Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 8.djvu/463

on aura

${\begin{aligned}{\frac {dx'}{dt}}=&{\frac {2}{3}}n\mathrm {F} e\cos(nt+\theta )+2ne(\mathrm {F+G} )nt\sin(nt+\theta )\\&-\mathrm {\sideset {^{2}}{}L} e'n^{2}t\sin(nt-B+\theta ')+{\frac {4n^{3}.\mathrm {\sideset {^{1}}{}H} }{n'^{2}-n^{2}}}\cos(n't+\theta ')\\&+en.\ \ \mathrm {M} \cos(\ \ n't+\mathrm {B} +\theta )\\&+en.\mathrm {\sideset {^{1}}{}M} \cos(\ \ n't-2nt+\ \ \mathrm {B} -\theta )\\&+en.\mathrm {\sideset {^{2}}{}M} \cos(2n't-\ \ nt+2\mathrm {B} +\theta )\\&+en.\mathrm {\sideset {^{3}}{}M} \cos(2n't-3nt+2\mathrm {B} -\theta )\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&+e'n.\ \ \mathrm {N} \cos(2n't-\ \ nt+\ \ \mathrm {B} +\theta ')\\&+e'n.\mathrm {\sideset {^{1}}{}N} \cos(\ \ n't+\ \ \theta '-\ \ \mathrm {B} )\\&+e'n.\mathrm {\sideset {^{2}}{}N} \cos(3n't-2nt+2\mathrm {B} +\theta ')\\&+e'n.\mathrm {\sideset {^{3}}{}N} \cos(\ \ n't-2nt+2\mathrm {B} -\theta ')\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\\\end{aligned}}$

Enfin, si, dans l’équation (18) de l’article XI, on substitue, au lieu de $\delta s,\alpha z\delta \mu ',$ que l’on néglige les quantités de l’ordre $\alpha ^{2},$ et que l’on fasse, pour abréger,

${\begin{aligned}\mathrm {\sideset {^{1}}{}P} =&-{\frac {n(n'+n)}{(n'-n)^{2}-n^{2}}}\left[\mathrm {\sideset {^{1}}{}F} +{\frac {1}{i^{2}}}+{\frac {2n}{n'-n}}\left(\mathrm {C} -{\frac {1}{i^{2}}}\right)\right]\\&-{\frac {n}{n'-n}}\left(\mathrm {C} -{\frac {1}{i^{2}}}\right)+{\frac {i}{4}}\left[2(b)+(b_{2})-{\frac {2}{i^{3}}}\right],\\\mathrm {\sideset {^{2}}{}P} =&\quad {\frac {n(3n-n')}{(n'-n)^{2}-n^{2}}}\left[\mathrm {\sideset {^{1}}{}F} +{\frac {1}{i^{2}}}+{\frac {2n}{n'-n}}\left(\mathrm {C} -{\frac {1}{i^{2}}}\right)\right]\\&+{\frac {n}{n'-n}}\left(\mathrm {C} -{\frac {1}{i^{2}}}\right)-{\frac {i}{4}}\left[2(b)+(b_{2})-{\frac {2}{i^{3}}}\right],\\\mathrm {\sideset {^{3}}{}P} =&-{\frac {2nn'}{(2n'-2n)^{2}-n^{2}}}\left({\frac {1}{2}}\mathrm {\sideset {^{2}}{}F} +{\frac {2n}{n'-n}}{\frac {1}{4}}\mathrm {\sideset {^{1}}{}C} \right)\\&-{\frac {n}{n'-n}}{\frac {1}{4}}\mathrm {\sideset {^{1}}{}C} +{\frac {i}{4}}\left[(b_{1})+(b_{3})\right],\\\mathrm {\sideset {^{4}}{}P} =&\quad {\frac {n(4n-2n')}{(2n'-2n)^{2}-n^{2}}}\left({\frac {1}{2}}\mathrm {\sideset {^{2}}{}F} +{\frac {2n}{n'-n}}{\frac {1}{4}}\mathrm {\sideset {^{1}}{}C} \right)\\&+{\frac {n}{n'-n}}{\frac {1}{4}}\mathrm {\sideset {^{1}}{}C} -{\frac {i}{4}}\left[(b_{1})+(b_{3})\right],\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\end{aligned}}$