Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/175

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

Pareillement on aura, par rapport aux variables $z,$ ces quatre équations (Article LXXXIX)

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