Page:Henri Poincaré - Les méthodes nouvelles de la mécanique céleste, Tome 1, 1892.djvu/242

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CHAPITRE IV.

Ces solutions s’écrivent

${\begin{aligned}\xi _{i}&={\frac {d\varphi _{i}}{dh}},&\eta _{i}&={\frac {d\psi _{i}}{dh}}\end{aligned}}$

et

${\begin{aligned}\xi _{i}&={\frac {d\varphi _{i}}{d\varepsilon }},&\eta _{i}&={\frac {d\psi _{i}}{d\varepsilon }}\end{aligned}}$

Après la différentiation il faut faire $h=\varepsilon =0.$

Or il vient

${\begin{aligned}\varphi _{i}(t,\,\mu ,\,\varepsilon )&=\,\theta _{i}\,\left[t(1+\varepsilon ),\,\mu ,\,\varepsilon \right],\\\psi _{i}(t,\,\mu ,\,\varepsilon )&=\Theta _{i}\left[t(1+\varepsilon ),\,\mu ,\,\varepsilon \right],\end{aligned}}$

d’où

${\begin{aligned}{\frac {d\varphi _{i}}{dh}}={\frac {d\varphi _{i}}{dt}}&=\,{\frac {d\theta _{i}}{du}}{\frac {du}{dt}}=\,{\frac {d\theta _{i}}{du}}(1+\varepsilon ),\\[0.5ex]{\frac {d\psi _{i}}{dh}}={\frac {d\psi _{i}}{dt}}&={\frac {d\Theta _{i}}{du}}{\frac {du}{dt}}={\frac {d\Theta _{i}}{du}}(1+\varepsilon ),\end{aligned}}$

et pour $\varepsilon =0$

${\begin{aligned}{\frac {d\varphi _{i}}{dh}}&={\frac {d\theta _{i}}{du}},&{\frac {d\psi _{i}}{dh}}&={\frac {d\Theta _{i}}{du}}\cdot \end{aligned}}$

D’autre part,

${\begin{aligned}{\frac {d\varphi _{i}}{d\varepsilon }}=\,{\frac {d\theta _{i}}{du}}{\frac {du}{d\varepsilon }}+\,{\frac {d\theta _{i}}{d\varepsilon }}&=t\,{\frac {d\theta _{i}}{du}}+\,{\frac {d\theta _{i}}{d\varepsilon }},\\{\frac {d\psi _{i}}{d\varepsilon }}={\frac {d\Theta _{i}}{du}}{\frac {du}{d\varepsilon }}+{\frac {d\Theta _{i}}{d\varepsilon }}&=t{\frac {d\Theta _{i}}{du}}+{\frac {d\Theta _{i}}{d\varepsilon }}\\\end{aligned}}$

ou, pour $\varepsilon =0,$

${\begin{aligned}{\frac {d\varphi _{i}}{d\varepsilon }}&=t{\frac {d\varphi _{i}}{dt}}+{\frac {d\theta _{i}}{d\varepsilon }},&{\frac {d\psi _{i}}{d\varepsilon }}&=t{\frac {d\psi _{i}}{dt}}+{\frac {d\Theta _{i}}{d\varepsilon }}\cdot \end{aligned}}$

Les solutions cherchées des équations (2) sont donc

${\begin{aligned}\xi _{i}&=\mathrm {S} ''_{i}={\frac {d\varphi _{i}}{dt}},&\eta _{i}&=\mathrm {T} ''_{i}={\frac {d\psi _{i}}{dt}}\end{aligned}}$

et

${\begin{aligned}\xi _{i}&=t\mathrm {S} ''_{i}+\mathrm {S} _{i}^{\star },&\eta _{i}&=t\mathrm {T} ''_{i}+\mathrm {T} _{i}^{\star }\end{aligned}}$

avec

${\begin{aligned}\mathrm {S} _{i}^{\star }&={\frac {d\theta _{i}}{d\varepsilon }},&\mathrm {T} _{i}^{\star }&={\frac {d\Theta _{i}}{d\varepsilon }}\cdot \end{aligned}}$