Page:Lagrange - Œuvres (1867) vol. 1.djvu/652

et

$\left(1+i\alpha \cos \mathrm {H} t+i^{2}\beta \cos 2\mathrm {H} t\right){\frac {dy}{dt}}$

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

deux équations à l’aide desquelles on éliminera ${\frac {dy}{dt}}.$

De l’intégration des équations

${\begin{aligned}(\mathrm {T} )\quad \qquad &{\frac {d^{2}y\ }{dt^{2}}}+\mathrm {K} ^{2}\ y\ +i\left(\mathrm {M} y'\cos \mathrm {H} t+\mathrm {N} {\frac {dy'}{dt}}\sin \mathrm {H} t\right)=\mathrm {T} ,\\(\mathrm {U} )\quad \qquad &{\frac {d^{2}y'}{dt^{2}}}+\mathrm {K} '^{2}y'+i\left(\mathrm {M} 'y\cos \mathrm {H} t-\mathrm {N} '{\frac {dy}{dt}}\sin \mathrm {H} t\right)=\mathrm {T} ',\end{aligned}}$

58. Soit fait, comme dans le numéro précédent,

$y\cos \mathrm {H} t=u,\ \ y\sin \mathrm {H} t=U,\ \ y\cos 2\mathrm {H} t=v,\ \ y\sin 2\mathrm {H} t=V,\ldots ,$

et de même

$y'\cos \mathrm {H} t=u',\ \ y'\sin \mathrm {H} t=U',\ \ y'\cos 2\mathrm {H} t=v',\ \ y'\sin 2\mathrm {H} t=V',\ldots ,$

on aura

${\begin{alignedat}{2}{\frac {dy}{dt}}\cos \mathrm {H} t=&{\frac {du}{dt}}+\mathrm {H} U,&{\frac {dy}{dt}}\cos 2\mathrm {H} t=&{\frac {dv}{dt}}+2\mathrm {H} V,\\{\frac {dy}{dt}}\sin \mathrm {H} t=&{\frac {dU}{dt}}-\mathrm {H} u,&{\frac {dy}{dt}}\sin 2\mathrm {H} t=&{\frac {dV}{dt}}-2\mathrm {H} v,\\{\frac {d^{2}y}{dt^{2}}}\cos \mathrm {H} t=&{\frac {d^{2}u}{dt^{2}}}+2\mathrm {H} {\frac {dU}{dt}}-\mathrm {H} ^{2}u,&{\frac {d^{2}y}{dt^{2}}}\cos 2\mathrm {H} t=&{\frac {d^{2}v}{dt^{2}}}+4\mathrm {H} {\frac {dV}{dt}}-4\mathrm {H} ^{2}v,\\{\frac {d^{2}y}{dt^{2}}}\sin \mathrm {H} t=&{\frac {d^{2}U}{dt^{2}}}-2\mathrm {H} {\frac {du}{dt}}-\mathrm {H} ^{2}U,\quad &{\frac {d^{2}y}{dt^{2}}}\sin 2\mathrm {H} t=&{\frac {d^{2}V}{dt^{2}}}-4\mathrm {H} {\frac {dv}{dt}}-4\mathrm {H} ^{2}V,\\\end{alignedat}}$