Page:Joseph Louis de Lagrange - Œuvres, Tome 5.djvu/182

${\frac {dt}{r^{\frac {3}{2}}}}$ est mis à la place de $dp,$

${\begin{aligned}d\mathrm {N} =&-\left(\mathrm {T} '{\frac {r^{3}}{2}}{\frac {d^{2}\mathrm {A} '}{dr^{2}}}+\mathrm {T} ''{\frac {r^{3}}{2}}{\frac {d^{2}\mathrm {A} ''}{dr^{2}}}+\ldots \right){\frac {\mathrm {M} dt}{r^{\frac {3}{2}}}}\\&-\left(\mathrm {T} 'r^{2}{\frac {d\mathrm {A} '}{dr}}+\mathrm {T} ''r^{2}{\frac {d\mathrm {A} ''}{dr}}+\ldots \right){\frac {mdt}{r^{\frac {3}{2}}}}\\\\&-\mathrm {T} '\left({\frac {rr'}{2}}{\frac {d\mathrm {B} '}{dr'}}+{\frac {r^{2}r'}{4}}{\frac {d^{2}\mathrm {B} '}{drdr'}}+{\frac {3r^{2}}{r'^{2}}}\right){\frac {\mathrm {M} 'dt}{r^{\frac {3}{2}}}}\\&-\mathrm {T} '\left(r\mathrm {B} '+{\frac {r^{2}}{2}}{\frac {d\mathrm {B} '}{dr}}-{\frac {3r^{2}}{2r'^{2}}}\right){\frac {m'dt}{r^{\frac {3}{2}}}}\\\\&-\mathrm {T} ''\left({\frac {rr''}{2}}{\frac {d\mathrm {B} ''}{dr''}}+{\frac {r^{2}r''}{4}}{\frac {d^{2}\mathrm {B} ''}{drdr''}}+{\frac {3r^{2}}{r''^{2}}}\right){\frac {\mathrm {M} ''dt}{r^{\frac {3}{2}}}}\\&-\mathrm {T} ''\left(r\mathrm {B} ''+{\frac {r^{2}}{2}}{\frac {d\mathrm {B} ''}{dr}}-{\frac {3r^{2}}{2r''^{2}}}\right){\frac {m''dt}{r^{\frac {3}{2}}}}\\&-\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\\d\mathrm {M} =&\left(\mathrm {T} '{\frac {r^{3}}{2}}{\frac {d^{2}\mathrm {A} '}{dr^{2}}}+\mathrm {T} ''{\frac {r^{3}}{2}}{\frac {d^{2}\mathrm {A} ''}{dr^{2}}}+\ldots \right){\frac {\mathrm {N} dt}{r^{\frac {3}{2}}}}\\&+\left(\mathrm {T} 'r^{2}{\frac {d\mathrm {A} '}{dr}}+\mathrm {T} ''r^{2}{\frac {d\mathrm {A} ''}{dr}}+\ldots \right){\frac {ndt}{r^{\frac {3}{2}}}}\\\\&+\mathrm {T} '\left({\frac {rr'}{2}}{\frac {d\mathrm {B} '}{dr'}}+{\frac {r^{2}r'}{4}}{\frac {d^{2}\mathrm {B} '}{drdr'}}+{\frac {3r^{2}}{r'^{2}}}\right){\frac {\mathrm {N} 'dt}{r^{\frac {3}{2}}}}\\&+\mathrm {T} '\left(r\mathrm {B} '+{\frac {r^{2}}{2}}{\frac {d\mathrm {B} '}{dr}}-{\frac {3r^{2}}{2r'^{2}}}\right){\frac {n'dt}{r^{\frac {3}{2}}}}\\\\&+\mathrm {T} ''\left({\frac {rr''}{2}}{\frac {d\mathrm {B} ''}{dr''}}+{\frac {r^{2}r''}{4}}{\frac {d^{2}\mathrm {B} ''}{drdr''}}+{\frac {3r^{2}}{r''^{2}}}\right){\frac {\mathrm {N} ''dt}{r^{\frac {3}{2}}}}\\&+\mathrm {T} ''\left(r\mathrm {B} ''+{\frac {r^{2}}{2}}{\frac {d\mathrm {B} ''}{dr}}-{\frac {3r^{2}}{2r''^{2}}}\right){\frac {n''dt}{r^{\frac {3}{2}}}}\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\end{aligned}}$

Ce sont les équations qui servent à déterminer les variations séculaires des éléments $\mathrm {M}$ et $\mathrm {N} \,;$ et l’on aura des équations semblables pour les variations séculaires de $\mathrm {M} '$ et $\mathrm {N} ',$ de $\mathrm {M} ''$ et $\mathrm {N} ''$ en marquant simplement