Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 3.djvu/266

$0=\left[1-(2c-g)^{2}\right]{\rm {B}}_{0}^{(11)}+{\frac {3}{4}}m^{2}{\frac {a}{a_{\text{ı}}}}$

$\times \left[{\begin{aligned}&2{\rm {B}}_{0}^{(11)}-5-10{\rm {A}}_{1}^{(11)}-(3-2m-2c+g){\rm {B}}_{1}^{(12)}\\&\ -(3-2m-g)(g+m-c){\frac {10+19m+8m^{2}}{2(2c+2m-2)}}{\rm {B}}_{1}^{(0)}\end{aligned}}\right],$

$0=\left[1-(2-2m-2c+g)^{2}\right]{\rm {B}}_{1}^{(12)}+{\frac {3}{4}}m^{2}{\frac {a}{a_{\text{ı}}}}$

$\times \left[{\begin{aligned}&2{\rm {B}}_{1}^{(12)}+(1-g){\frac {10+19m+8m^{2}}{4}}\\&\ 10{\rm {A}}_{1}^{(1)}-4{\rm {A}}_{1}^{(11)}-2{\rm {B}}_{0}^{(11)}\end{aligned}}\right],$

$0=\left[1-(2c+g-2+2m)^{2}\right]{\rm {B}}_{1}^{(13)}+{\frac {3}{4}}m^{2}{\frac {a}{a_{\text{ı}}}}$

$\times \left[{\begin{aligned}&{\frac {10+19m+8m^{2}}{2}}+2{\rm {B}}_{1}^{(13)}\\&\ 10{\rm {A}}_{1}^{(1)}-4{\rm {A}}_{1}^{(11)}-5{\rm {B}}_{-}^{(0)}1\end{aligned}}\right],$

$0=\left[1-(g+m-1)^{2}\right]{\rm {B}}_{2}^{(14)}+{\frac {3}{4}}m^{2}{\frac {a}{a_{\text{ı}}}}\left(3+2{\rm {B}}_{2}^{(14)}\right),$

$0=\left[1-(g+1-m)^{2}\right]{\rm {B}}_{2}^{(15)}+{\frac {3}{4}}m^{2}{\frac {a}{a_{\text{ı}}}}\left({\frac {5}{2}}+2{\rm {B}}_{2}^{(15)}\right).$

15. Il nous reste présentement à déterminer la valeur du temps $t$ en fonction de $v.$ Pour cela, reprenons la première des équations (L) du n^o 1,

$dt={\frac {dv}{hu^{2}{\sqrt {1+{\frac {2}{h^{2}}}\int {\frac {\partial {\rm {Q}}}{\partial v}}{\frac {dv}{u^{2}}}}}}}.$

Il faut, par le n^o 6, y substituer, au lieu de $u,$ la fonction

${\frac {1}{a}}\left[{\begin{aligned}&1+e^{2}+{\frac {1}{4}}\gamma ^{2}+{\text{ϐ}}+e\left(1+e^{2}\right)\cos(cv-\varpi )\\&-{\frac {1}{4}}\gamma ^{2}\left(1+e^{2}-{\frac {1}{4}}\gamma ^{2}\right)\cos(2gv-2\theta )\end{aligned}}\right]$

On aura d’abord, en développant le facteur ${\frac {dv}{hu^{2}}},$ un terme indépendant de cosinus, et qui, par la nature du mouvement elliptique, doit être ${\frac {a^{2}dv}{\sqrt {a_{\text{ı}}}}}$ (Livre II, n^o 16) ; on aura ensuite

$dt={\frac {a^{2}dv}{\sqrt {a_{\text{ı}}}}}\times$

$\left\{{\begin{aligned}&\left\{{\begin{aligned}&1-2e\left(1-{\frac {1}{4}}\gamma ^{2}\right)\cos(cv-\varpi )\\&+e^{2}\left({\frac {3}{2}}+{\frac {1}{4}}e^{2}-{\frac {3}{4}}\gamma ^{2}\right)\cos(2cv-2\varpi )\\&+{\frac {1}{2}}\gamma ^{2}\left(1+{\frac {3}{2}}e^{2}-{\frac {1}{2}}\gamma ^{2}\right)\cos(2gv-2\theta )-e^{3}\cos(3cv-3\varpi )\\&-{\frac {3}{4}}e\gamma ^{2}\left[\cos(2gv-cv-2\theta +\varpi )+\cos(2gv+cv-2\theta -\varpi )\right]\end{aligned}}\right\}\\\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \times \left\{{\begin{aligned}&1-{\frac {1}{h^{2}}}\int {\frac {\partial {\rm {Q}}}{\partial v}}{\frac {dv}{u^{2}}}\\\\&+{\frac {3}{2h^{4}}}\left(\int {\frac {\partial {\rm {Q}}}{\partial v}}{\frac {dv}{u^{2}}}\right)^{2}\\&-\ldots \ldots \ldots \end{aligned}}\right\}\\&-2a\delta u\left[{\begin{aligned}&1+{\frac {1}{2}}e^{2}-{\frac {1}{4}}\gamma ^{2}-3e\cos(cv-\varpi )+3e^{2}\cos(2cv-2\varpi )\\&+{\frac {3}{4}}\gamma ^{2}\cos(2gv-2\theta )-{\frac {3}{2}}e\gamma ^{2}\cos(2gv-cv-2\theta +\varpi )\end{aligned}}\right]\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \times \left[{\begin{aligned}&1-{\frac {1}{h^{2}}}\int {\frac {\partial {\rm {Q}}}{\partial v}}{\frac {dv}{u^{2}}}\\\\&+\ldots \ldots \ldots \end{aligned}}\right]\\\\&+3a^{2}(\delta u)^{2}\left[1-4e\cos(cv-\varpi )\right][1-\ldots ]\end{aligned}}\right\}$