Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/762

24. Telles sont les valeurs qu’il faut substituer dans les formules du n^o 6, lesquelles deviendront par là

${\begin{aligned}{\frac {d\Omega }{da}}dt=&-{\frac {na}{2}}dc-{\frac {1}{2}}na{\sqrt {1-b^{2}}}df-{\frac {1}{2}}na{\sqrt {1-b^{2}}}\cos h\,dg,\\{\frac {d\Omega }{db}}dt=&{\frac {na^{2}b}{\sqrt {1-b^{2}}}}df+{\frac {na^{2}b}{\sqrt {1-b^{2}}}}\cos h\,dg,\\{\frac {d\Omega }{dc}}dt=&{\frac {na}{2}}da,\\{\frac {d\Omega }{df}}dt=&{\frac {1}{2}}na{\sqrt {1-b^{2}}}da-{\frac {na^{2}b}{\sqrt {1-b^{2}}}}db,\\{\frac {d\Omega }{dg}}dt=&{\frac {1}{2}}na{\sqrt {1-b^{2}}}\cos h\,da-{\frac {na^{2}b}{\sqrt {1-b^{2}}}}\cos hdb-na^{2}{\sqrt {1-b^{2}}}\sin h\,dh,\\{\frac {d\Omega }{dh}}dt=&na^{2}{\sqrt {1-b^{2}}}\sin h\,dg\,;\end{aligned}}$

d’où l’on tire facilement

${\begin{aligned}{\frac {da}{dt}}=&{\frac {2}{na}}{\frac {d\Omega }{dc}},\\{\frac {db}{dt}}=&{\frac {1-b^{2}}{na^{2}b}}{\frac {d\Omega }{dc}}-{\frac {\sqrt {1-b^{2}}}{na^{2}b}}{\frac {d\Omega }{df}},\\{\frac {dc}{dt}}=&-{\frac {2}{na}}{\frac {d\Omega }{da}}-{\frac {1-b^{2}}{na^{2}b}}{\frac {d\Omega }{db}},\\{\frac {df}{dt}}=&{\frac {\sqrt {1-b^{2}}}{na^{2}b}}{\frac {d\Omega }{db}}-{\frac {\cos h}{na^{2}{\sqrt {1-b^{2}}}\sin h}}{\frac {d\Omega }{dh}},\\{\frac {dg}{dt}}=&{\frac {1}{na^{2}{\sqrt {1-b^{2}}}\sin h}}{\frac {d\Omega }{dh}},\\{\frac {dh}{dt}}=&{\frac {\cos h}{na^{2}{\sqrt {1-b^{2}}}\sin h}}{\frac {d\Omega }{df}}-{\frac {1}{na^{2}{\sqrt {1-b^{2}}}\sin h}}{\frac {d\Omega }{dg}}.\end{aligned}}$

Voilà les formules des variations des six éléments elliptiques $a,b,c,$$f,g,h,$ exprimées par les différences partielles de la même fonction $\Omega ,$ relativement à ces mêmes éléments ; ce qui est infiniment commode pour