Page:Lagrange - Œuvres (1867) vol. 1.djvu/651

ou, divisant par $e^{\mathrm {R} t{\sqrt {-1}}},$ et changeant les exponentielles imaginaires en sinus et cosinus,

${\frac {dy}{dt}}+i\alpha {\frac {du}{dt}}+i\left(\mathrm {N+2H\alpha +R^{2}} {\mathfrak {A}}\right)U+i^{2}\beta {\frac {dv}{dt}}+i^{2}\left(\mathrm {{\frac {N}{2}}\alpha +4H\beta +R^{2}} {\mathfrak {B}}\right)V$

$-\left[\left(1-{\frac {i\mathrm {N} }{2}}{\mathfrak {A}}\right)y+i(\alpha +2\mathrm {H} {\mathfrak {A}})u-i{\mathfrak {A}}{\frac {dU}{dt}}\right.$

$\left.+i^{2}\left({\frac {\mathrm {N} }{2}}{\mathfrak {A}}+\beta +4\mathrm {H} {\mathfrak {B}}\right)v-i^{2}{\mathfrak {B}}{\frac {dV}{dt}}\right]\mathrm {R} {\sqrt {-1}}$

${\begin{aligned}=\cos \mathrm {R} t\int \mathrm {T} &\left[\left(1+i\alpha \cos \mathrm {H} t+i^{2}\beta \cos 2\mathrm {H} t\right)\cos \mathrm {R} t\right.\\&\qquad \left.-i({\mathfrak {A}}\sin \mathrm {H} t+i{\mathfrak {B}}\sin 2\mathrm {H} t)\mathrm {R} \sin \mathrm {R} t\right]dt\\+\sin \mathrm {R} t\int \mathrm {T} &\left[\left(1+i\alpha \cos \mathrm {H} t+i^{2}\beta \cos 2\mathrm {H} t\right)\sin \mathrm {R} t\right.\\&\qquad \left.+i({\mathfrak {A}}\sin \mathrm {H} t+i{\mathfrak {B}}\sin 2\mathrm {H} t)\mathrm {R} \cos \mathrm {R} t\right]dt\\-\sin \mathrm {R} t\int \mathrm {T} &\left[\left(1+i\alpha \cos \mathrm {H} t+i^{2}\beta \cos 2\mathrm {H} t\right)\cos \mathrm {R} t\right.\\&\qquad \left.-i({\mathfrak {A}}\sin \mathrm {H} t+i{\mathfrak {B}}\sin 2\mathrm {H} t)\mathrm {R} \sin \mathrm {R} t\right]dt{\sqrt {-1}}\\+\cos \mathrm {R} t\int \mathrm {T} &\left[\left(1+i\alpha \cos \mathrm {H} t+i^{2}\beta \cos 2\mathrm {H} t\right)\sin \mathrm {R} t\right.\\&\qquad \left.+i({\mathfrak {A}}\sin \mathrm {H} t+i{\mathfrak {B}}\sin 2\mathrm {H} t)\mathrm {R} \cos \mathrm {R} t\right]dt{\sqrt {-1}}.\end{aligned}}$

Donc, si l’on remet pour $u,U,v$ et $V$ leurs valeurs, et qu’on compare les imaginaires avec les imaginaires, et les réelles avec les réelles, on aura

$\left[1-{\frac {i\mathrm {N} }{2}}{\mathfrak {A}}+i(\alpha +\mathrm {H} {\mathfrak {A}})\cos \mathrm {H} t+i^{2}\left(1-{\frac {\mathrm {N} }{2}}{\mathfrak {A}}+\beta +2\mathrm {H} {\mathfrak {B}}\right)\cos 2\mathrm {H} t\right]y$

$-i({\mathfrak {A}}\sin \mathrm {H} t+i{\mathfrak {B}}\sin 2\mathrm {H} t){\frac {dy}{dt}}$

${\begin{aligned}=\sin \mathrm {R} t\int \mathrm {T} &\left[\left(1+i\alpha \cos \mathrm {H} t+i^{2}\beta \cos 2\mathrm {H} t\right){\frac {\cos \mathrm {R} t}{\mathrm {R} }}\right.\\&\qquad \left.-i({\mathfrak {A}}\sin \mathrm {H} t+i{\mathfrak {B}}\sin 2\mathrm {H} t)\sin \mathrm {R} t{\begin{aligned}\\\\\end{aligned}}\right]dt\\-\cos \mathrm {R} t\int \mathrm {T} &\left[\left(1+i\alpha \cos \mathrm {H} t+i^{2}\beta \cos 2\mathrm {H} t\right){\frac {\sin \mathrm {R} t}{\mathrm {R} }}\right.\\&\qquad \left.+i({\mathfrak {A}}\sin \mathrm {H} t+i{\mathfrak {B}}\sin 2\mathrm {H} t)\cos \mathrm {R} t{\begin{aligned}\\\\\end{aligned}}\right]dt.\end{aligned}}$