Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/188

Maintenant, il est visible par l’équation (1^o) que, si $n$ était $=0,$ on aurait

$\mathrm {V_{1}^{2}+M_{1}^{2}} =0,$

c’est-à-dire, à cause de $\mathrm {M} _{1}^{2}=\mu _{1}^{2}(1-n{\text{ϐ}})$ (Article LXXIX

$\mathrm {V} _{1}^{2}+\mu _{1}^{2}=0\quad {\text{et}}\quad \mathrm {V} _{1}^{2}=-\mu _{1}^{2}.$

Supposons donc, en général,

$\mathrm {V} _{1}^{2}=-\mu _{1}^{2}(1-nv_{1}),$

et l’équation dont nous parlons se changera en celle-ci

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

XCVII.

L’équation

$\mathrm {V} _{1}^{2}=-\mu _{1}^{2}(1-nv_{1})$

donne

$\mathrm {V} _{1}=\mu _{1}\left(1-{\frac {n}{2}}v_{1}\right){\sqrt {-1}}\,;\quad {\text{donc}}\quad \mathrm {V} _{1}{\sqrt {-1}}=-\mu _{1}\left(1-{\frac {1}{2}}nv_{1}\right)\,;$

donc, substituant cette valeur dans l’équation $(\mathrm {Q} ),$ aussi bien que celle de $\mathrm {M} _{2}^{2},$ qui est $\mu _{2}^{2}(1-n{\text{ϐ}}_{2})$ (Article XXIX), on aura

${\begin{aligned}&\left[\mu _{2}^{2}(1-n{\text{ϐ}}_{2})-\left[\mu _{2}-\mu _{1}\mp \mu _{1}\left(1-{\frac {1}{2}}nv_{1}\right)\right]^{2}\right]\left(\mathrm {A} _{1}\pm \alpha _{1}{\sqrt {-1}}\right)\\&\ \ -\ \ \ nf_{1}\chi _{2}\left[{\overset {\circ }{\Psi }}_{1}(a_{1},a_{2})-\left[\mu _{2}-\mu _{1}\mp \mu _{1}\left(1-{\frac {1}{2}}nv_{1}\right)\right]{\overset {\circ }{\Pi }}_{1}(a_{1},a_{2})\right]\\&\ \ -{\frac {1}{2}}nf_{3}\chi _{2}\left[{\overset {\circ }{\Psi }}_{1}(a_{3},a_{2})-\left[\mu _{2}-\mu _{1}\mp \mu _{1}\left(1-{\frac {1}{2}}nv_{1}\right)\right]{\overset {\circ }{\Pi }}_{1}(a_{3},a_{2})\right]\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \times \left(\mathrm {B} _{1}\pm \beta _{1}{\sqrt {-1}}\right)\\&\ \ -{\frac {1}{2}}nf_{4}\chi _{2}\left[{\overset {\circ }{\Psi }}_{1}(a_{4},a_{2})-\left[\mu _{2}-\mu _{1}\mp \mu _{1}\left(1-{\frac {1}{2}}nv_{1}\right)\right]{\overset {\circ }{\Pi }}_{1}(a_{4},a_{2})\right]\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \times \left(\mathrm {C} _{1}\pm \gamma _{1}{\sqrt {-1}}\right)=0.\end{aligned}}$