Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/183

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

XCV.

Cela fait, je transforme les expressions intégrales

$\int {\frac {d^{2}\mathrm {x} }{dt^{2}}}e^{{\text{V}}_{1}t}dt,\quad \int {\frac {d^{2}p}{dt^{2}}}e^{{\text{V}}_{1}t}dt,\ldots$

en leurs équivalentes

$\left({\frac {d\mathrm {x} _{1}}{dt}}-\mathrm {V_{1}x_{1}} \right)e^{{\text{V}}_{1}t}+\mathrm {V} _{1}^{2}\int \mathrm {x} _{1}\varepsilon ^{{\text{V}}_{1}t}dt,\quad \left({\frac {dp}{dt}}-\mathrm {V} _{1}p\right)e^{{\text{V}}_{1}t}+\mathrm {V} _{1}^{2}\int pe^{{\text{V}}_{1}t}dt,\ldots .$