Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 3.djvu/262

$-{\frac {3m^{2}}{4}}{\frac {a}{a_{\text{ı}}}}e'\gamma \left\{{\begin{aligned}&\left[{\begin{aligned}&(2-m-g){\rm {B}}_{1}^{(9)}\\&+{\frac {7}{2}}(2-2m-g){\rm {B}}_{1}^{(0)}\end{aligned}}\right]\sin(gv-c'mv-\theta +\varpi ')\\\\+&\left[{\begin{aligned}&(2-3m-g){\rm {B}}_{1}^{(10)}\\&-{\frac {1}{2}}(2-2m-g){\rm {B}}_{1}^{(0)}\end{aligned}}\right]\sin(gv+c'mv-\theta +\varpi ')\\\\+&(g-m){\rm {B}}_{1}^{(8)}\sin(2v-2mv-gv+c'mv+\theta -\varpi ')\\+&(g+m){\rm {B}}_{1}^{(7)}\sin(2v-2mv-gv-c'mv+\theta +\varpi ')\end{aligned}}\right\}$

$-{\frac {3m^{2}}{4}}{\frac {a}{a_{\text{ı}}}}e^{2}\gamma \left\{{\begin{aligned}&5{\rm {A}}_{1}^{(1)}-2{\rm {A}}_{1}^{(11)}+(2-2m-2c+g){\rm {B}}_{1}^{(12)}\\&\ \ -{\frac {10+19m+8m^{2}}{4}}(2-2m-g){\rm {B}}_{1}^{(0)}\end{aligned}}\right\}$

$\times \sin(2cv-gv-2\varpi +\theta )$

$-{\frac {3m^{2}}{4}}{\frac {a}{a_{\text{ı}}}}{\rm {B}}_{0}^{(11)}e^{2}\gamma \sin(2v-2mv-2cv+gv+2\varpi -2\theta ).$

Enfin la fonction $\left({\frac {d^{2}s}{dv^{2}}}+s\right){\frac {2}{h^{2}}}\int {\frac {\partial {\rm {Q}}}{\partial v}}{\frac {dv}{u^{2}}}$ donne, par sa variation, les termes

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

$\times \left\{{\begin{aligned}&{\frac {1}{1-m}}\gamma \sin(gv-\theta )\\+&{\frac {10+19m+8m^{2}}{2(2c-2+2m)}}e^{2}\gamma \sin(2cv-gv-2\varpi +\theta )\end{aligned}}\right\}.$

Les termes dépendants du cube de la force perturbatrice sont insensibles.

13. En rassemblant tous ces termes, la troisième des équations (L) du n^o 1 deviendra

(L") $0={\frac {d^{2}s}{dv^{2}}}+s+{\frac {3}{2}}m^{2}{\frac {a}{a_{\text{ı}}}}$

$\times \left\{{\begin{aligned}1&+2e^{2}-{\frac {1}{2}}\gamma ^{2}+{\frac {3}{2}}e'^{2}\\&-{\frac {1}{2}}\left[{\frac {(3-2m-g)(g+m)}{1-m}}{\rm {B}}_{1}^{(0)}+4{\rm {A}}_{2}^{(0)}\right]\left(1-{\frac {5}{2}}e'^{2}\right)\\\\&-{\frac {7}{4}}(3-3m-g){\rm {B}}_{1}^{(10)}e'^{2}+{\frac {1}{4}}(3-m-g){\rm {B}}_{1}^{(9)}e'^{2}\\&+{\frac {3}{2}}\left({\rm {B_{1}^{(7)}+B_{1}^{(8)}}}\right)e'^{2}\end{aligned}}\right\}$

$\times \gamma \sin(gv-\theta )$

$-{\frac {3}{4}}m^{2}{\frac {a}{a_{\text{ı}}}}\left\{{\begin{aligned}&(1+g)\left(1+2e^{2}-{\frac {2+m}{4}}\gamma ^{2}-{\frac {5}{2}}e'^{2}\right)\\\\&\ {\frac {1-g^{2}}{1-m}}-4{\rm {A}}_{2}^{(0)}+10{\rm {A}}_{1}^{(1)}e^{2}-2{\rm {B}}_{1}^{(0)}\end{aligned}}\right\}$

$\times \gamma \sin(2v-2mv-gv+\theta )$

$+{\frac {3}{2}}m^{2}{\frac {a}{a_{\text{ı}}}}\left({\frac {1-g}{2}}+{\rm {B}}_{2}^{(1)}\right)\gamma \sin(2v-2mv+gv-\theta )$