Page:Lagrange - Œuvres (1867) vol. 1.djvu/655

et

$(\mathrm {X} )\left\{{\begin{aligned}&\left[\lambda {\frac {dy}{dt}}+\lambda '{\frac {dy}{dt}}+\lambda '{\frac {dy}{dt}}+\mu {\frac {du}{dt}}+{\mathfrak {M}}{\frac {dU}{dt}}+\mu '{\frac {du'}{dt}}\right.\\&\qquad {\mathfrak {M}}'{\frac {dU'}{dt}}\nu {\frac {dv}{dt}}{\mathfrak {N}}{\frac {dV}{dt}}\nu '{\frac {dv'}{dt}}{\mathfrak {N}}'{\frac {dV'}{dt}}\\&\qquad +\left(-\lambda \rho +{\frac {i\mathrm {N} '}{2}}{\mathfrak {M}}'\right)y+\left(-\lambda '\rho +{\frac {i\mathrm {N} }{2}}{\mathfrak {M}}\right)y'\\&\qquad +(-\mu \rho -2\mathrm {H} {\mathfrak {M}})u+(-i\mathrm {N} '\lambda '+2\mathrm {H} \mu -{\mathfrak {M}}\pi )U\\&\qquad +(-\mu '\rho -2\mathrm {H} {\mathfrak {M}}')u'+(i\mathrm {N} \lambda +2\mathrm {H} \mu '-{\mathfrak {M}}'\pi )U'\\&\qquad +\left({\frac {i\mathrm {N} '}{2}}{\mathfrak {M}}'-\nu \rho -4\mathrm {H} {\mathfrak {N}}\right)v+\left(-{\frac {i\mathrm {N} '}{2}}\mu '+4\mathrm {H} \nu -{\mathfrak {N}}\rho \right)V\\&\qquad \left.+\left(-{\frac {i\mathrm {N} }{2}}{\mathfrak {M}}-\nu '\rho -4\mathrm {H} {\mathfrak {N}}'\right)v'+\left({\frac {i\mathrm {N} }{2}}\mu +4\mathrm {H} \nu '-{\mathfrak {N}}'\rho \right)V'\right]e^{\rho t}\\&=\int \left[\mathrm {T} \left(\lambda +\mu \cos \mathrm {H} t+{\mathfrak {M}}\sin \mathrm {H} t+\nu \cos 2\mathrm {H} t+{\mathfrak {N}}\sin 2\mathrm {H} t\right)\right.\\&\quad \left.-\mathrm {T} '\left(\lambda '+\mu '\cos \mathrm {H} t+{\mathfrak {M}}'\sin \mathrm {H} t+\nu '\cos 2\mathrm {H} t+{\mathfrak {N}}'\sin 2\mathrm {H} t\right)\right]e^{\rho t}dt.\end{aligned}}\right.$

59. Qu’on multiplie la quatrième, la sixième, la huitième et la dixième des équations (V) par $\pm {\sqrt {-1}},$ et qu’on les ajoute ensuite chacune à sa précédente, on aura, au lieu des dix équations (V), les six suivantes :

${\begin{aligned}&\left(\rho ^{2}+\mathrm {K} ^{2}\ \right)\lambda \ +i\left(\mathrm {{\frac {M'}{2}}\rho '+{\frac {N'}{2}}} {\mathfrak {M}}'\rho \right)=0,\\&\left(\rho ^{2}+\mathrm {K} '^{2}\right)\lambda '+i\left(\mathrm {{\frac {M}{2}}\ \rho \ -{\frac {N}{2}}} \ {\mathfrak {M}}\ \rho \right)=0,\\&i\mathrm {\left[M'+N'\left(H\pm \rho {\sqrt {-1}}\right)\right]} \lambda '+\mathrm {\left[K^{2}\ -\left(H\pm \rho {\sqrt {-1}}\right)^{2}\right]} \left(\mu \ \pm {\mathfrak {M}}\ {\sqrt {-1}}\right)=0,\\&i\mathrm {\left[M\ -N\ \left(H\pm \rho {\sqrt {-1}}\right)\right]} \lambda \ +\mathrm {\left[K'^{2}-\left(H\pm \rho {\sqrt {-1}}\right)^{2}\right]} \left(\mu '\pm {\mathfrak {M}}'{\sqrt {-1}}\right)=0,\\&i\mathrm {\left[M'+N'\left(2H\pm \rho {\sqrt {-1}}\right)\right]} \left(\mu '\pm {\mathfrak {M}}'{\sqrt {-1}}\right)\\&\qquad +2\mathrm {\left[K^{2}\ -\left(2H\pm \rho {\sqrt {-1}}\right)^{2}\right]} \left(\nu \ \pm {\mathfrak {N}}\ {\sqrt {-1}}\right)=0,\\&i\mathrm {\left[M-N\left(2H\pm \rho {\sqrt {-1}}\right)\right]} \left(\mu \pm {\mathfrak {M}}{\sqrt {-1}}\right)\\&\qquad +2\mathrm {\left[K'^{2}-\left(2H\pm \rho {\sqrt {-1}}\right)^{2}\right]} \left(\nu '\pm {\mathfrak {N}}'{\sqrt {-1}}\right)=0.\end{aligned}}$

Les deux premières donnent, ou bien

(Y)

$-\rho ^{2}=\mathrm {K} ^{2}+{\frac {i}{\lambda }}\left({\frac {\mathrm {M} '}{2}}\mu '+{\frac {\mathrm {N} '}{2}}{\mathfrak {M}}'\rho \right)$