Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/189

Donc :

1^o Si l’on prend le signe supérieur, et qu’on néglige les termes affectés de $n,$ on aura

$\left[\mu _{2}^{2}-(\mu _{2}-2\mu _{1})^{2}\right]\left(\mathrm {A} _{1}+\alpha _{1}{\sqrt {-1}}\right)=0\,;$

ce qui donne

$\mathrm {A} _{1}+\alpha _{1}{\sqrt {-1}}=0\quad {\text{et}}\quad \alpha _{1}{\sqrt {-1}}=-\mathrm {A} _{1}.$

2^o Si l’on prend le signe inférieur, et qu’après avoir ôté ce qui se détruit on divise toute l’équation par $n,$ on aura, en négligeant toujours les termes affectés de $n,$

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

On tirera de même de l’équation $(\mathrm {R} )$

$\beta _{1}{\sqrt {-1}}=-\mathrm {B} _{1}$

et

${\begin{aligned}(\mathrm {U} )\quad \mu _{3}^{2}\left({\frac {\mu _{1}}{\mu _{3}}}v_{1}-{\text{ϐ}}_{3}\right)&\left(\mathrm {B} _{1}-\beta _{1}{\sqrt {-1}}\right)-f_{1}\chi _{3}\left[{\overset {\circ }{\Psi }}_{1}(a_{1},a_{3})-\mu _{3}{\overset {\circ }{\Pi }}_{1}(a_{1},a_{3})\right]\\-&{\frac {1}{2}}f_{2}\chi _{3}\left[{\overset {\circ }{\Psi }}_{1}(a_{2},a_{3})-\mu _{3}{\overset {\circ }{\Pi }}_{1}(a_{2},a_{3})\right]\left(\mathrm {A} _{1}-\beta _{1}{\sqrt {-1}}\right)\\-&{\frac {1}{2}}f_{4}\chi _{3}\left[{\overset {\circ }{\Psi }}_{1}(a_{4},a_{3})-\mu _{3}{\overset {\circ }{\Pi }}_{1}(a_{4},a_{3})\right]\left(\mathrm {C} _{1}-\gamma _{1}{\sqrt {-1}}\right)=0\,;\end{aligned}}$

et de l’équation $(\mathrm {S} )$

$\gamma _{1}{\sqrt {-1}}=-\mathrm {C} _{1}$

et

${\begin{aligned}(\mathrm {W} )\quad \mu _{4}^{2}\left({\frac {\mu _{1}}{\mu _{4}}}v_{1}-{\text{ϐ}}_{4}\right)&\left(\mathrm {C} _{1}-\gamma _{1}{\sqrt {-1}}\right)-f_{1}\chi _{4}\left[{\overset {\circ }{\Psi }}_{1}(a_{1},a_{4})-\mu _{4}{\overset {\circ }{\Pi }}_{1}(a_{1},a_{4})\right]\\-&{\frac {1}{2}}f_{2}\chi _{4}\left[{\overset {\circ }{\Psi }}_{1}(a_{2},a_{4})-\mu _{4}{\overset {\circ }{\Pi }}_{1}(a_{2},a_{4})\right]\left(\mathrm {A} _{1}-\beta _{1}{\sqrt {-1}}\right)\\-&{\frac {1}{2}}f_{3}\chi _{4}\left[{\overset {\circ }{\Psi }}_{1}(a_{3},a_{4})-\mu _{4}{\overset {\circ }{\Pi }}_{1}(a_{3},a_{4})\right]\left(\mathrm {B} _{1}-\beta _{1}{\sqrt {-1}}\right)=0.\end{aligned}}$