Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/110

XXXI.

Multipliant la valeur de $\mathrm {Q} _{1}$ par $1+nx_{1},$ et faisant, pour abréger,

${\begin{aligned}{\overset {\backsim }{\Pi }}_{1}(a_{1},a_{2})=&{\widehat {\Pi }}_{1}(a_{1},a_{2})+{\widehat {\Gamma }}_{1}(a_{1},a_{2}),\\{\overset {\backsim }{\Pi }}_{2}(a_{1},a_{2})=&{\widehat {\Pi }}_{2}(a_{1},a_{2})+{\widehat {\Gamma }}_{2}(a_{1},a_{2}),\\{\overset {\backsim }{\Pi }}_{3}(a_{1},a_{2})=&{\widehat {\Pi }}_{3}(a_{1},a_{2})+{\widehat {\Gamma }}_{3}(a_{1},a_{2}),\\\ldots \ldots \ldots \ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\end{aligned}}$

on aura, après l’intégration et la substitution,

${\begin{aligned}&\mathrm {Y} _{1}=\mathrm {H} _{1}\\&+{\frac {\chi _{2}}{1+n\chi _{1}}}\left[{\frac {{\widehat {\Gamma }}_{1}(a_{1},a_{2})}{\mu _{2}-\mu _{1}}}\cos(\mu _{2}-\mu _{1})t+{\frac {{\widehat {\Gamma }}_{2}(a_{1},a_{2})}{2(\mu _{2}-\mu _{1})}}\cos 2(\mu _{2}-\mu _{1})t+\ldots \right]\\&+{\frac {\chi _{3}}{1+n\chi _{1}}}\left[{\frac {{\widehat {\Gamma }}_{1}(a_{1},a_{3})}{\mu _{3}-\mu _{1}}}\cos(\mu _{3}-\mu _{1})t+{\frac {{\widehat {\Gamma }}_{2}(a_{1},a_{3})}{2(\mu _{3}-\mu _{1})}}\cos 2(\mu _{3}-\mu _{1})t+\ldots \right]\\&+{\frac {\chi _{4}}{1+n\chi _{1}}}\left[{\frac {{\widehat {\Gamma }}_{1}(a_{1},a_{4})}{\mu _{4}-\mu _{1}}}\cos(\mu _{4}-\mu _{1})t+{\frac {{\widehat {\Gamma }}_{2}(a_{1},a_{4})}{2(\mu _{4}-\mu _{1})}}\cos 2(\mu _{4}-\mu _{1})t+\ldots \right]\\&-{\frac {\mathrm {K} _{1}}{1+n\chi _{1}}}{\frac {3}{4(m-\mu _{1})}}\cos 2(m-\mu _{1})t\\&-{\frac {n\chi _{2}}{1+n\chi _{1}}}\int x_{1}\left[{\overset {\backsim }{\Pi }}_{1}(a_{1},a_{2})\sin(\mu _{2}-\mu _{1})t+{\overset {\backsim }{\Pi }}_{2}(a_{1},a_{2})\sin 2(\mu _{2}-\mu _{1})t+\ldots \right]dt\\&-{\frac {n\chi _{3}}{1+n\chi _{1}}}\int x_{1}\left[{\overset {\backsim }{\Pi }}_{1}(a_{1},a_{3})\sin(\mu _{3}-\mu _{1})t+{\overset {\backsim }{\Pi }}_{2}(a_{1},a_{3})\sin 2(\mu _{3}-\mu _{1})t+\ldots \right]dt\\&-{\frac {n\chi _{4}}{1+n\chi _{1}}}\int x_{1}\left[{\overset {\backsim }{\Pi }}_{1}(a_{1},a_{4})\sin(\mu _{4}-\mu _{1})t+{\overset {\backsim }{\Pi }}_{2}(a_{1},a_{4})\sin 2(\mu _{4}-\mu _{1})t+\ldots \right]dt\\&+{\frac {n\mathrm {K} _{1}}{1+n\chi _{1}}}\int x_{1}\times 3\sin 2(m-\mu _{1})tdt\\&-{\frac {n\chi _{2}}{1+n\chi _{1}}}\int x_{2}\left[{\widehat {\Psi }}_{1}(a_{1},a_{2})\sin(\mu _{2}-\mu _{1})t+{\widehat {\Psi }}_{2}(a_{1},a_{2})\sin 2(\mu _{2}-\mu _{1})t+\ldots \right]dt\\&-{\frac {n\chi _{3}}{1+n\chi _{1}}}\int x_{3}\left[{\widehat {\Psi }}_{1}(a_{1},a_{3})\sin(\mu _{3}-\mu _{1})t+{\widehat {\Psi }}_{2}(a_{1},a_{3})\sin 2(\mu _{3}-\mu _{1})t+\ldots \right]dt\\&-{\frac {n\chi _{4}}{1+n\chi _{1}}}\int x_{4}\left[{\widehat {\Psi }}_{1}(a_{1},a_{4})\sin(\mu _{4}-\mu _{1})t+{\widehat {\Psi }}_{2}(a_{1},a_{4})\sin 2(\mu _{4}-\mu _{1})t+\ldots \right]dt\\&-{\frac {n\mathrm {K} _{1}}{1+n\chi _{1}}}\int \xi \times {\frac {9}{2}}\sin 2(m-\mu _{1})tdt\\&-{\frac {n\chi _{2}}{1+n\chi _{1}}}\int (y_{2}-y_{1})\left[{\widehat {\Gamma }}_{1}(a_{1},a_{2})\cos(\mu _{2}-\mu _{1})t+2{\widehat {\Gamma }}_{2}(a_{1},a_{2})\cos 2(\mu _{2}-\mu _{1})t+\ldots \right]dt\end{aligned}}$