Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 3.djvu/68

Les variations de $\varphi ,\theta ,\varphi '$ et $\theta '$ dépendent ainsi des variations de $\gamma$ et de $\Pi .$ On a, par le n^o 12,

${\frac {d\gamma }{dt}}=-{\frac {m{\sqrt {a}}+m'{\sqrt {a'}}}{m'{\sqrt {a'}}}}m'an\left\{{\begin{aligned}&{\frac {\partial {\rm {P}}}{\partial \gamma }}\cos(5n't-2nt+5\varepsilon '-2\varepsilon )\\-&{\frac {\partial {\rm {P}}'}{\partial \gamma }}\sin(5n't-2nt+5\varepsilon '-2\varepsilon )\end{aligned}}\right\},$

${\frac {\gamma d\Pi }{dt}}=-{\frac {m{\sqrt {a}}+m'{\sqrt {a'}}}{m'{\sqrt {a'}}}}m'an\left\{{\begin{aligned}&{\frac {\partial {\rm {P}}}{\partial \gamma }}\sin(5n't-2nt+5\varepsilon '-2\varepsilon )\\+&{\frac {\partial {\rm {P}}'}{\partial \gamma }}\cos(5n't-2nt+5\varepsilon '-2\varepsilon )\end{aligned}}\right\}.$

De là on tire, en négligeant les quantités périodiques dont l’effet est insensible, et en observant que

${\frac {\partial {\rm {P'}}}{\partial \gamma }}{\frac {\partial ^{2}{\rm {P}}}{\partial \gamma ^{2}}}-{\frac {\partial {\rm {P}}}{\partial \gamma }}{\frac {\partial ^{2}{\rm {P'}}}{\partial \gamma ^{2}}}=0,$

${\begin{aligned}\delta \gamma =&-{\frac {3m'^{2}a^{2}n^{3}}{(5n'-2n)^{2}}}{\frac {m{\sqrt {a}}+m'{\sqrt {a'}}}{m'{\sqrt {a'}}}}{\frac {5m{\sqrt {a}}+2m'{\sqrt {a'}}}{m'{\sqrt {a'}}}}t\left({\rm {P}}{\frac {\partial {\rm {P'}}}{\partial \gamma }}-{\rm {P}}'{\frac {\partial {\rm {P}}}{\partial \gamma }}\right)\\\\&+{\frac {m'^{2}a^{2}n^{2}}{5n'-2n}}{\frac {m{\sqrt {a}}+m'{\sqrt {a'}}}{m'{\sqrt {a'}}}}t\left({\frac {\partial {\rm {P'}}}{\partial e}}{\frac {\partial ^{2}{\rm {P}}}{\partial e\partial \gamma }}-{\frac {\partial {\rm {P}}}{\partial e}}{\frac {\partial ^{2}{\rm {P'}}}{\partial e\partial \gamma }}\right)\\\\&+{\frac {mm'aa'nn't}{5n'-2n}}{\frac {m{\sqrt {a}}+m'{\sqrt {a'}}}{m'{\sqrt {a'}}}}t\left({\frac {\partial {\rm {P'}}}{\partial e'}}{\frac {\partial ^{2}{\rm {P}}}{\partial e'\partial \gamma }}-{\frac {\partial {\rm {P}}}{\partial e'}}{\frac {\partial ^{2}{\rm {P'}}}{\partial e'\partial \gamma }}\right),\end{aligned}}$

${\begin{aligned}\delta \Pi =&{\frac {3m'^{2}a^{2}n^{3}t}{\gamma (5n'-2n)^{2}}}{\frac {m{\sqrt {a}}+m'{\sqrt {a'}}}{m'{\sqrt {a'}}}}{\frac {5m{\sqrt {a}}+2m'{\sqrt {a'}}}{m'{\sqrt {a'}}}}t\left({\rm {P}}{\frac {\partial {\rm {P}}}{\partial \gamma }}+{\rm {P}}'{\frac {\partial {\rm {P'}}}{\partial \gamma }}\right)\\\\&+{\frac {m'^{2}a^{2}n^{2}}{\gamma (5n'-2n)}}{\frac {m{\sqrt {a}}+m'{\sqrt {a'}}}{m'{\sqrt {a'}}}}t\left\{{\begin{aligned}&{\frac {\partial {\rm {P}}}{\partial e}}{\frac {\partial ^{2}{\rm {P}}}{\partial e\partial \gamma }}+{\frac {\partial {\rm {P'}}}{\partial e}}{\frac {\partial ^{2}{\rm {P'}}}{\partial e\partial \gamma }}\\\\+&{\frac {\partial {\rm {P}}}{\partial \gamma }}{\frac {\partial ^{2}{\rm {P}}}{\partial \gamma ^{2}}}+{\frac {\partial {\rm {P'}}}{\partial \gamma }}{\frac {\partial ^{2}{\rm {P'}}}{\partial \gamma ^{2}}}\end{aligned}}\right\}\\&+{\frac {mm'aa'nn'}{\gamma (5n'-2n)}}{\frac {m{\sqrt {a}}+m'{\sqrt {a'}}}{m'{\sqrt {a'}}}}t\left\{{\begin{aligned}&{\frac {\partial {\rm {P}}}{\partial e'}}{\frac {\partial ^{2}{\rm {P}}}{\partial e'\partial \gamma }}+{\frac {\partial {\rm {P'}}}{\partial e'}}{\frac {\partial ^{2}{\rm {P'}}}{\partial e'\partial \gamma }}\\\\+&{\frac {\partial {\rm {P}}}{\partial \gamma }}{\frac {\partial ^{2}{\rm {P}}}{\partial \gamma ^{2}}}+{\frac {\partial {\rm {P'}}}{\partial \gamma }}{\frac {\partial ^{2}{\rm {P'}}}{\partial \gamma ^{2}}}\end{aligned}}\right\}\\\end{aligned}}$

15. Si l’on voulait déterminer pour un temps quelconque les éléments des orbites planétaires, il faudrait intégrer les équations différentielles (A) et (C) des n^o 55 et 59 du Livre 11, par la méthode ex-