Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/178

De là je tire, par la différentiation, les formules suivantes

${\begin{aligned}{\frac {d^{2}\mathrm {x} _{2}}{dt^{2}}}\sin(\mu _{2}-\mu _{1})t=&{\frac {d^{2}p}{dt^{2}}}-2(\mu _{2}-\mu _{1}){\frac {d\mathrm {P} }{dt}}-(\mu _{2}-\mu _{1})^{2}p,\\{\frac {d^{2}\mathrm {x} _{2}}{dt^{2}}}\cos(\mu _{2}-\mu _{1})t=&{\frac {d^{2}\mathrm {P} }{dt^{2}}}+2(\mu _{2}-\mu _{1}){\frac {dp}{dt}}-(\mu _{2}-\mu _{1})^{2}\mathrm {P} ,\\{\frac {d^{2}\mathrm {x} _{3}}{dt^{2}}}\sin(\mu _{3}-\mu _{1})t=&{\frac {d^{2}q}{dt^{2}}}-2(\mu _{3}-\mu _{1}){\frac {d\mathrm {Q} }{dt}}-(\mu _{3}-\mu _{1})^{2}q,\\{\frac {d^{2}\mathrm {x} _{3}}{dt^{2}}}\cos(\mu _{3}-\mu _{1})t=&{\frac {d^{2}\mathrm {Q} }{dt^{2}}}+2(\mu _{3}-\mu _{1}){\frac {dq}{dt}}-(\mu _{3}-\mu _{1})^{2}\mathrm {Q} ,\\{\frac {d^{2}\mathrm {x} _{4}}{dt^{2}}}\sin(\mu _{4}-\mu _{1})t=&{\frac {d^{2}r}{dt^{2}}}-2(\mu _{4}-\mu _{1}){\frac {d\mathrm {R} }{dt}}-(\mu _{4}-\mu _{1})^{2}r,\\{\frac {d^{2}\mathrm {x} _{4}}{dt^{2}}}\cos(\mu _{4}-\mu _{1})t=&{\frac {d^{2}\mathrm {R} }{dt^{2}}}+2(\mu _{4}-\mu _{1}){\frac {dr}{dt}}-(\mu _{4}-\mu _{1})^{2}\mathrm {R} .\end{aligned}}$

Cela posé, je multiplie d’abord l’équation $(\mathrm {M} _{2})$ par $\sin(\mu _{2}-\mu _{1})t,$ j’ai

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

Je ne conserve dans cette équation que les termes analogues à ceux de l’équation $(\mathrm {M} _{1}),$ c’est-à-dire, les termes qui, en faisant pour $x_{1},x_{2},\ldots$ les substitutions de l’Article LXXXVII, en donneraient d’autres où le coef-