Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 3.djvu/56

$a'{\rm {N}}^{(4)}=-{\frac {e^{3}}{768}}$

$\times \left\{{\begin{aligned}-&\left(11840e^{2}+152000e'^{2}\right)\alpha b_{\frac {1}{2}}^{(5)}-\left(6560e^{2}+65168e'^{2}\right)\alpha {\frac {db_{\frac {1}{2}}^{(5)}}{d\alpha }}\\\\-&\left(592e^{2}+4720e'^{2}\right)\alpha ^{2}{\frac {d^{2}b_{\frac {1}{2}}^{(5)}}{d\alpha ^{2}}}+\left(152e^{2}+920e'^{2}\right)\alpha ^{3}{\frac {d^{3}b_{\frac {1}{2}}^{(5)}}{d\alpha ^{3}}}\\\\+&\left(26e^{2}+128e'^{2}\right)\alpha ^{4}{\frac {d^{4}b_{\frac {1}{2}}^{(5)}}{d\alpha ^{4}}}+\left(e^{2}+4e'^{2}\right)\alpha ^{5}{\frac {d^{5}b_{\frac {1}{2}}^{(5)}}{d\alpha ^{5}}}\end{aligned}}\right\}$

$-{\frac {e^{3}\gamma ^{2}}{384}}\left\{{\begin{aligned}&554\alpha \left(b_{\frac {3}{2}}^{(4)}+b_{\frac {3}{2}}^{(6)}\right)+222\alpha ^{2}\left({\frac {db_{\frac {3}{2}}^{(4)}}{d\alpha }}+{\frac {db_{\frac {3}{2}}^{(6)}}{d\alpha }}\right)\\\\&+27\alpha ^{3}\left({\frac {d^{2}b_{\frac {3}{2}}^{(4)}}{d\alpha ^{2}}}+{\frac {d^{2}b_{\frac {3}{2}}^{(6)}}{d\alpha ^{2}}}\right)+\alpha ^{4}\left({\frac {d^{3}b_{\frac {3}{2}}^{(4)}}{d\alpha ^{3}}}+{\frac {d^{3}b_{\frac {3}{2}}^{(6)}}{d\alpha ^{3}}}\right)\end{aligned}}\right\},$

$a'{\rm {N}}^{(5)}=-{\frac {e'e^{4}}{768}}\left\{{\begin{aligned}&41448b_{\frac {1}{2}}^{(6)}+18392\alpha {\frac {db_{\frac {1}{2}}^{(6)}}{d\alpha }}+1780\alpha ^{2}{\frac {d^{2}b_{\frac {1}{2}}^{(6)}}{d\alpha ^{2}}}\\\\&-156\alpha ^{3}{\frac {d^{3}b_{\frac {1}{2}}^{(6)}}{d\alpha ^{3}}}-29\alpha ^{4}{\frac {d^{4}b_{\frac {1}{2}}^{(6)}}{d\alpha ^{4}}}-\alpha ^{5}{\frac {d^{5}b_{\frac {1}{2}}^{(6)}}{d\alpha ^{5}}}\end{aligned}}\right\}$

$a'{\rm {N}}^{(6)}=-{\frac {e'^{2}e\gamma ^{2}}{128}}\left\{{\begin{aligned}&-85\alpha b_{\frac {3}{2}}^{(2)}+85\alpha ^{2}{\frac {db_{\frac {3}{2}}^{(2)}}{d\alpha }}+21\alpha ^{3}{\frac {d^{2}b_{\frac {3}{2}}^{(2)}}{d\alpha ^{2}}}+\alpha ^{4}{\frac {d^{3}b_{\frac {3}{2}}^{(2)}}{d\alpha ^{3}}}\end{aligned}}\right\},$

$a'{\rm {N}}^{(7)}=-{\frac {e'\gamma ^{2}}{128}}\left\{{\begin{aligned}&\left(56e^{2}+842e'^{2}\right)\alpha b_{\frac {3}{2}}^{(3)}+\left(4e^{2}+87e'^{2}\right)\alpha ^{2}{\frac {db_{\frac {3}{2}}^{(3)}}{d\alpha }}\\\\&-\left(16e^{2}+2e'^{2}\right)\alpha ^{3}{\frac {d^{2}b_{\frac {3}{2}}^{(3)}}{d\alpha ^{2}}}-\left(2e^{2}+e'^{2}\right)\alpha ^{4}{\frac {d^{3}b_{\frac {3}{2}}^{(3)}}{d\alpha ^{3}}}\end{aligned}}\right\},$

$a'{\rm {N}}^{(8)}=-{\frac {e\gamma ^{2}}{128}}\left\{{\begin{aligned}&-\left(174e^{2}+196e'^{2}\right)\alpha b_{\frac {3}{2}}^{(4)}-\left(50e^{2}+180e'^{2}\right)\alpha ^{2}{\frac {db_{\frac {3}{2}}^{(4)}}{d\alpha }}\\\\&+\left(14e'^{2}-e^{2}\right)\alpha ^{3}{\frac {d^{2}b_{\frac {3}{2}}^{(4)}}{d\alpha ^{2}}}+\left(2e'^{2}+e^{2}\right)\alpha ^{4}{\frac {d^{3}b_{\frac {3}{2}}^{(4)}}{d\alpha ^{3}}}\end{aligned}}\right\},$

$a'{\rm {N}}^{(9)}=-{\frac {e'e^{2}\gamma ^{2}}{128}}\left\{{\begin{aligned}&580\alpha b_{\frac {3}{2}}^{(5)}+86\alpha ^{2}{\frac {db_{\frac {3}{2}}^{(5)}}{d\alpha }}-8\alpha ^{3}{\frac {d^{2}b_{\frac {3}{2}}^{(5)}}{d\alpha ^{2}}}-\alpha ^{4}{\frac {d^{3}b_{\frac {3}{2}}^{(5)}}{d\alpha ^{3}}}\end{aligned}}\right\}.$

Lorsque l’on considère l’action de $m'$ sur $m$, il faut, par le n^o 4, augmenter, dans $a'{\rm {N}}^{(0)}\ b_{\frac {1}{2}}^{(1)}$ de $-{\frac {a}{a'}}$ ou de $-\alpha ,$ ce qui augmente $a'{\rm {N}}^{(0)}$ de