Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 8.djvu/453

de la planète $\mathrm {P} ,$

${\frac {arr'}{\left[r^{2}+r'^{2}+\left(r's'-rs\right)^{2}\right]^{\frac {3}{2}}}}$

$\times {\frac {i}{\left(1+i^{2}\right)^{\frac {3}{2}}}}\left\{{\begin{aligned}1&+{\frac {\alpha e\left(i^{2}-2\right)}{1+i^{2}}}\cos(nt+\theta )\\&-{\frac {\alpha ^{2}ee'}{2}}{\frac {2-11i^{2}+2i^{4}}{\left(1+i^{2}\right)^{2}}}\cos(n't-nt+\theta '-\theta )\\&+{\frac {\alpha e'\left(1-2i^{2}\right)}{1+i^{2}}}\cos(n't+\theta ')\\&+{\frac {3}{2}}{\frac {\alpha ^{2}\gamma \gamma '}{1+i^{2}}}\cos(n't-nt+\theta '-\theta )\end{aligned}}\right\}\,;$

on aura ensuite

$h={\frac {2i}{1+i^{2}}}\,;$

$\alpha h'={\frac {2i}{1+i^{2}}}$

$\times \left\{{\begin{aligned}\alpha e&{\frac {i^{2}-1}{1+i^{2}}}\cos(nt+\theta )-{\frac {1}{2}}\alpha ^{2}ee'{\frac {1-6i^{2}+i^{4}}{\left(1+i^{2}\right)^{2}}}\cos(n't-nt+\theta '-\theta )\\&+\alpha e'{\frac {1-i^{2}}{1+i^{2}}}\cos(n't+\theta ')+{\frac {\alpha ^{2}\gamma \gamma 'i}{1+i^{2}}}\cos(n't-nt+\theta '-\theta )\end{aligned}}\right\}\,;$

partant,

${\frac {\alpha ^{2}h'^{2}}{2}}=-{\frac {2\alpha ^{2}ee'i^{2}\left(1-i^{2}\right)^{2}}{\left(1+i^{2}\right)^{4}}}\cos(n't-nt+\theta '-\theta ).$

De là, on conclura, en faisant, pour abréger,

$\mathrm {A'-A+\theta '-\theta =B,\qquad A'-A=V}$

et

$\mathrm {A'-A+\theta '-\theta -\varpi '+\varpi =U} ,$

${\frac {arr'\sin(\varphi '-\varphi )}{\sideset {^{1}}{^{3}}v}}=i\left\{{\begin{aligned}&\left[(b)-{\frac {1}{2}}(b_{2})\right]\sin(n't-nt+\mathrm {B} )\\+{\frac {1}{2}}&\left[(b_{1})-\ \ (b_{3})\right]\sin 2(n't-nt+\mathrm {B} )\\+{\frac {1}{2}}&\left[(b_{2})-\ \ (b_{4})\right]\sin 3(n't-nt+\mathrm {B} )\\+\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \end{aligned}}\right\}$

$+\left\{{\begin{aligned}&2\alpha ei\quad \sin(nt+\theta )\\-&2\alpha e'i\sin(n't+\theta ')\end{aligned}}\right\}\left\{{\begin{aligned}&\left[(b)-{\frac {1}{2}}(b_{2})\right]\cos(n't-nt+\mathrm {B} )\\+{\frac {2}{2}}&\left[(b_{1})-\ \ (b_{3})\right]\cos 2(n't-nt+\mathrm {B} )\\+{\frac {3}{2}}&\left[(b_{2})-\ \ (b_{4})\right]\cos 3(n't-nt+\mathrm {B} )\\+\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \end{aligned}}\right\}$

$+\left\{{\begin{aligned}&{\frac {\alpha ei(i^{2}-2)}{1+i^{2}}}\quad \cos(nt+\theta )\\+&{\frac {\alpha e'i(1-2i^{2})}{1+i^{2}}}\cos(n't+\theta ')\end{aligned}}\right\}\left\{{\begin{aligned}&\left[(b)-{\frac {1}{2}}(b_{2})\right]\sin(n't-nt+\mathrm {B} )\\+{\frac {1}{2}}&\left[(b_{1})-\ \ (b_{3})\right]\sin 2(n't-nt+\mathrm {B} )\\+\ \ &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \end{aligned}}\right\}$