Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/155

${\begin{aligned}&+n\chi _{3}f_{1}z_{1}\left[{\overset {\backsim }{\Gamma }}_{1}(a_{1},a_{3})+{\frac {2\mu _{1}}{\mu _{3}-\mu _{1}}}{\widehat {\Gamma }}_{1}(a_{1},a_{3})\right]\cos(\mu _{3}-\mu _{1})t\\&+n\chi _{3}f_{1}z_{1}\left[{\overset {\backsim }{\Gamma }}_{2}(a_{1},a_{3})+{\frac {2\mu _{1}}{2(\mu _{3}-\mu _{1})}}{\widehat {\Gamma }}_{2}(a_{1},a_{3})\right]\cos 2(\mu _{3}-\mu _{1})t\ldots \\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&+n\chi _{4}f_{1}z_{1}\left[{\overset {\backsim }{\Gamma }}_{1}(a_{1},a_{4})+{\frac {2\mu _{1}}{\mu _{4}-\mu _{1}}}{\widehat {\Gamma }}_{1}(a_{1},a_{4})\right]\cos(\mu _{4}-\mu _{1})t\\&+n\chi _{4}f_{1}z_{1}\left[{\overset {\backsim }{\Gamma }}_{2}(a_{1},a_{4})+{\frac {2\mu _{1}}{2(\mu _{4}-\mu _{1})}}{\widehat {\Gamma }}_{2}(a_{1},a_{4})\right]\cos 2(\mu _{4}-\mu _{1})t\ldots \\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&+n\mathrm {K} _{1}f_{1}z_{1}\left[{\frac {3}{2}}-{\frac {3\mu _{1}}{2(m-\mu _{1})}}\right]\cos 2(m-\mu _{1})t\\-&n\chi _{2}f_{1}z_{2}\\&\ \ \times \left[{\overset {\circ }{\Gamma }}(a_{1},a_{2})+{\overset {\circ }{\Gamma }}_{1}(a_{1},a_{2})\cos(\mu _{2}-\mu _{1})t+{\overset {\circ }{\Gamma }}_{2}(a_{1},a_{2})\cos 2(\mu _{2}-\mu _{1})t\ldots \right]\\-&n\chi _{3}f_{1}z_{3}\\&\ \ \times \left[{\overset {\circ }{\Gamma }}(a_{1},a_{3})+{\overset {\circ }{\Gamma }}_{1}(a_{1},a_{3})\cos(\mu _{3}-\mu _{1})t+{\overset {\circ }{\Gamma }}_{2}(a_{1},a_{3})\cos 2(\mu _{3}-\mu _{1})t\ldots \right]\\-&n\chi _{4}f_{1}z_{4}\\&\ \ \times \left[{\overset {\circ }{\Gamma }}(a_{1},a_{4})+{\overset {\circ }{\Gamma }}_{1}(a_{1},a_{4})\cos(\mu _{4}-\mu _{1})t+{\overset {\circ }{\Gamma }}_{2}(a_{1},a_{4})\cos 2(\mu _{4}-\mu _{1})t\ldots \right]=0.\end{aligned}}$

LXXVII.

Si l’on rejette dans les équations $(\mathrm {G} )$ et $(\mathrm {H} )$ tous les termes affectés de $n,$ comme aussi tous les termes constants qui doivent être nuls par les conditions de l’Article XXXII on a

${\frac {d^{2}x_{1}}{dt^{2}}}+\mathrm {M} _{1}^{2}x_{1}=0,\quad {\frac {dy_{1}}{dt}}+2\mu _{1}x_{1}=0.$

D’où l’on tire

$x_{1}=\varepsilon _{1}\cos(\mathrm {M} _{1}t+\omega _{1}),\quad y_{1}=-{\frac {2\mu _{1}}{\mathrm {M} _{1}}}\varepsilon _{1}\sin(\mathrm {M} _{1}t+\omega _{1}),$

ce qui donne pour $x_{1}$ et $y_{1}$ les mêmes valeurs que nous avons déjà trouvées (Article LXXV).

La quantité $-{\frac {2\mu _{1}}{\mathrm {M} _{1}}}\varepsilon _{1}\sin(\mathrm {M} _{1}t+\omega _{1})$ n’est que le premier terme de l’é-