Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 3.djvu/115

jupiter et uranus.

$\alpha ={\frac {a^{\rm {iv}}}{a^{\rm {vi}}}}=0{,}27112980,$

d’où l’on a conclu

${\begin{aligned}b_{-{\frac {1}{2}}}^{(0)}&=2{,}03692776,\\b_{-{\frac {1}{2}}}^{(1)}&=-0{,}26861497.\end{aligned}}$

Ensuite,

${\begin{alignedat}{3}b_{\frac {1}{2}}^{(0)}&=2{,}038359,&\qquad b_{\frac {1}{2}}^{(1)}&=0{,}278966,&\qquad b_{\frac {1}{2}}^{(2)}&=0{,}056906,\\b_{\frac {1}{2}}^{(3)}&=0{,}012879,&b_{\frac {1}{2}}^{(4)}&=0{,}003058,&b_{\frac {1}{2}}^{(5)}&=0{,}000745,\\b_{\frac {1}{2}}^{(6)}&=0{,}000185\,;\\\\{\frac {db_{\frac {1}{2}}^{(0)}}{d\alpha }}&=0{,}295410,&{\frac {db_{\frac {1}{2}}^{(1)}}{d\alpha }}&=1{,}089551,&{\frac {db_{\frac {1}{2}}^{(2)}}{d\alpha }}&=0{,}433630,\\{\frac {db_{\frac {1}{2}}^{(3)}}{d\alpha }}&=0{,}145398,&{\frac {db_{\frac {1}{2}}^{(4)}}{d\alpha }}&=0{,}045930,&{\frac {db_{\frac {1}{2}}^{(5)}}{d\alpha }}&=0{,}015410\,;\\\\{\frac {d^{2}b_{\frac {1}{2}}^{(0)}}{d\alpha ^{2}}}&=1{,}283434,&{\frac {d^{2}b_{\frac {1}{2}}^{(1)}}{d\alpha ^{2}}}&=0{,}714932,&{\frac {d^{2}b_{\frac {1}{2}}^{(2)}}{d\alpha ^{2}}}&=1{,}815451,\\{\frac {d^{2}b_{\frac {1}{2}}^{(3)}}{d\alpha ^{2}}}&=1{,}133359\,;\\\\b_{\frac {3}{2}}^{(0)}&=2{,}372983,&b_{\frac {3}{2}}^{(1)}&=0{,}938794,&b_{\frac {3}{2}}^{(2)}&=0{,}315186,\\b_{\frac {3}{2}}^{(3)}&=0{,}099260.\end{alignedat}}$

saturne et uranus.

$\alpha ={\frac {a^{\rm {v}}}{a^{\rm {vi}}}}=0{,}49719638,$

d’où l’on a conclu

${\begin{aligned}b_{-{\frac {1}{2}}}^{(0)}&=2{,}12564287,\\b_{-{\frac {1}{2}}}^{(1)}&=-0{,}48131675.\end{aligned}}$