Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 11.djvu/167

153

THÉORIE DE JUPITER ET DE SATURNE.

et l’on trouvera

${\begin{aligned}\mathrm {M} ^{(0)}=&{\frac {1}{48}}\left(-236\mathrm {A} ^{(2)}+111a'{\frac {\partial \mathrm {A} ^{(2)}}{\partial a'}}-18a'^{2}{\frac {\partial ^{2}\mathrm {A} ^{(2)}}{\partial a'^{2}}}+a'^{3}{\frac {\partial ^{3}\mathrm {A} ^{(2)}}{\partial a'^{3}}}\right),\\\mathrm {M} ^{(1)}=&{\frac {1}{16}}\left(306\mathrm {A} ^{(3)}+51a{\frac {\partial \mathrm {A} ^{(3)}}{\partial a}}-84a'{\frac {\partial \mathrm {A} ^{(3)}}{\partial a'}}-14aa'{\frac {\partial ^{2}\mathrm {A} ^{(3)}}{\partial a\partial a'}}\right.\\&\qquad \qquad \qquad \qquad \qquad \qquad \left.+6a'^{2}{\frac {\partial ^{2}\mathrm {A} ^{(3)}}{\partial a'^{2}}}+aa'^{2}{\frac {\partial ^{3}\mathrm {A} ^{(3)}}{\partial a\partial a'^{2}}}\right),\\\mathrm {M} ^{(2)}=&{\frac {1}{16}}\left(-352\mathrm {A} ^{(4)}-112a{\frac {\partial \mathrm {A} ^{(4)}}{\partial a}}+44a'{\frac {\partial \mathrm {A} ^{(4)}}{\partial a'}}-8a^{2}{\frac {\partial ^{2}\mathrm {A} ^{(4)}}{\partial a^{2}}}\right.\\&\qquad \qquad \qquad \qquad \qquad \qquad \left.+14aa'{\frac {\partial ^{2}\mathrm {A} ^{(4)}}{\partial a\partial a'}}+a^{2}a'{\frac {\partial ^{3}\mathrm {A} ^{(4)}}{\partial a^{2}\partial a'}}\right),\\\mathrm {M} ^{(3)}=&{\frac {1}{48}}\left(380\mathrm {A} ^{(5)}+174a{\frac {\partial \mathrm {A} ^{(5)}}{\partial a}}+24a^{2}{\frac {\partial ^{2}\mathrm {A} ^{(5)}}{\partial a^{2}}}+a^{3}{\frac {\partial ^{3}\mathrm {A} ^{(5)}}{\partial a^{3}}}\right),\\\mathrm {M} ^{(4)}=&{\frac {aa'}{16}}\left(7\mathrm {L} ^{(3)}-a'{\frac {\partial \mathrm {L} ^{(3)}}{\partial a'}}\right),\\\mathrm {M} ^{(5)}=&{\frac {aa'}{16}}\left(7\mathrm {L} ^{(4)}+a{\frac {\partial \mathrm {L} ^{(4)}}{\partial a}}\right).\end{aligned}}$

De là on tire, par l’article XV,

${\begin{aligned}a'\mathrm {M} ^{(0)}=&\quad \ {\frac {1}{48}}\left(389b_{\frac {1}{2}}^{(2)}+201\alpha {\frac {db_{\frac {1}{2}}^{(2)}}{d\alpha }}+27\alpha ^{2}{\frac {d^{2}b_{\frac {1}{2}}^{(2)}}{d\alpha ^{2}}}+\alpha ^{3}{\frac {d^{3}b_{\frac {1}{2}}^{(2)}}{d\alpha ^{3}}}\right),\\a'\mathrm {M} ^{(1)}=&-{\frac {1}{16}}\left(402b_{\frac {1}{2}}^{(3)}+193\alpha {\frac {db_{\frac {1}{2}}^{(3)}}{d\alpha }}+26\alpha ^{2}{\frac {d^{2}b_{\frac {1}{2}}^{(3)}}{d\alpha ^{2}}}+\alpha ^{3}{\frac {d^{3}b_{\frac {1}{2}}^{(3)}}{d\alpha ^{3}}}\right),\\a'\mathrm {M} ^{(2)}=&\quad \ {\frac {1}{16}}\left(396b_{\frac {1}{2}}^{(4)}+184\alpha {\frac {db_{\frac {1}{2}}^{(4)}}{d\alpha }}+25\alpha ^{2}{\frac {d^{2}b_{\frac {1}{2}}^{(4)}}{d\alpha ^{2}}}+\alpha ^{3}{\frac {d^{3}b_{\frac {1}{2}}^{(4)}}{d\alpha ^{3}}}\right),\\a'\mathrm {M} ^{(3)}=&-{\frac {1}{48}}\left(380b_{\frac {1}{2}}^{(5)}+174\alpha {\frac {db_{\frac {1}{2}}^{(5)}}{d\alpha }}+24\alpha ^{2}{\frac {d^{2}b_{\frac {1}{2}}^{(5)}}{d\alpha ^{2}}}+\alpha ^{3}{\frac {d^{3}b_{\frac {1}{2}}^{(5)}}{d\alpha ^{3}}}\right),\\a'\mathrm {M} ^{(4)}=&\quad \ {\frac {\alpha }{16}}\left(\ \ 10b_{\frac {3}{2}}^{(3)}+\alpha {\frac {db_{\frac {3}{2}}^{(3)}}{d\alpha }}\right),\\a'\mathrm {M} ^{(5)}=&-{\frac {\alpha }{16}}\left(\quad 7b_{\frac {3}{2}}^{(4)}+\alpha {\frac {db_{\frac {3}{2}}^{(4)}}{d\alpha }}\right).\end{aligned}}$

On aura ensuite

${\begin{aligned}k\ =&\quad \ \mathrm {M} ^{(0)}e'^{3}\sin 3\varpi '+\mathrm {M} ^{(1)}e'^{2}e\sin(2\varpi '+\varpi \,)+\mathrm {M} ^{(2)}e'e^{2}\sin(\varpi '+2\varpi )\\&+\mathrm {M} ^{(3)}e^{3}\ \sin 3\varpi \ +\mathrm {M} ^{(4)}e'\gamma ^{2}\sin(2\Pi \,+\varpi ')+\mathrm {M} ^{(5)}e\ \gamma ^{2}\sin(2\Pi +\varpi ),\\k'=&-\mathrm {M} ^{(0)}e'^{3}\cos 3\varpi '-\mathrm {M} ^{(1)}e'^{2}e\cos(2\varpi '+\varpi )-\mathrm {M} ^{(2)}e'e^{2}\cos(\varpi '+2\varpi )\\&-\mathrm {M} ^{(3)}e^{3}\ \cos 3\varpi \ -\mathrm {M} ^{(4)}e'\gamma ^{2}\cos(2\Pi +\varpi ')-\mathrm {M} ^{(5)}e\gamma ^{2}\cos(2\Pi +\varpi ).\end{aligned}}$