Page:Joseph Louis de Lagrange - Œuvres, Tome 5.djvu/253

1^o Pour les mouvements annuels des nœuds par rapport à l’écliptique vraie

${\begin{aligned}&{\frac {d\Omega }{dt}}\ \ \ =-(0,1)-(0,2)-(0,3)-(0,4)-(0,5)-(3,0)\\&\qquad +{\bigl [}(0,1)-(3,1){\bigr ]}{\frac {\theta '\cos(\omega '-\omega )}{\theta }}+{\bigl [}(0,2)-(3,2){\bigr ]}{\frac {\theta ''\cos(\omega ''-\omega )}{\theta }}\\&\qquad +{\bigl [}(0,4)-(3,4){\bigr ]}{\frac {\theta ^{\scriptscriptstyle {\text{IV}}}\cos(\omega ^{\scriptscriptstyle {\text{IV}}}-\omega )}{\theta }}+{\bigl [}(0,5)-(3,5){\bigr ]}{\frac {\theta ^{\scriptscriptstyle {\text{V}}}\cos(\omega ^{\scriptscriptstyle {\text{V}}}-\omega )}{\theta }},\\\\&{\frac {d\Omega '}{dt}}\ \ =-(1,0)-(1,2)-(1,3)-(1,4)-(1,5)-(3,1)\\&\qquad +{\bigl [}(1,0)-(3,0){\bigr ]}{\frac {\theta \cos(\omega -\omega ')}{\theta '}}+{\bigl [}(1,2)-(3,2){\bigr ]}{\frac {\theta ''\cos(\omega ''-\omega ')}{\theta '}}\\&\qquad +{\bigl [}(1,4)-(3,4){\bigr ]}{\frac {\theta ^{\scriptscriptstyle {\text{IV}}}\cos(\omega ^{\scriptscriptstyle {\text{IV}}}-\omega ')}{\theta '}}+{\bigl [}(1,5)-(3,5){\bigr ]}{\frac {\theta ^{\scriptscriptstyle {\text{V}}}\cos(\omega ^{\scriptscriptstyle {\text{V}}}-\omega ')}{\theta '}},\\\\&{\frac {d\Omega ''}{dt}}\ \ =-(2,0)-(2,1)-(2,3)-(2,4)-(2,5)-(3,2)\\&\qquad +{\bigl [}(2,0)-(3,0){\bigr ]}{\frac {\theta \cos(\omega -\omega '')}{\theta ''}}+{\bigl [}(2,1)-(3,1){\bigr ]}{\frac {\theta '\cos(\omega '-\omega '')}{\theta ''}}\\&\qquad +{\bigl [}(2,4)-(3,4){\bigr ]}{\frac {\theta ^{\scriptscriptstyle {\text{IV}}}\cos(\omega ^{\scriptscriptstyle {\text{IV}}}-\omega '')}{\theta ''}}+{\bigl [}(2,5)-(3,5){\bigr ]}{\frac {\theta ^{\scriptscriptstyle {\text{V}}}\cos(\omega ^{\scriptscriptstyle {\text{V}}}-\omega '')}{\theta ''}},\\\\&{\frac {d\Omega ^{\scriptscriptstyle {\text{IV}}}}{dt}}=-(4,0)-(4,1)-(4,2)-(4,3)-(4,5)-(3,4)\\&\qquad +{\bigl [}(4,0)-(3,0){\bigr ]}{\frac {\theta \cos(\omega -\omega ^{\scriptscriptstyle {\text{IV}}})}{\theta ^{\scriptscriptstyle {\text{IV}}}}}+{\bigl [}(4,1)-(3,1){\bigr ]}{\frac {\theta '\cos(\omega '-\omega ^{\scriptscriptstyle {\text{IV}}})}{\theta ^{\scriptscriptstyle {\text{IV}}}}}\\&\qquad +{\bigl [}(4,2)-(3,2){\bigr ]}{\frac {\theta ''\cos(\omega ''-\omega ^{\scriptscriptstyle {\text{IV}}})}{\theta ^{\scriptscriptstyle {\text{IV}}}}}+{\bigl [}(4,5)-(3,5){\bigr ]}{\frac {\theta ^{\scriptscriptstyle {\text{V}}}\cos(\omega ^{\scriptscriptstyle {\text{V}}}-\omega ^{\scriptscriptstyle {\text{IV}}})}{\theta ^{\scriptscriptstyle {\text{IV}}}}},\\\\&{\frac {d\Omega ^{\scriptscriptstyle {\text{V}}}}{dt}}\ =-(5,0)-(5,1)-(5,2)-(5,3)-(5,4)-(3,5)\\&\qquad +{\bigl [}(5,0)-(3,0){\bigr ]}{\frac {\theta \cos(\omega -\omega ^{\scriptscriptstyle {\text{V}}})}{\theta ^{\scriptscriptstyle {\text{V}}}}}+{\bigl [}(5,1)-(3,1){\bigr ]}{\frac {\theta '\cos(\omega '-\omega ^{\scriptscriptstyle {\text{V}}})}{\theta ^{\scriptscriptstyle {\text{V}}}}}\\&\qquad +{\bigl [}(5,2)-(3,2){\bigr ]}{\frac {\theta ''\cos(\omega ''-\omega ^{\scriptscriptstyle {\text{V}}})}{\theta ^{\scriptscriptstyle {\text{V}}}}}+{\bigl [}(5,4)-(3,4){\bigr ]}{\frac {\theta ^{\scriptscriptstyle {\text{IV}}}\cos(\omega ^{\scriptscriptstyle {\text{IV}}}-\omega ^{\scriptscriptstyle {\text{V}}})}{\theta ^{\scriptscriptstyle {\text{V}}}}}.\end{aligned}}$