Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/176

${\begin{aligned}(\mathrm {N} _{3})\qquad \qquad {\frac {d^{2}z_{2}}{dt^{2}}}&+\mathrm {N} _{2}^{2}z_{2}\\&-nf_{3}\chi _{1}{\overset {\circ }{\Gamma }}_{1}(a_{3},a_{1})z_{1}\cos(\mu _{1}-\mu _{3})t\\&-nf_{3}\chi _{2}{\overset {\circ }{\Gamma }}_{1}(a_{3},a_{2})z_{2}\cos(\mu _{2}-\mu _{3})t\\&-nf_{3}\chi _{4}{\overset {\circ }{\Gamma }}_{1}(a_{3},a_{4})z_{4}\cos(\mu _{4}-\mu _{3})t=0,\\\\(\mathrm {N} _{4})\qquad \qquad {\frac {d^{2}z_{4}}{dt^{2}}}&+\mathrm {N} _{4}^{2}z_{4}\\&-nf_{4}\chi _{1}{\overset {\circ }{\Gamma }}_{1}(a_{4},a_{1})z_{1}\cos(\mu _{1}-\mu _{4})t\\&-nf_{4}\chi _{2}{\overset {\circ }{\Gamma }}_{1}(a_{4},a_{2})z_{2}\cos(\mu _{2}-\mu _{4})t\\&-nf_{4}\chi _{3}{\overset {\circ }{\Gamma }}_{1}(a_{4},a_{3})z_{3}\cos(\mu _{3}-\mu _{4})t=0.\end{aligned}}$

§ III. — Où l’on donne une nouvelle méthode pour intégrer les équations précédentes.

XCII.

Je fais

${\begin{alignedat}{2}\mathrm {x} _{2}\cos(\mu _{2}-\mu _{1})t=&\mathrm {P} ,\qquad &\mathrm {x} _{2}\sin(\mu _{2}-\mu _{1})t=&p,\\\mathrm {x} _{3}\cos(\mu _{3}-\mu _{1})t=&\mathrm {Q} ,&\mathrm {x} _{3}\sin(\mu _{3}-\mu _{1})t=&q,\\\mathrm {x} _{4}\cos(\mu _{4}-\mu _{1})t=&\mathrm {R} ,&\mathrm {x} _{4}\sin(\mu _{4}-\mu _{1})t=&r,\end{alignedat}}$

d’où je tire

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