Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/115

on trouvera les trois équations suivantes

$\left.{\begin{aligned}1^{\circ }&\ \ {\frac {d^{2}x_{1}}{dt^{2}}}+\mathrm {M} _{1}^{2}x_{1}+f_{1}\mathrm {L} _{1}\\&-\chi _{2}f_{1}\left[{\breve {\Gamma }}_{1}(a_{1},a_{2})+{\frac {2\mu _{1}}{\mu _{2}-\mu _{1}}}{\widehat {\Gamma }}_{1}(a_{1},a_{2})\right]\cos(\mu _{2}-\mu _{1})t\\&-\chi _{2}f_{1}\left[{\breve {\Gamma }}_{2}(a_{1},a_{2})+{\frac {2\mu _{1}}{2(\mu _{2}-\mu _{1})}}{\widehat {\Gamma }}_{2}(a_{1},a_{2})\right]\cos 2(\mu _{2}-\mu _{1})t+\ldots \\&-\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&-\chi _{3}f_{1}\left[{\breve {\Gamma }}_{1}(a_{1},a_{3})+{\frac {2\mu _{1}}{\mu _{3}-\mu _{1}}}{\widehat {\Gamma }}_{1}(a_{1},a_{3})\right]\cos(\mu _{3}-\mu _{1})t\\&-\chi _{3}f_{1}\left[{\breve {\Gamma }}_{2}(a_{1},a_{3})+{\frac {2\mu _{1}}{2(\mu _{3}-\mu _{1})}}{\widehat {\Gamma }}_{2}(a_{1},a_{3})\right]\cos 2(\mu _{3}-\mu _{1})t+\ldots \\&-\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&-\chi _{4}f_{1}\left[{\breve {\Gamma }}_{1}(a_{1},a_{4})+{\frac {2\mu _{1}}{\mu _{4}-\mu _{1}}}{\widehat {\Gamma }}_{1}(a_{1},a_{4})\right]\cos(\mu _{4}-\mu _{1})t\\&-\chi _{4}f_{1}\left[{\breve {\Gamma }}_{2}(a_{1},a_{4})+{\frac {2\mu _{1}}{2(\mu _{4}-\mu _{1})}}{\widehat {\Gamma }}_{2}(a_{1},a_{4})\right]\cos 2(\mu _{4}-\mu _{1})t+\ldots \\&-\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\\end{aligned}}\right\}=0\,;$

$\left.{\begin{aligned}2^{\circ }&\ \ {\frac {d^{2}y_{1}}{dt^{2}}}+2\mu _{1}x_{1}-f_{1}h_{1}\\&-\chi _{2}f_{1}\left[{\frac {{\widehat {\Gamma }}_{1}(a_{1},a_{2})}{\mu _{2}-\mu _{1}}}\cos(\mu _{2}-\mu _{1})t+{\frac {{\widehat {\Gamma }}_{2}(a_{1},a_{2})}{2(\mu _{2}-\mu _{1})}}\cos 2(\mu _{2}-\mu _{1})t+\ldots \right]\\&-\chi _{3}f_{1}\left[{\frac {{\widehat {\Gamma }}_{1}(a_{1},a_{3})}{\mu _{3}-\mu _{1}}}\cos(\mu _{3}-\mu _{1})t+{\frac {{\widehat {\Gamma }}_{2}(a_{1},a_{3})}{2(\mu _{3}-\mu _{1})}}\cos 2(\mu _{3}-\mu _{1})t+\ldots \right]\\&-\chi _{4}f_{1}\left[{\frac {{\widehat {\Gamma }}_{1}(a_{1},a_{4})}{\mu _{4}-\mu _{1}}}\cos(\mu _{4}-\mu _{1})t+{\frac {{\widehat {\Gamma }}_{2}(a_{1},a_{4})}{2(\mu _{4}-\mu _{1})}}\cos 2(\mu _{4}-\mu _{1})t+\ldots \right]\\&+\mathrm {K} _{1}{\frac {3f_{1}}{4(m-\mu _{1})}}\cos 2(m-\mu _{1})t\\\end{aligned}}\right\}=0\,;$

$3^{\circ }\qquad \qquad \qquad \qquad \qquad z_{1}{\frac {d^{2}t}{dt^{2}}}\mathrm {N} _{1}z_{1}=0.$