Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/179

ficient de $t$ serait presque égal à $\mu _{1},$ et qui sont les seuls auxquels nous devions avoir égard dans l’intégration de l’équation de l’Article XCII.

J’aurai donc simplement

${\begin{aligned}{\frac {d^{2}\mathrm {x} _{2}}{dt^{2}}}&\sin(\mu _{2}-\mu _{1})t+\mathrm {M} _{2}^{2}\mathrm {x} _{2}\sin(\mu _{2}-\mu _{1})t+{\frac {n}{2}}f_{2}\chi _{1}{\overset {\circ }{\Pi }}_{1}(a_{2},a_{1}){\frac {d\mathrm {x} _{1}}{dt}}\\-&{\frac {n}{2}}f_{2}\chi _{3}{\overset {\circ }{\Psi }}_{1}(a_{2},a_{3})\mathrm {x} _{3}\sin(\mu _{3}-\mu _{1})t+{\frac {n}{2}}f_{2}\chi _{3}{\overset {\circ }{\Pi }}_{1}(a_{2},a_{3}){\frac {d\mathrm {x} _{3}}{dt}}\cos(\mu _{3}-\mu _{1})t\\-&{\frac {n}{2}}f_{2}\chi _{4}{\overset {\circ }{\Psi }}_{1}(a_{2},a_{4})\mathrm {x} _{4}\sin(\mu _{4}-\mu _{1})t+{\frac {n}{2}}f_{2}\chi _{4}{\overset {\circ }{\Pi }}_{1}(a_{2},a_{4}){\frac {d\mathrm {x} _{4}}{dt}}\cos(\mu _{4}-\mu _{1})t=0.\end{aligned}}$

Je substitue au lieu de

${\frac {d^{2}\mathrm {x} _{2}}{dt^{2}}}\sin(\mu _{2}-\mu _{1})t,\quad \mathrm {x} _{2}\cos(\mu _{2}-\mu _{1})t,\ldots$

leurs valeurs en $\mathrm {P} ,p,\ldots \,;$ j’ai

(1^o)

${\begin{aligned}{\frac {d^{2}p}{dt^{2}}}&-2(\mu _{2}-\mu _{1}){\frac {d\mathrm {P} }{dt}}\left[\mathrm {M} _{2}^{2}-(\mu _{2}-\mu _{1})^{2}\right]p+{\frac {n}{2}}f_{2}\chi _{1}{\overset {\circ }{\Pi }}_{1}(a_{2},a_{1}){\frac {d\mathrm {x} _{1}}{dt}}\\&-{\frac {n}{2}}f_{2}\chi _{3}\left[{\overset {\circ }{\Psi }}_{1}(a_{2},a_{3})-(\mu _{3}-\mu _{1}){\overset {\circ }{\Pi }}_{1}(a_{2},a_{3})\right]q+{\frac {n}{2}}f_{2}\chi _{3}{\overset {\circ }{\Pi }}_{1}(a_{2},a_{3}){\frac {d\mathrm {Q} }{dt}}\\&-{\frac {n}{2}}f_{2}\chi _{4}\left[{\overset {\circ }{\Psi }}_{1}(a_{2},a_{4})-(\mu _{4}-\mu _{1}){\overset {\circ }{\Pi }}_{1}(a_{2},a_{4})\right]r+{\frac {n}{2}}f_{2}\chi _{4}{\overset {\circ }{\Pi }}_{1}(a_{2},a_{4}){\frac {d\mathrm {R} }{dt}}=0.\end{aligned}}$

Je multiplie en second lieu la même équation $(\mathrm {M} _{2})$ par $\cos(\mu _{2}-\mu _{1})t\,;$ j’ai

${\begin{aligned}{\frac {d^{2}\mathrm {x} _{2}}{dt^{2}}}&\cos(\mu _{2}-\mu _{1})t+\mathrm {M} _{2}^{2}\mathrm {x} _{2}\cos(\mu _{2}-\mu _{1})t\\&-{\frac {n}{2}}f_{2}\chi _{1}{\overset {\circ }{\Psi }}_{1}(a_{2},a_{1})\mathrm {x} _{1}\left[1+\cos 2(\mu _{2}-\mu _{1})t\right]\\&+{\frac {n}{2}}f_{2}\chi _{1}{\overset {\circ }{\Pi }}_{1}(a_{2},a_{1}){\frac {d\mathrm {x} _{1}}{dt}}\sin 2(\mu _{2}-\mu _{1})t\\&-{\frac {n}{2}}f_{2}\chi _{3}{\overset {\circ }{\Psi }}_{1}(a_{2},a_{3})\ \mathrm {x} _{3}\ \left[\cos(\mu _{3}-\mu _{1})t+\cos(\mu _{3}-2\mu _{2}+\mu _{1})t\right]\\&-{\frac {n}{2}}f_{2}\chi _{3}{\overset {\circ }{\Pi }}_{1}(a_{2},a_{3}){\frac {d\mathrm {x} _{3}}{dt}}\left[\sin(\mu _{3}-\mu _{1})t+\sin(\mu _{3}-2\mu _{2}+\mu _{1})t\right]\end{aligned}}$