Maintenant, il est visible par l’équation (1o) que, si
était
on aurait
![{\displaystyle \mathrm {V_{1}^{2}+M_{1}^{2}} =0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/655d182b94d2b68fa487aa66d01058a4b0f1ef82)
c’est-à-dire, à cause de
(Article LXXIX
![{\displaystyle \mathrm {V} _{1}^{2}+\mu _{1}^{2}=0\quad {\text{et}}\quad \mathrm {V} _{1}^{2}=-\mu _{1}^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c9f4832a1b543f804d575adcc6fecf2d0e6d998)
Supposons donc, en général,
![{\displaystyle \mathrm {V} _{1}^{2}=-\mu _{1}^{2}(1-nv_{1}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6ec668e95b903603db1c031531df91c0169aaab)
et l’équation dont nous parlons se changera en celle-ci
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80b4cb29fc2810127ff99b1a3a23efb568115759)
XCVII.
L’équation
![{\displaystyle \mathrm {V} _{1}^{2}=-\mu _{1}^{2}(1-nv_{1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68d2717922b40e1b1b38d291630deebe388fd390)
donne
![{\displaystyle \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)\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0617a55e97c3b021cb99bb6c9ea06451512a2efd)
donc, substituant cette valeur dans l’équation
aussi bien que celle de
qui est
(Article XXIX), on aura
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33f3dbc511631ed192c33468db22477e2c939571)