Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 3.djvu/66

tion du centre,

${\begin{aligned}&-{\frac {3m'^{2}a^{2}n^{3}}{(5n'-2n)^{3}}}{\frac {5m{\sqrt {a}}+4m'{\sqrt {a'}}}{m'{\sqrt {a'}}}}\left({\rm {P}}{\frac {\partial {\rm {P'}}}{\partial e}}+{\rm {P}}'{\frac {\partial {\rm {P}}}{\partial e}}\right)\\&\qquad \qquad \qquad \qquad \qquad \qquad \times \cos(5nt-10n't+5\varepsilon -10\varepsilon '-\varpi )\\\\&-{\frac {3m'^{2}a^{2}n^{3}}{(5n'-2n)^{3}}}{\frac {5m{\sqrt {a}}+4m'{\sqrt {a'}}}{m'{\sqrt {a'}}}}\left({\rm {P'}}{\frac {\partial {\rm {P'}}}{\partial e}}-{\rm {P}}{\frac {\partial {\rm {P}}}{\partial e}}\right)\\&\qquad \qquad \qquad \qquad \qquad \qquad \times \sin(5nt-10n't+5\varepsilon -10\varepsilon '-\varpi ).\end{aligned}}$

Cette inégalité peut être mise sous la forme suivante : si l’on représente par ${\rm {K}}\sin(5n't-3nt+5\varepsilon '-3\varepsilon +{\rm {B)}}$ l’inégalité de $m$ dépendante de $3nt-5n't+3\varepsilon -5\varepsilon ',$ et par ${\overline {\rm {H}}}\sin(5n't-2nt+5\varepsilon '-2\varepsilon +{\overline {\rm {A}}})$ la grande inégalité, l’inégalité précédente sera, par le n^o 69 du Livre II,

${\frac {1}{4}}{\frac {5m{\sqrt {a}}+4m'{\sqrt {a'}}}{m'{\sqrt {a'}}}}{\overline {\rm {H}}}{\rm {K}}\sin(5nt-10n't+5\varepsilon -10\varepsilon '-{\rm {B}}-{\overline {\rm {A}}}).$

On trouvera pareillement, en n’ayant égard qu’aux variations séculaires dépendantes du carré de la force perturbatrice,

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

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

On trouve encore que le mouvement de $m'$ en longitude est affecté de l’inégalité

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

$\times \left\{{\begin{aligned}&\left({\rm {P}}{\frac {\partial {\rm {P'}}}{\partial e'}}+{\rm {P}}'{\frac {\partial {\rm {P}}}{\partial e'}}\right)\cos(4nt-9n't+4\varepsilon -9\varepsilon '-\varpi ')\\\\+&\left({\rm {P'}}{\frac {\partial {\rm {P'}}}{\partial e'}}+{\rm {P}}{\frac {\partial {\rm {P}}}{\partial e'}}\right)\sin(4nt-9n't+4\varepsilon -9\varepsilon '-\varpi ')\end{aligned}}\right\}.$