Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/185

(3^o)

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

(4^o)

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

(5^o)

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

(6^o)

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