Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 3.djvu/64

que cela a eoeore lieu pour le moyen mouvement de Saturne et pour le grand axe de son orbite.

13. Considérons présentement les variations des excentricités et des périhélies. Nous avons donné, dans le n^o 60 du Livre II, les expressions des accroissements de ${\frac {de}{dt}},{\frac {d\varpi }{dt}},{\frac {de'}{dt}},{\frac {d\varpi '}{dt}},$ dus aux deux grandes inégalités de Jupiter et de Saturne, et nous avons observé dans le même numéro que les variations de $e,\varpi ,e',\varpi ',$ relatives à l’angle $5n't-2nt,$ introduisent dans ces expressiosns des variations semblables à celles qu’y produisent les deux grandes inégalités. En appliquant ces considérations à Jupiter et à Saturne, et désignant par $\delta e$ et $\delta \varpi$ les variations dues au carré de la force perturbatrice, on trouve

$\delta e=-{\frac {3m'^{2}a^{2}n^{3}}{(5n'-2n)^{2}}}{\frac {5m{\sqrt {a}}+2m'{\sqrt {a'}}}{m'{\sqrt {a'}}}}$

$\left\{{\begin{aligned}\left({\rm {P}}{\frac {\partial {\rm {P'}}}{\partial e}}-{\rm {P}}'{\frac {\partial {\rm {P}}}{\partial e}}\right)t&+{\frac {{\rm {P}}{\frac {\partial {\rm {P'}}}{\partial e}}+{\rm {P}}'{\frac {\partial {\rm {P}}}{\partial e}}}{2(5n'-2n)}}.\sin 2(5n't-2nt+5\varepsilon '-2\varepsilon )\\\\&+{\frac {{\rm {P'}}{\frac {\partial {\rm {P'}}}{\partial e}}-{\rm {P}}{\frac {\partial {\rm {P}}}{\partial e}}}{2(5n'-2n)}}.\cos 2(5n't-2nt+5\varepsilon '-2\varepsilon )\end{aligned}}\right\}$

$+{\frac {m'^{2}a^{2}n^{2}}{5n'-2n}}\left\{{\begin{aligned}&\left({\frac {\partial {\rm {P'}}}{\partial e}}{\frac {\partial ^{2}{\rm {P}}}{\partial e^{2}}}-{\frac {\partial {\rm {P}}}{\partial e}}{\frac {\partial ^{2}{\rm {P'}}}{\partial e^{2}}}+{\frac {\partial {\rm {P'}}}{\partial \gamma }}{\frac {\partial ^{2}{\rm {P}}}{\partial e\partial \gamma }}-{\frac {\partial {\rm {P}}}{\partial \gamma }}{\frac {\partial ^{2}{\rm {P'}}}{\partial e\partial \gamma }}\right)t\\\\&+{\frac {\left({\frac {\partial {\rm {P}}}{\partial e}}\right)^{2}-\left({\frac {\partial {\rm {P'}}}{\partial e}}\right)^{2}}{4e(5n'-2n)}}.\cos 2(5n't-2nt+5\varepsilon '-2\varepsilon )\\\\&-{\frac {{\frac {\partial {\rm {P}}}{\partial e}}{\frac {\partial {\rm {P'}}}{\partial e}}}{2e(5n'-2n)}}.\sin 2(5n't-2nt+5\varepsilon '-2\varepsilon )\end{aligned}}\right\}$

$+{\frac {mm'aa'nn't}{5n'-2n}}\left({\frac {\partial {\rm {P'}}}{\partial e'}}{\frac {\partial ^{2}{\rm {P}}}{\partial e\partial e'}}-{\frac {\partial {\rm {P}}}{\partial e'}}{\frac {\partial ^{2}{\rm {P'}}}{\partial e\partial e'}}+{\frac {\partial {\rm {P'}}}{\partial \gamma }}{\frac {\partial ^{2}{\rm {P}}}{\partial e\partial \gamma }}-{\frac {\partial {\rm {P}}}{\partial \gamma }}{\frac {\partial ^{2}{\rm {P'}}}{\partial e\partial \gamma }}\right),$

$\delta \varpi ={\frac {3m'^{2}a^{2}n^{3}}{(5n'-2n)^{2}e}}{\frac {5m{\sqrt {a}}+2m'{\sqrt {a'}}}{m'{\sqrt {a'}}}}$

$\left\{{\begin{aligned}\left({\rm {P}}{\frac {\partial {\rm {P}}}{\partial e}}-{\rm {P}}'{\frac {\partial {\rm {P'}}}{\partial e}}\right)t&+{\frac {{\rm {P}}{\frac {\partial {\rm {P}}}{\partial e}}-{\rm {P}}'{\frac {\partial {\rm {P'}}}{\partial e}}}{2(5n'-2n)}}.\sin 2(5n't-2nt+5\varepsilon '-2\varepsilon )\\\\&+{\frac {{\rm {P'}}{\frac {\partial {\rm {P}}}{\partial e}}+{\rm {P}}{\frac {\partial {\rm {P'}}}{\partial e}}}{2(5n'-2n)}}.\cos 2(5n't-2nt+5\varepsilon '-2\varepsilon )\end{aligned}}\right\}$

$+{\frac {m'^{2}a^{2}n^{2}}{(5n'-2n)e}}\left\{{\begin{aligned}&\left({\frac {\partial {\rm {P}}}{\partial e}}{\frac {\partial ^{2}{\rm {P}}}{\partial e^{2}}}+{\frac {\partial {\rm {P'}}}{\partial e}}{\frac {\partial ^{2}{\rm {P'}}}{\partial e^{2}}}+{\frac {\partial {\rm {P}}}{\partial \gamma }}{\frac {\partial ^{2}{\rm {P}}}{\partial e\partial \gamma }}+{\frac {\partial {\rm {P'}}}{\partial \gamma }}{\frac {\partial ^{2}{\rm {P'}}}{\partial e\partial \gamma }}\right)t\\\\&+{\frac {\left({\frac {\partial {\rm {P}}}{\partial e}}\right)^{2}-\left({\frac {\partial {\rm {P'}}}{\partial e}}\right)^{2}}{2e(5n'-2n)}}.\sin 2(5n't-2nt+5\varepsilon '-2\varepsilon )\\\\&+{\frac {{\frac {\partial {\rm {P}}}{\partial e}}{\frac {\partial {\rm {P'}}}{\partial e}}}{e(5n'-2n)}}.\cos 2(5n't-2nt+5\varepsilon '-2\varepsilon )\end{aligned}}\right\}$

$+{\frac {mm'aa'nn't}{(5n'-2n)e}}\left({\frac {\partial {\rm {P}}}{\partial e'}}{\frac {\partial ^{2}{\rm {P}}}{\partial e\partial e'}}+{\frac {\partial {\rm {P'}}}{\partial e'}}{\frac {\partial ^{2}{\rm {P'}}}{\partial e\partial e'}}+{\frac {\partial {\rm {P}}}{\partial \gamma }}{\frac {\partial ^{2}{\rm {P}}}{\partial e\partial \gamma }}+{\frac {\partial {\rm {P'}}}{\partial \gamma }}{\frac {\partial ^{2}{\rm {P'}}}{\partial e\partial \gamma }}\right).$