Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/173

en son équivalente

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

$-(\mu _{2}-\mu _{1})^{2}\int \mathrm {x} _{2}\sin(\mu _{2}-\mu _{1})tdt,\ \ \qquad$

et il me vient l’équation

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

$+\left[\mathrm {M} _{2}^{2}-(\mu _{2}-\mu _{1})^{2}\right]\int \mathrm {x} _{2}\sin(\mu _{2}-\mu _{1})tdt=0\,;$

d’où je tire

$\int \mathrm {x} _{2}\sin(\mu _{2}-\mu _{1})tdt={\frac {(\mu _{2}-\mu _{1})\mathrm {x} _{2}\cos(\mu _{2}-\mu _{1})t-{\cfrac {d\mathrm {x} _{2}}{dt}}\sin(\mu _{2}-\mu _{1})t}{\mathrm {M} _{2}^{2}-(\mu _{2}-\mu _{1})^{2}}}.$

Je trouve, de la même manière,

${\begin{aligned}\int \mathrm {x} _{3}\sin(\mu _{3}-\mu _{1})tdt={\frac {(\mu _{3}-\mu _{1})\mathrm {x} _{3}\cos(\mu _{3}-\mu _{1})t-{\cfrac {d\mathrm {x} _{3}}{dt}}\sin(\mu _{3}-\mu _{1})t}{\mathrm {M} _{3}^{2}-(\mu _{3}-\mu _{1})^{2}}}.\\\int \mathrm {x} _{4}\sin(\mu _{4}-\mu _{1})tdt={\frac {(\mu _{4}-\mu _{1})\mathrm {x} _{4}\cos(\mu _{4}-\mu _{1})t-{\cfrac {d\mathrm {x} _{4}}{dt}}\sin(\mu _{4}-\mu _{1})t}{\mathrm {M} _{4}^{2}-(\mu _{4}-\mu _{1})^{2}}}.\end{aligned}}$

On fera toutes ces substitutions dans l’équation $(\mathrm {L} ),$ moyennant quoi elle n’aura plus que des termes de la forme de

$\mathrm {x} _{2}\cos(\mu _{2}-\mu _{1})t\quad {\text{et}}\quad {\frac {d\mathrm {x} _{2}}{dt}}\sin(\mu _{2}-\mu _{1})t.$

XCI.

Donc si l’on fait, pour abréger,

${\begin{aligned}{\overset {\circ }{\Psi }}_{1}=&{\widehat {\Psi }}_{1}(a_{1},a_{2})-4{\frac {\mu _{1}\mu _{2}}{\mathrm {M} _{2}^{2}}}{\widehat {\Gamma }}_{1}(a_{1},a_{2})\\&\quad -\left[2{\widehat {\Psi }}_{1}(a_{1},a_{2})+4{\frac {\mu _{2}(\mu _{2}-\mu _{1})}{\mathrm {M} _{2}^{2}}}{\widehat {\Gamma }}_{1}(a_{1},a_{2})\right]{\frac {\mu _{1}(\mu _{2}-\mu _{1})}{\mathrm {M} _{2}^{2}-(\mu _{2}-\mu _{1})^{2}}},\\{\overset {\circ }{\Psi }}_{1}=&{\widehat {\Psi }}_{1}(a_{1},a_{3})-4{\frac {\mu _{1}\mu _{3}}{\mathrm {M} _{3}^{2}}}{\widehat {\Gamma }}_{1}(a_{1},a_{3})\\&\quad -\left[2{\widehat {\Psi }}_{1}(a_{1},a_{3})+4{\frac {\mu _{3}(\mu _{3}-\mu _{1})}{\mathrm {M} _{3}^{2}}}{\widehat {\Gamma }}_{1}(a_{1},a_{3})\right]{\frac {\mu _{1}(\mu _{3}-\mu _{1})}{\mathrm {M} _{3}^{2}-(\mu _{3}-\mu _{1})^{2}}},\\{\overset {\circ }{\Psi }}_{1}=&{\widehat {\Psi }}_{1}(a_{1},a_{4})-4{\frac {\mu _{1}\mu _{4}}{\mathrm {M} _{4}^{2}}}{\widehat {\Gamma }}_{1}(a_{1},a_{4})\\&\quad -\left[2{\widehat {\Psi }}_{1}(a_{1},a_{4})+4{\frac {\mu _{4}(\mu _{4}-\mu _{1})}{\mathrm {M} _{4}^{2}}}{\widehat {\Gamma }}_{1}(a_{1},a_{4})\right]{\frac {\mu _{1}(\mu _{4}-\mu _{1})}{\mathrm {M} _{4}^{2}-(\mu _{4}-\mu _{1})^{2}}},\\\end{aligned}}$