overline {r}}'\right]-{\frac {\mathrm {T} ''}{n''}}\left[{\overline {r}},{\overline {r}}''\right]-\ldots \right){\overline {p}}.\end{aligned}}$

Or il faut (8) que la valeur de $\varpi ,$ c’est-à-dire

$\Sigma +3{\overline {r}}\int d\,{\overline {p}}\int (d\Omega ),$

ne contienne aucun terme proportionnel à ${\overline {p}}\,;$ égalant donc à zéro la somme des coefficients de ces sortes de termes dans $\Sigma$ et dans

$3{\overline {r}}\int d\,{\overline {p}}\int (d\Omega ),$

on aura l’équation

${\begin{aligned}&-2{\overline {r}}^{2}\left(\mathrm {T} '{\frac {d\left[{\overline {r}},{\overline {r}}'\right]}{d\,{\overline {r}}}}+\mathrm {T} ''{\frac {d\left[{\overline {r}},{\overline {r}}''\right]}{d\,{\overline {r}}}}+\ldots \right)\\&+3{\overline {r}}\left(\chi -{\frac {\mathrm {T} '}{n'}}\left[{\overline {r}},{\overline {r}}'\right]-{\frac {\mathrm {T} ''}{n''}}\left[{\overline {r}},{\overline {r}}''\right]-\ldots \right)=0,\end{aligned}}$

d’où l’on tire

$\chi =\mathrm {T} '\left({\frac {\left[{\overline {r}},{\overline {r}}'\right]}{n'}}+{\frac {2{\overline {r}}}{3}}{\frac {d\left[{\overline {r}},{\overline {r}}'\right]}{d\,{\overline {r}}}}\right)+\mathrm {T} ''\left({\frac {\left[{\overline {r}},{\overline {r}}''\right]}{n''}}+{\frac {2{\overline {r}}}{3}}{\frac {d\left[{\overline {r}},{\overline {r}}''\right]}{d\,{\overline {r}}}}\right)+\ldots .$