Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/182

Voilà, comme on voit, six équations différentielles de la même nature que l’équation de l’Article XCII, et qui, étant combinées avec cette dernière équation, suffiront pour déterminer les sept variables $\mathrm {x} _{1},\mathrm {P} ,p,\mathrm {Q} ,q,\mathrm {R} ,r.$

XCIV.

Pour cet effet, je multiplie l’équation de l’Article XCII par $e^{\mathrm {V} _{1}t},$ l’équation (1^o) de l’Article précédent par $\alpha _{1}e^{\mathrm {V} _{1}t}dt,$ l’équation (2^o) par $\mathrm {A} _{1}e^{\mathrm {V} _{1}t}dt,$ l’équation (3^o) par $\beta _{1}e^{\mathrm {V} _{1}t}dt,$ l’équation (4^o) par $\mathrm {B} _{1}e^{\mathrm {V} _{1}t}dt,$ l’équation (5^o) par $\gamma _{1}e^{\mathrm {V} _{1}t}dt,$ l’équation (6^o) par $\mathrm {C} _{1}e^{\mathrm {V} _{1}t}dt,$ ($\alpha _{1},\beta _{1},\gamma _{1},$ $\mathrm {A_{1},B_{1},C_{1}}$ et $\mathrm {V} _{1}$ sont des constantes indéterminées), et après en avoir fait une somme, j’en prends l’intégrale ; j’ai

${\begin{aligned}\int &\left({\frac {d^{2}\mathrm {x} _{1}}{dt^{2}}}+\alpha _{1}{\frac {d^{2}p}{dt^{2}}}+\mathrm {A} _{1}{\frac {d^{2}\mathrm {P} }{dt^{2}}}+\beta _{1}{\frac {d^{2}q}{dt^{2}}}+\mathrm {B} _{1}{\frac {d^{2}\mathrm {Q} }{dt^{2}}}+\gamma _{1}{\frac {d^{2}\mathrm {R} }{dt^{2}}}+\mathrm {C} _{1}{\frac {d^{2}r}{dt^{2}}}\right)e^{\mathrm {V} _{1}t}dt\\+\int &{\Biggl [}{\frac {n}{2}}\left(\alpha _{1}f_{2}\chi _{1}{\overset {\circ }{\Pi }}_{1}(a_{2},a_{1})+\beta _{1}f_{3}\chi _{1}{\overset {\circ }{\Pi }}_{1}(a_{3},a_{1})+\gamma _{1}f_{4}\chi _{1}{\overset {\circ }{\Pi }}_{1}(a_{4},a_{1})\right){\frac {d\mathrm {x} _{1}}{dt}}{\Biggr .}\\&\quad +\left(-nf_{1}\chi _{2}{\overset {\circ }{\Pi }}_{1}(a_{1},a_{2})+2\mathrm {A} _{1}(\mu _{2}-\mu _{1})-{\frac {n}{2}}\mathrm {B} _{1}f_{3}\chi _{2}{\overset {\circ }{\Pi }}_{1}(a_{3},a_{2})\right.\\&\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left.-{\frac {n}{2}}\mathrm {C} _{1}f_{4}\chi _{2}{\overset {\circ }{\Pi }}_{1}(a_{4},a_{2})\right){\frac {dp}{dt}}\\&\quad +\left(-2\alpha _{1}(\mu _{2}-\mu _{1})+{\frac {n}{2}}\beta _{1}f_{3}\chi _{2}{\overset {\circ }{\Pi }}_{1}(a_{3},a_{2})+{\frac {n}{2}}\gamma _{1}f_{4}\chi _{2}{\overset {\circ }{\Pi }}_{1}(a_{4},a_{2})\right){\frac {d\mathrm {P} }{dt}}\\&\quad +\left(-nf_{1}\chi _{3}{\overset {\circ }{\Pi }}_{1}(a_{1},a_{3})-{\frac {n}{2}}\mathrm {A} _{1}f_{2}\chi _{3}{\overset {\circ }{\Pi }}_{1}(a_{2},a_{3})+2\mathrm {B} _{1}(\mu _{3}-\mu _{1})\right.\\&\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left.-{\frac {n}{2}}\mathrm {C} _{1}f_{4}\chi _{3}{\overset {\circ }{\Pi }}_{1}(a_{4},a_{3})\right){\frac {dq}{dt}}\\&\quad +\left({\frac {n}{2}}\alpha _{1}f_{2}\chi _{3}{\overset {\circ }{\Pi }}_{1}(a_{2},a_{3})-2\beta _{1}(\mu _{3}-\mu _{1})+{\frac {n}{2}}\gamma _{1}f_{4}\chi _{3}{\overset {\circ }{\Pi }}_{1}(a_{4},a_{3})\right){\frac {d\mathrm {Q} }{dt}}\\&\quad +\left(-nf_{1}\chi _{4}{\overset {\circ }{\Pi }}_{1}(a_{1},a_{4})-{\frac {n}{2}}\mathrm {A} _{1}f_{2}\chi _{4}{\overset {\circ }{\Pi }}_{1}(a_{2},a_{4})-{\frac {n}{2}}\mathrm {B} _{1}f_{3}\chi _{4}{\overset {\circ }{\Pi }}_{1}(a_{3},a_{4})\right.\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad {\biggl .}+2\mathrm {C} _{1}(\mu _{4}-\mu _{1}){\biggr )}{\frac {dr}{dt}}\\&\qquad {\Biggl .}+\left({\frac {n}{2}}\alpha _{1}f_{2}\chi _{4}{\overset {\circ }{\Pi }}_{1}(a_{2},a_{4})+{\frac {n}{2}}\beta _{1}f_{3}\chi _{4}{\overset {\circ }{\Pi }}_{1}(a_{3},a_{4})+2\gamma _{1}(\mu _{4}-\mu _{1})\right){\frac {d\mathrm {R} }{dt}}{\Biggr ]}e^{\mathrm {V} _{1}t}dt\\+\int &{\Biggl [}\left(\mathrm {M} _{1}^{2}-{\frac {n}{2}}\mathrm {A} _{1}f_{2}\chi _{1}{\overset {\circ }{\Psi }}_{1}(a_{2},a_{1})-{\frac {n}{2}}\mathrm {B} _{1}f_{3}\chi _{1}{\overset {\circ }{\Psi }}_{1}(a_{3},a_{1})\right.\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left.-{\frac {n}{2}}\mathrm {C} _{1}f_{4}\chi _{1}{\overset {\circ }{\Psi }}_{1}(a_{4},a_{1})\right)\mathrm {x} _{1}{\Biggr .}\\&\quad +{\Biggl (}\alpha _{1}\left[\mathrm {M} _{2}^{2}-(\mu _{2}-\mu _{1})^{2}\right]-{\frac {n}{2}}\beta _{1}f_{3}\chi _{2}\left[{\overset {\circ }{\Psi }}_{1}(a_{3},a_{2})-(\mu _{2}-\mu _{1}){\overset {\circ }{\Pi }}_{1}(a_{3},a_{2})\right]{\Biggr .}\\&\qquad \qquad \qquad \qquad \qquad \qquad {\Biggl .}-{\frac {n}{2}}\gamma _{1}f_{4}\chi _{2}\left[{\overset {\circ }{\Psi }}_{1}(a_{4},a_{2})-(\mu _{2}-\mu _{1}){\overset {\circ }{\Pi }}_{1}(a_{4},a_{2})\right]{\Biggr )}p\end{aligned}}$