Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/171

donc, substituant ces valeurs de $\mathrm {y_{2},y_{3},y_{4}}$ dans l’équation $(\mathrm {L} )$ de l’Article précédent, on changera les termes

${\begin{aligned}&nf_{1}\chi _{2}{\breve {\Gamma }}_{1}\,(a_{1},a_{2})\mathrm {y} _{2}\sin(\mu _{2}-\mu _{1})t\\+&nf_{1}\chi _{3}{\breve {\Gamma }}_{1}\,(a_{1},a_{3})\mathrm {y} _{3}\sin(\mu _{3}-\mu _{1})t\\+&nf_{1}\chi _{4}{\breve {\Gamma }}_{1}\,(a_{1},a_{4})\mathrm {y} _{4}\sin(\mu _{4}-\mu _{1})t\end{aligned}}$

en ceux-ci

${\begin{aligned}&2nf_{1}\chi _{2}{\frac {\mu _{2}}{\mathrm {M} _{2}^{2}}}{\breve {\Gamma }}_{1}(a_{1},a_{2}){\frac {d\mathrm {x} _{2}}{dt}}\sin(\mu _{2}-\mu _{1})t\\+&2nf_{1}\chi _{3}{\frac {\mu _{3}}{\mathrm {M} _{3}^{2}}}{\breve {\Gamma }}_{1}(a_{1},a_{3}){\frac {d\mathrm {x} _{3}}{dt}}\sin(\mu _{3}-\mu _{1})t\\+&2nf_{1}\chi _{4}{\frac {\mu _{4}}{\mathrm {M} _{4}^{2}}}{\breve {\Gamma }}_{1}(a_{1},a_{4}){\frac {d\mathrm {x} _{4}}{dt}}\sin(\mu _{4}-\mu _{1})t,\end{aligned}}$

et les termes

${\begin{aligned}&2nf_{1}\mu _{1}\chi _{2}{\widehat {\Gamma }}_{1}\,(a_{1},a_{2})\int \mathrm {y} _{2}\cos(\mu _{2}-\mu _{1})tdt\\+&2nf_{1}\mu _{1}\chi _{3}{\widehat {\Gamma }}_{1}\,(a_{1},a_{3})\int \mathrm {y} _{3}\cos(\mu _{3}-\mu _{1})tdt\\+&2nf_{1}\mu _{1}\chi _{4}{\widehat {\Gamma }}_{1}\,(a_{1},a_{4})\int \mathrm {y} _{4}\cos(\mu _{4}-\mu _{1})tdt\end{aligned}}$

en ceux-ci

${\begin{aligned}&4nf_{1}\mu _{1}\chi _{2}{\frac {\mu _{2}}{\mathrm {M} _{2}^{2}}}{\widehat {\Gamma }}_{1}(a_{1},a_{2})\int {\frac {d\mathrm {x} _{2}}{dt}}\cos(\mu _{2}-\mu _{1})tdt\\+&4nf_{1}\mu _{1}\chi _{3}{\frac {\mu _{3}}{\mathrm {M} _{3}^{2}}}{\widehat {\Gamma }}_{1}(a_{1},a_{3})\int {\frac {d\mathrm {x} _{3}}{dt}}\cos(\mu _{3}-\mu _{1})tdt\\+&4nf_{1}\mu _{1}\chi _{4}{\frac {\mu _{4}}{\mathrm {M} _{4}^{2}}}{\widehat {\Gamma }}_{1}(a_{1},a_{4})\int {\frac {d\mathrm {x} _{4}}{dt}}\cos(\mu _{4}-\mu _{1})tdt,\end{aligned}}$