et
étant deux constantes arbitraires que nous déterminerons dans un moment.
On aura donc par là
![{\displaystyle {\begin{aligned}(\gamma )\quad \mu _{1}\mathrm {x} _{1}&-\mathrm {A} _{1}\left({\frac {dp}{dt}}-(2\mu _{2}-\mu _{1})\mathrm {P} \right)-\mathrm {B} _{1}\left({\frac {dq}{dt}}-(2\mu _{3}-\mu _{1})\mathrm {Q} \right)\\&-\mathrm {C} _{1}\left({\frac {dr}{dt}}-(2\mu _{4}-\mu _{1})\mathrm {R} \right)\\&=\mathrm {D} _{1}\sin \left(\mu _{1}-{\frac {n}{2}}\rho \right)t-\mathrm {E} _{1}\cos \left(\mu _{1}-{\frac {n}{2}}\rho \right)t,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/149cb432ee874a28600e8c8f4aa7e35a34be550f)
équation qui suffira pour trouver la valeur de
comme on le verra ci-après.
CII.
Soit, lorsque
![{\displaystyle \mathrm {x} _{1}=\mathrm {X} _{1},\quad \mathrm {x} _{2}=\mathrm {X} _{2},\quad \mathrm {x} _{3}=\mathrm {X} _{3},\quad \mathrm {x} _{4}=\mathrm {X} _{4},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33467b9b642d8c53c9e78b830a5f5e6bb436484f)
![{\displaystyle {\frac {d\mathrm {x} _{1}}{dt}}=\mathrm {Y} _{1},\quad {\frac {d\mathrm {x} _{2}}{dt}}=\mathrm {Y} _{2},\quad {\frac {d\mathrm {x} _{3}}{dt}}=\mathrm {Y} _{3},\quad {\frac {d\mathrm {x} _{4}}{dt}}=\mathrm {Y} _{4}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43bb05b47159a0100b65fa49e2a48967465380ca)
On aura (Article XCII)
![{\displaystyle \mathrm {P=X_{2},\quad Q=X_{3},\quad R=X_{4}} ,\quad p=0,\quad q=0,\quad r=0\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/541f58b365ff7f2f3b49f785c848a495d7d80127)
ensuite
![{\displaystyle {\frac {d\mathrm {P} }{dt}}=\mathrm {Y} _{2},\quad {\frac {d\mathrm {Q} }{dt}}=\mathrm {Y} _{3},\quad {\frac {d\mathrm {R} }{dt}}=\mathrm {Y} _{4},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa1bd6a403f542672715c7e51a902bf7305f3d8b)
![{\displaystyle {\frac {dp}{dt}}=(\mu _{2}-\mu _{1})\mathrm {X} _{2},\quad {\frac {dq}{dt}}=(\mu _{3}-\mu _{1})\mathrm {X} _{3},\quad {\frac {dr}{dt}}=(\mu _{4}-\mu _{1})\mathrm {X} _{4}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/398b54c3c3ef2efe18b627992b63dfe3b4298b07)
Donc, substituant ces valeurs dans les équations
et
et faisant
on aura
![{\displaystyle {\begin{aligned}&\mathrm {D_{1}=Y_{1}+A_{1}Y_{2}+B_{1}Y_{3}+C_{1}Y_{4}} ,\\&\mathrm {E_{1}\ =-\left(\mu _{1}X_{1}+A_{1}\mu _{2}X_{2}+B_{1}\mu _{3}X_{3}+C_{1}\mu _{4}X_{4}\right)} .\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/876513935e3cbeda448e8dffe9df02cf2d87f20a)
CIII.
Soit maintenant, pour abréger,
![{\displaystyle {\frac {dp}{dt}}-(2\mu _{2}-\mu _{1})\mathrm {P=(P)} ,\quad {\frac {dq}{dt}}-(2\mu _{3}-\mu _{1})\mathrm {Q=(Q)} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d773209e15745d245b579202e984c4d848eef424)
![{\displaystyle {\frac {dr}{dt}}-(2\mu _{4}-\mu _{1})\mathrm {R=(R)} \,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fac61983b542c5295be223a411adf6f7c4b83eb0)