Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 3.djvu/159

133

SECONDE PARTIE. — LIVRE VI.

La partie de $\delta \nu ^{\rm {iv}}$ donnée dans le n^o 18,

$-{\frac {6m^{\rm {v}}{\rm {{v}n^{\rm {iv^{2}}}}}}{\left(5n^{\rm {v}}-2n^{\rm {iv}}\right)^{2}}}\left\{{\begin{aligned}&\left[{\begin{aligned}a^{\rm {iv}}{\rm {P}}'&+{\frac {2a^{\rm {iv}}d{\rm {P}}}{\left(5n^{\rm {v}}-2n^{\rm {iv}}\right)dt}}-{\frac {3a^{\rm {iv}}d^{2}{\rm {P}}'}{\left(5n^{\rm {v}}-2n^{\rm {iv}}\right)^{2}dt^{2}}}\\\\&+t\left(a^{\rm {iv}}{\frac {d{\rm {P}}'}{dt}}+{\frac {2a^{\rm {iv}}d^{2}{\rm {P}}}{\left(5n^{\rm {v}}-2n^{\rm {iv}}\right)dt^{2}}}\right)\\\\&+{\frac {1}{2}}t^{2}a^{\rm {iv}}{\frac {d^{2}P'}{dt^{2}}}\end{aligned}}\right]\\&\qquad \qquad \qquad \times \sin .\left(5n^{\rm {v}}-2n^{\rm {iv}}+5\varepsilon ^{\rm {v}}-2\varepsilon ^{\rm {iv}}\right)\\\\-&\left[{\begin{aligned}a^{\rm {iv}}{\rm {P}}&-{\frac {2a^{\rm {iv}}d{\rm {P}}'}{\left(5n^{\rm {v}}-2n^{\rm {iv}}\right)dt}}-{\frac {3a^{\rm {iv}}d^{2}{\rm {P}}}{\left(5n^{\rm {v}}-2n^{\rm {iv}}\right)^{2}dt^{2}}}\\\\&+t\left(a^{\rm {iv}}{\frac {d{\rm {P}}}{dt}}+{\frac {2a^{\rm {iv}}d^{2}{\rm {P}}'}{\left(5n^{\rm {v}}-2n^{\rm {iv}}\right)dt^{2}}}\right)\\\\&+{\frac {1}{2}}t^{2}a^{\rm {iv}}{\frac {d^{2}P}{dt^{2}}}\end{aligned}}\right]\\&\qquad \qquad \qquad \times \cos .\left(5n^{\rm {v}}-2n^{\rm {iv}}+5\varepsilon ^{\rm {v}}-2\varepsilon ^{\rm {iv}}\right)\\\\\end{aligned}}\right\},$

devient ainsi, en la réduisant en nombres,

${\begin{aligned}&\left(3900''{,}616270-t.0''{,}025982-t^{2}.0''{,}000059403\right)\sin \left(5n^{\rm {v}}t-2n^{\rm {iv}}t+5\varepsilon ^{\rm {v}}-2\varepsilon ^{\rm {iv}}\right)\\+&\left(\ \ 368''{,}910343-t.1''{,}461994+t^{2}.0''{,}000242476\right)\cos \left(5n^{\rm {v}}t-2n^{\rm {iv}}t+5\varepsilon ^{\rm {v}}-2\varepsilon ^{\rm {iv}}\right)\end{aligned}}$

La grande inégalité de Jupiter se compose de plusieurs autres parties :

elle renferme encore, par le n^o 8, la fonction

$-{\frac {2m^{\rm {v}}n^{\rm {iv}}}{5n^{\rm {v}}-2n^{\rm {iv}}}}\left[{\begin{aligned}&a^{\rm {iv^{2}}}{\frac {\partial {\rm {P}}}{\partial a^{\rm {iv}}}}\cos .\left(5n^{\rm {v}}-2n^{\rm {iv}}+5\varepsilon ^{\rm {v}}-2\varepsilon ^{\rm {iv}}\right)\\\\-&a^{\rm {iv^{2}}}{\frac {\partial {\rm {P}}'}{\partial a^{\rm {iv}}}}\sin .\left(5n^{\rm {v}}-2n^{\rm {iv}}+5\varepsilon ^{\rm {v}}-2\varepsilon ^{\rm {iv}}\right)\end{aligned}}\right]$

Pour la réduire en nombres, il faut calculer les valeurs de $a^{\rm {{v}^{2}}}{\frac {\partial {\rm {{\rm {M}}(O)}}}{\partial a^{\rm {iv}}}},$$a^{\rm {{v}^{2}}}{\frac {\partial {\rm {{\rm {M}}(1)}}}{\partial a^{\rm {iv}}}},$ … On a trouvé

${\begin{aligned}&a^{\rm {{v}^{2}}}{\frac {\partial {\rm {{\rm {M}}(O)}}}{\partial a^{\rm {iv}}}}=-26{,}46390.m^{\rm {v}},\\\\&a^{\rm {{v}^{2}}}{\frac {\partial {\rm {{\rm {M}}(1)}}}{\partial a^{\rm {iv}}}}=\ \ \ 65{,}75870.m^{\rm {v}},\\\\&a^{\rm {{v}^{2}}}{\frac {\partial {\rm {{\rm {M}}(1)}}}{\partial a^{\rm {iv}}}}=-50{,}22714.m^{\rm {v}},\\\\&a^{\rm {{v}^{2}}}{\frac {\partial {\rm {{\rm {M}}(1)}}}{\partial a^{\rm {iv}}}}=\ \ \ 12{,}11696.m^{\rm {v}},\\\\&a^{\rm {{v}^{2}}}{\frac {\partial {\rm {{\rm {M}}(1)}}}{\partial a^{\rm {iv}}}}=-6{,}75963.m^{\rm {v}},\\\\&a^{\rm {{v}^{2}}}{\frac {\partial {\rm {{\rm {M}}(1)}}}{\partial a^{\rm {iv}}}}=\ \ \ 4{,}13173.m^{\rm {v}},\\\\\end{aligned}}$