Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/186

(7^o)

${\begin{aligned}\mathrm {C} _{1}\mathrm {V} _{1}^{2}&-\left[{\frac {n}{2}}\alpha _{1}f_{2}\chi _{4}{\overset {\circ }{\Pi }}_{1}(a_{2},a_{4})+{\frac {n}{2}}\beta _{1}f_{3}\chi _{4}{\overset {\circ }{\Pi }}_{1}(a_{3},a_{4})-2\gamma _{1}(\mu _{4}-\mu _{1})\right]\mathrm {V} _{1}\\&-{\frac {n}{2}}f_{1}\chi _{4}\left[{\overset {\circ }{\Psi }}_{1}(a_{1},a_{4})-(\mu _{4}-\mu _{1}){\overset {\circ }{\Pi }}_{1}(a_{1},a_{4})\right]\\&-{\frac {n}{2}}\mathrm {A} _{1}f_{2}\chi _{4}\left[{\overset {\circ }{\Psi }}_{1}(a_{2},a_{4})-(\mu _{4}-\mu _{1}){\overset {\circ }{\Pi }}_{1}(a_{2},a_{4})\right]\\&-{\frac {n}{2}}\mathrm {B} _{1}f_{3}\chi _{4}\left[{\overset {\circ }{\Psi }}_{1}(a_{3},a_{4})-(\mu _{4}-\mu _{1}){\overset {\circ }{\Pi }}_{1}(a_{3},a_{4})\right]\\&\ \ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad +\mathrm {C} _{1}\left[\mathrm {M} _{4}^{2}-(\mu _{4}-\mu _{1})^{2}\right]=0.\end{aligned}}$

Ensuite j’ai l’équation intégrale

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