Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 1.djvu/339

valeur $-{\rm {A}}^{(i-1)}-a{\frac {\partial {\rm {A}}^{(i-1)}}{\partial a}},$ on aura

${\begin{aligned}{\rm {C}}&=a^{2}{\frac {\partial {\rm {A}}^{(0)}}{\partial a}}+{\frac {1}{2}}a^{3}{\frac {\partial ^{2}{\rm {A}}^{(0)}}{\partial a^{2}}},\\\\{\rm {D}}&=a{\rm {A}}^{(1)}-a^{2}{\frac {\partial {\rm {A}}^{(1)}}{\partial a}}-{\frac {1}{2}}a^{3}{\frac {\partial ^{2}{\rm {A}}^{(1)}}{\partial a^{2}}},\\\\{\rm {D}}^{(i)}&={\frac {(i-1)(2i-1)n}{n-i(n-n')}}a{\rm {A}}^{(i-1)}+{\frac {i^{2}(n-n')-n}{n-i(n-n')}}a^{2}{\frac {\partial {\rm {A}}^{(i-1)}}{\partial a}}\\&\qquad \qquad \qquad \qquad \qquad \qquad -{\frac {1}{2}}a^{3}{\frac {\partial ^{2}{\rm {A}}^{(i-1)}}{\partial a^{2}}},\\f&={\frac {2}{3}}a^{2}{\frac {\partial {\rm {A}}^{(0)}}{\partial a}}+{\frac {1}{4}}a^{3}{\frac {\partial ^{2}{\rm {A}}^{(0)}}{\partial a^{2}}},\\\\f'&={\frac {1}{4}}\left(a{\rm {A}}^{(1)}-a^{2}{\frac {\partial {\rm {A}}^{(1)}}{\partial a}}-a^{3}{\frac {\partial ^{2}{\rm {A}}^{(1)}}{\partial a^{2}}}\right).\end{aligned}}$

Soit, de plus,

${\begin{aligned}\mathrm {E} ^{(i)}&=-{\frac {3n}{n-n'}}a{\rm {A}}^{(i)}+{\frac {i^{2}(n-n')\left[n+i(n-n')\right]-3n^{2}}{i^{2}(n-n')^{2}-n^{2}}}\\&\qquad \qquad \times \left(a^{2}{\frac {\partial {\rm {A}}^{(i)}}{\partial a}}+{\frac {2n}{n-n'}}a{\rm {A}}^{(i)}\right)+{\frac {1}{2}}a^{3}{\frac {\partial ^{2}{\rm {A}}^{(i)}}{\partial a^{2}}},\\\\\mathrm {F} ^{(i)}&={\frac {(i-1)n}{n-n'}}a{\rm {A}}^{(i)}+{\frac {{\frac {in}{2}}\left[n+i(n-n')\right]-3n^{2}}{i^{2}(n-n')^{2}-n^{2}}}\\&\qquad \qquad \times \left(a^{2}{\frac {\partial {\rm {A}}^{(i)}}{\partial a}}+{\frac {2n}{n-n'}}a{\rm {A}}^{(i)}\right)-{\frac {2n^{2}\mathrm {E} ^{(i)}}{n^{2}-\left[n-i(n-n')\right]^{2}}},\\\\\mathrm {G} ^{(i)}&={\frac {(i-1)(2i-1)na{\rm {A}}^{(i-1)}+(i-1)na^{2}{\frac {\partial {\rm {A}}^{(i-1)}}{\partial a}}}{2\left[n-i(n-n')\right]}}\\&\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad -{\frac {2n^{2}\mathrm {D} ^{(i)}}{n^{2}-\left[n-i(n-n')\right]^{2}}},\end{aligned}}$

on aura

${\frac {\partial r}{a}}={\frac {m'}{6}}a^{2}{\frac {\partial {\rm {A}}^{(0)}}{\partial a}}+{\frac {m'n^{2}}{2}}\Sigma {\frac {a^{2}{\frac {\partial {\rm {A}}^{(i)}}{\partial a}}+{\frac {2n}{n-n'}}a{\rm {A}}^{(i)}}{i^{2}(n-n')^{2}-n^{2}}}$

$\times \cos i(n't-nt+\varepsilon '-\varepsilon )$

${\begin{aligned}&-m'fe\cos(nt+\varepsilon -\varpi )-m'f'e'\cos(nt+\varepsilon -\varpi ')\\&+{\frac {1}{2}}m'{\rm {C}}nte\sin(nt+\varepsilon -\varpi )+{\frac {1}{2}}m'{\rm {D}}nte'\sin(nt+\varepsilon -\varpi ')\\&+n^{2}m'\Sigma \left\{{\begin{aligned}&{\frac {\mathrm {E} ^{(i)}}{n^{2}-\left[n-i(n-n')\right]^{2}}}\\&\qquad \qquad \times e\cos \left[i(n't-nt+\varepsilon '-\varepsilon )+nt+\varepsilon -\varpi \right]\\+&{\frac {\mathrm {D} ^{(i)}}{n^{2}-\left[n-i(n-n')\right]^{2}}}\\&\qquad \qquad \times e'\cos \left[i(n't-nt+\varepsilon '-\varepsilon )+nt+\varepsilon -\varpi '\right]\end{aligned}}\right\}\end{aligned}}$