Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 8.djvu/271

on aura [voir les Recherches de M. de Lagrange sur Jupiter et Saturne (*)]

${\begin{aligned}b_{2}=&{\frac {\left(1+z^{2}\right)b_{1}-2b\mu z}{z(2-\mu )}},\\b_{3}=&{\frac {2\left(1+z^{2}\right)b_{2}-b_{1}z(1+\mu )}{z(3-\mu )}},\\b_{4}=&{\frac {3\left(1+z^{2}\right)b_{3}-zb_{2}(2+\mu )}{z(4-\mu )}},\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \\b'=&{\frac {b\left(1+z^{2}\right)+{\cfrac {\mu -1}{\mu }}b_{1}z}{\left(1-z^{2}\right)^{2}}},\\b'_{1}=&{\frac {4bz+{\cfrac {\mu -1}{\mu }}b_{1}\left(1+z^{2}\right)}{\left(1-z^{2}\right)^{2}}}.\end{aligned}}$

De là, en faisant $\mu ={\frac {3}{2}},$ j’ai conclu

${\begin{aligned}b_{2}=&{\frac {2\left(1+z^{2}\right)}{z}}b_{1}-6b,\\b'=&{\frac {b\left(1+z^{2}\right)}{\left(1-z^{2}\right)^{2}}}+{\frac {{\cfrac {1}{3}}b_{1}z}{\left(1-z^{2}\right)^{2}}},\\b'_{1}=&{\frac {4bz}{\left(1-z^{2}\right)^{2}}}+{\frac {{\cfrac {1}{3}}b_{1}\left(1+z^{2}\right)}{\left(1-z^{2}\right)^{2}}},\\b'_{2}=&{\frac {2b\left(1+z^{2}\right)}{\left(1-z^{2}\right)^{2}}}+{\frac {{\cfrac {10}{3}}b_{1}z-{\cfrac {2}{3}}{\cfrac {b_{1}\left(1-z^{2}\right)^{3}}{z^{2}}}}{\left(1-z^{2}\right)^{2}}},\\b'_{3}=&{\frac {{\cfrac {8b\left(1+z^{2}\right)^{2}}{z}}-28bz}{\left(1-z^{2}\right)^{2}}}+{\frac {11b_{1}(1+z^{2})-{\cfrac {8}{3}}{\cfrac {b_{1}\left(1+z^{2}\right)^{3}}{z^{2}}}}{\left(1-z^{2}\right)^{4}}},\\b''=&{\frac {{\cfrac {12}{5}}bz^{2}+b\left(1+z^{2}\right)^{2}}{\left(1-z^{2}\right)^{4}}}+{\frac {{\cfrac {8}{15}}b_{1}z\left(1+z^{2}\right)}{\left(1-z^{2}\right)^{4}}},\\b''_{1}=&{\frac {{\cfrac {32}{5}}bz\left(1+z^{2}\right)}{\left(1-z^{2}\right)^{4}}}+{\frac {{\cfrac {4}{3}}b_{1}z^{2}+{\cfrac {1}{5}}b_{1}\left(1+z^{2}\right)^{2}}{\left(1-z^{2}\right)^{4}}},\end{aligned}}$