c’est-à-dire en mettant au lieu de e ( μ 1 − 1 2 n ρ ) t − 1 {\displaystyle e^{\left(\mu _{1}-{\frac {1}{2}}n\rho \right)t{\sqrt {-1}}}} sa valeur cos ( μ 1 − 1 2 n ρ ) t + sin ( μ 1 − 1 2 n ρ ) t × − 1 , {\displaystyle \cos \left(\mu _{1}-{\frac {1}{2}}n\rho \right)t+\sin(\mu _{1}-{\frac {1}{2}}n\rho )t\times {\sqrt {-1}},}
équation qui, à cause du radical − 1 , {\displaystyle {\sqrt {-1}},} lequel peut avoir indifféremment les signes + {\displaystyle +} ou − {\displaystyle -} se décompose en ces deux-ci
et
sa valeur 
![{\displaystyle {\begin{aligned}\left[{\frac {d\mathrm {x} _{1}}{dt}}+\mathrm {A} _{1}\left({\frac {d\mathrm {P} }{dt}}+(2\mu _{2}-\mu _{1})p\right)+\mathrm {B} _{1}\left({\frac {d\mathrm {Q} }{dt}}+(2\mu _{3}-\mu _{1})q\right)\right.&\\\left.+\mathrm {C} _{1}\left({\frac {d\mathrm {R} }{dt}}+(2\mu _{4}-\mu _{1})r\right)\right]\cos \left(\mu _{1}-{\frac {n}{2}}\rho \right)t&\\+\left[\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)\right.&\\\left.-\mathrm {C} _{1}\left({\frac {dr}{dt}}-(2\mu _{4}-\mu _{1})\mathrm {R} \right)\right]\sin \left(\mu _{1}-{\frac {n}{2}}\rho \right)t&\\+\left[{\frac {d\mathrm {x} _{1}}{dt}}+\mathrm {A} _{1}\left({\frac {d\mathrm {P} }{dt}}+(2\mu _{2}-\mu _{1})p\right)+\mathrm {B} _{1}\left({\frac {d\mathrm {Q} }{dt}}+(2\mu _{3}-\mu _{1})q\right)\right.&\\\left.+\mathrm {C} _{1}\left({\frac {d\mathrm {R} }{dt}}+(2\mu _{4}-\mu _{1})r\right)\right]\sin \left(\mu _{1}-{\frac {n}{2}}\rho \right)t&\times {\sqrt {-1}}\\-\left[\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)\right.&\\\left.-\mathrm {C} _{1}\left({\frac {dr}{dt}}-(2\mu _{4}-\mu _{1})\mathrm {R} \right)\right]\cos \left(\mu _{1}-{\frac {n}{2}}\rho \right)t&\times {\sqrt {-1}}=\mathrm {const} .,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fea20eeb629c4229158590fb60511e1e4293bbbc)
lequel peut avoir indifféremment les signes 
ou 
se décompose en ces deux-ci![{\displaystyle {\begin{aligned}(\alpha )\quad \left[{\frac {d\mathrm {x} _{1}}{dt}}+\mathrm {A} _{1}\left({\frac {d\mathrm {P} }{dt}}+(2\mu _{2}-\mu _{1})p\right)+\mathrm {B} _{1}\left({\frac {d\mathrm {Q} }{dt}}+(2\mu _{3}-\mu _{1})q\right)\right.&\\\left.+\mathrm {C} _{1}\left({\frac {d\mathrm {R} }{dt}}+(2\mu _{4}-\mu _{1})r\right)\right]\cos \left(\mu _{1}-{\frac {n}{2}}\rho \right)t&\\+\left[\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)\right.&\\\left.-\mathrm {C} _{1}\left({\frac {dr}{dt}}-(2\mu _{4}-\mu _{1})\mathrm {R} \right)\right]\sin \left(\mu _{1}-{\frac {n}{2}}\rho \right)t&=\mathrm {D} _{1},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af66a642fc20f8f4c53f4c2d2f33559d0c94e684)
![{\displaystyle {\begin{aligned}(\beta )\quad \left[{\frac {d\mathrm {x} _{1}}{dt}}+\mathrm {A} _{1}\left({\frac {d\mathrm {P} }{dt}}+(2\mu _{2}-\mu _{1})p\right)+\mathrm {B} _{1}\left({\frac {d\mathrm {Q} }{dt}}+(2\mu _{3}-\mu _{1})q\right)\right.&\\\left.+\mathrm {C} _{1}\left({\frac {d\mathrm {R} }{dt}}+(2\mu _{4}-\mu _{1})r\right)\right]\sin \left(\mu _{1}-{\frac {n}{2}}\rho \right)t&\\-\left[\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)\right.&\\\left.-\mathrm {C} _{1}\left({\frac {dr}{dt}}-(2\mu _{4}-\mu _{1})\mathrm {R} \right)\right]\cos \left(\mu _{1}-{\frac {n}{2}}\rho \right)t&=\mathrm {E} _{1},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06d9368472a8f17410fc626f458cb31e23095766)
