Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/154

${\begin{aligned}&-2n\mathrm {K} _{1}f_{1}x_{1}\times {\frac {3}{4(m-\mu _{1})}}\cos 2(m-\mu _{1})t\\&+n\chi _{2}f_{1}\\&\quad \times \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\\&+n\chi _{3}f_{1}\\&\quad \times \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\\&+n\chi _{4}f_{1}\\&\quad \times \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\\\\&+n\mathrm {K} _{1}f_{1}\int x_{1}\times 3\sin 2(m-\mu _{1})tdt\\&+n\chi _{2}f_{1}\\&\quad \times \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\\&+n\chi _{3}f_{1}\\&\quad \times \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\\&+n\chi _{4}f_{1}\\&\quad \times \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\\\\&+n\mathrm {K} _{1}f_{1}\int \xi \times {\frac {9}{2}}\sin 2(m-\mu _{1})tdt\\+&n\chi _{2}f_{1}\\&\ \ \times \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\\+&n\chi _{3}f_{1}\\&\ \ \times \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\\+&n\chi _{4}f_{1}\\&\ \ \times \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\\\\&-n\mathrm {K} _{1}f_{1}\int (\mathrm {J} -y_{1})\times 3\cos 2(m-\mu _{1})tdt=0,\\\\(\mathrm {K} )&\ \ {\frac {d^{2}z_{1}}{dt^{2}}}+\mathrm {N} _{1}^{2}-4n\mu _{1}^{2}z_{1}x_{1}+2n{\frac {dx_{1}dz_{1}}{dt^{2}}}\\&+n\chi _{2}f_{1}z_{1}\left[{\overset {\backsim }{\Gamma }}_{1}(a_{1},a_{2})+{\frac {2\mu _{1}}{\mu _{2}-\mu _{1}}}{\widehat {\Gamma }}_{1}(a_{1},a_{2})\right]\cos(\mu _{2}-\mu _{1})t\\&+n\chi _{2}f_{1}z_{1}\left[{\overset {\backsim }{\Gamma }}_{2}(a_{1},a_{2})+{\frac {2\mu _{1}}{2(\mu _{2}-\mu _{1})}}{\widehat {\Gamma }}_{2}(a_{1},a_{2})\right]\cos 2(\mu _{2}-\mu _{1})t\ldots \\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \end{aligned}}$