Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/111

${\begin{aligned}&-{\frac {n\chi _{3}}{1+n\chi _{1}}}\int (y_{3}-y_{1})\left[{\widehat {\Gamma }}_{1}(a_{1},a_{3})\cos(\mu _{3}-\mu _{1})t+2{\widehat {\Gamma }}_{2}(a_{1},a_{3})\cos 2(\mu _{3}-\mu _{1})t+\ldots \right]dt\\&-{\frac {n\chi _{4}}{1+n\chi _{1}}}\int (y_{4}-y_{1})\left[{\widehat {\Gamma }}_{1}(a_{1},a_{4})\cos(\mu _{4}-\mu _{1})t+2{\widehat {\Gamma }}_{2}(a_{1},a_{4})\cos 2(\mu _{4}-\mu _{1})t+\ldots \right]dt\\&+{\frac {n\mathrm {K} _{1}}{1+n\chi _{1}}}\int (\mathrm {J} -y_{1})\times 3\cos 2(m-\mu _{1})tdt.\end{aligned}}$

XXXII.

Enfin, si l’on fait

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

et ainsi de suite, on trouvera

${\begin{aligned}&\mathrm {Z} _{1}=\\&\quad {\frac {n\chi _{2}}{1+n\chi _{1}}}z_{1}\left[{\overset {\backsim }{\Gamma }}(a_{1},a_{2})+{\overset {\backsim }{\Gamma }}_{1}(a_{1},a_{2})\cos(\mu _{2}-\mu _{1})t+{\overset {\backsim }{\Gamma }}_{2}(a_{1},a_{2})\cos 2(\mu _{2}-\mu _{1})t+\ldots \right]\\&+{\frac {n\chi _{3}}{1+n\chi _{1}}}z_{1}\left[{\overset {\backsim }{\Gamma }}(a_{1},a_{3})+{\overset {\backsim }{\Gamma }}_{1}(a_{1},a_{3})\cos(\mu _{3}-\mu _{1})t+{\overset {\backsim }{\Gamma }}_{2}(a_{1},a_{3})\cos 2(\mu _{3}-\mu _{1})t+\ldots \right]\\&+{\frac {n\chi _{4}}{1+n\chi _{1}}}z_{1}\left[{\overset {\backsim }{\Gamma }}(a_{1},a_{4})+{\overset {\backsim }{\Gamma }}_{1}(a_{1},a_{4})\cos(\mu _{4}-\mu _{1})t+{\overset {\backsim }{\Gamma }}_{2}(a_{1},a_{4})\cos 2(\mu _{4}-\mu _{1})t+\ldots \right]\\&+{\frac {n\mathrm {K} _{1}}{1+n\chi _{1}}}{\frac {2}{5}}z_{1}\\&-{\frac {n\chi _{2}}{1+n\chi _{1}}}z_{2}\left[{\overset {\circ }{\Gamma }}(a_{1},a_{2})+{\overset {\circ }{\Gamma }}_{1}(a_{1},a_{2})\cos(\mu _{2}-\mu _{1})t+{\overset {\circ }{\Gamma }}_{2}(a_{1},a_{2})\cos 2(\mu _{2}-\mu _{1})t+\ldots \right]\\&-{\frac {n\chi _{3}}{1+n\chi _{1}}}z_{3}\left[{\overset {\circ }{\Gamma }}(a_{1},a_{3})+{\overset {\circ }{\Gamma }}_{1}(a_{1},a_{3})\cos(\mu _{3}-\mu _{1})t+{\overset {\circ }{\Gamma }}_{2}(a_{1},a_{3})\cos 2(\mu _{3}-\mu _{1})t+\ldots \right]\\&-{\frac {n\chi _{4}}{1+n\chi _{1}}}z_{4}\left[{\overset {\circ }{\Gamma }}(a_{1},a_{4})+{\overset {\circ }{\Gamma }}_{1}(a_{1},a_{4})\cos(\mu _{4}-\mu _{1})t+{\overset {\circ }{\Gamma }}_{2}(a_{1},a_{4})\cos 2(\mu _{4}-\mu _{1})t+\ldots \right].\end{aligned}}$