Page:Lagrange - Œuvres (1867) vol. 1.djvu/650

${\begin{aligned}&\mathrm {\left({\frac {M}{2}}-NH\right)} \alpha -\mathrm {{\frac {N}{2}}R^{2}} {\mathfrak {A}}+\mathrm {\left(-R^{2}+K^{2}-4H^{2}\right)} \beta -4\mathrm {HR} ^{2}{\mathfrak {B}}=0,\\-&\mathrm {\frac {N}{2}} \alpha +\mathrm {\left({\frac {M}{2}}-NH\right)} {\mathfrak {A}}-4\mathrm {H} \beta +\mathrm {\left(-R^{2}+K^{2}-4H^{2}\right)} {\mathfrak {B}}=0,\end{aligned}}$

d’où l’on tire

${\begin{aligned}\mathrm {R} ^{2}=&{\frac {\mathrm {K} ^{2}+{\frac {1}{2}}i^{2}\mathrm {M} \alpha }{1-{\frac {1}{2}}i^{2}\mathrm {N} {\mathfrak {A}}}},\\\alpha =&\mathrm {\frac {(M-NH)\left(R^{2}-K^{2}+H^{2}\right)+2NHR^{2}}{\left(R^{2}-K^{2}+H^{2}\right)^{2}-4H^{2}R^{2}}} ,\\{\mathfrak {A}}=&\mathrm {\frac {N\left(R^{2}-K^{2}+H^{2}\right)+2(M-NH)H}{\left(R^{2}-K^{2}+H^{2}\right)^{2}-4H^{2}R^{2}}} ,\\\beta =&\mathrm {\frac {\left({\cfrac {1}{2}}M-NH\right)\left(R^{2}-K^{2}+4H^{2}\right)+2NHR^{2}}{\left(R^{2}-K^{2}+4H^{2}\right)^{2}-16H^{2}R^{2}}} \alpha ,\\&-\mathrm {\frac {{\cfrac {1}{2}}N\left(R^{2}-K^{2}+4H^{2}\right)R^{2}+4\left({\cfrac {1}{2}}M-NH\right)HR}{\left(R^{2}-K^{2}+4H^{2}\right)^{2}-16H^{2}R^{2}}} {\mathfrak {A}},\\{\mathfrak {B}}=&\mathrm {\frac {\left({\cfrac {1}{2}}M-NH\right)\left(R^{2}-K^{2}+4H^{2}\right)+2NHR^{2}}{\left(R^{2}-K^{2}+4H^{2}\right)^{2}-16H^{2}R^{2}}} {\mathfrak {A}},\\&-\mathrm {\frac {{\cfrac {1}{2}}N\left(R^{2}-K^{2}+4H^{2}\right)R^{2}+4\left({\cfrac {1}{2}}M-NH\right)H}{\left(R^{2}-K^{2}+4H^{2}\right)^{2}-16H^{2}R^{2}}} \alpha \,;\end{aligned}}$

et ensuite

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

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

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

$=\int \mathrm {T} \left[1+i\alpha \cos \mathrm {H} t+i^{2}\beta \cos 2\mathrm {H} t\right.$

$\left.+\left(i{\mathfrak {A}}\sin \mathrm {H} t+i^{2}{\mathfrak {B}}\sin 2\mathrm {H} t\right)\mathrm {R} {\sqrt {-1}}\right]e^{\mathrm {R} t{\sqrt {-1}}}dt\,;$