Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 3.djvu/261

Le terme ${\frac {3m'u'^{3}s}{2h^{2}u^{4}}}\cos(2v-2v')$ donne, par sa variation, les suivants

${\frac {3m'u'^{3}\delta s}{2h^{2}u^{4}}}\cos(2v-2v')-{\frac {6m'u'^{3}s\delta u}{h^{2}u^{5}}}\cos(2v-2v')$

$+{\frac {3m'u'^{3}s\delta v'}{2h^{2}u^{4}}}\sin(2v-2v'),$

et il en résulte la fonction

$-{\frac {3m^{2}}{4}}{\frac {a}{a_{\text{ı}}}}\left({\rm {B}}_{1}^{(0)}+4{\rm {A}}_{2}^{(0)}+{\frac {7}{2}}{\rm {B}}_{1}^{(10)}e'^{2}-{\frac {1}{2}}{\rm {B}}_{1}^{(0)}e'^{2}\right)\left(1-{\frac {5}{2}}e'^{2}\right)\gamma \sin(gv-\theta )$

$+{\frac {3m^{2}}{4}}{\frac {a}{a_{\text{ı}}}}e\gamma \left\{{\begin{aligned}&\left[(1+m){\rm {B_{1}^{(0)}-A_{1}^{(1)}}}\right]\sin(gv-cv-\theta +\varpi )\\+&\left[(1-m){\rm {B_{1}^{(0)}-A_{1}^{(1)}}}\right]\sin(gv+cv-\theta -\varpi )\end{aligned}}\right\}$

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

$-{\frac {3m^{2}}{4}}{\frac {a}{a_{\text{ı}}}}e^{2}\gamma \left\{{\begin{aligned}&5{\rm {A_{1}^{(1)}-2A_{1}^{(11)}+B_{1}^{(12)}}}\\&-{\frac {10+19m+8m^{2}}{4}}{\rm {B}}_{1}^{(0)}\end{aligned}}\right\}\sin(2cv-gv-2\varpi +\theta )$

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

Le terme $-{\frac {3m'u'^{3}}{2h^{2}u^{4}}}{\frac {ds}{dv}}\sin(2v-2v')$ donne, par sa variation,

${\begin{aligned}-{\frac {3m'u'^{3}}{2h^{2}u^{4}}}{\frac {d\delta s}{dv}}\sin(2v-2v')&+{\frac {6m'u'^{3}}{h^{2}u^{5}}}\delta u{\frac {ds}{dv}}\sin(2v-2v')\\&+{\frac {3m'u'^{3}}{h^{2}u^{4}}}{\frac {ds}{dv}}\delta v'\cos(2v-2v').\end{aligned}}$

De là résulte la fonction

$-{\frac {3m^{2}}{4}}{\frac {a}{a_{\text{ı}}}}\left[{\begin{aligned}&(2-2m-g){\rm {B}}_{1}^{(0)}+{\frac {1}{2}}(2-3m-g){\rm {B}}_{1}^{(10)}e'^{2}\\&\qquad {\frac {1}{2}}(2-m-g){\rm {B}}_{1}^{(9)}e'^{2}\end{aligned}}\right]$

$\times \left(1-{\frac {5}{2}}e'^{2}\right)\gamma \sin(gv-\theta )$

$+{\frac {3m^{2}}{2}}{\frac {a}{a_{\text{ı}}}}e\gamma \left\{{\begin{aligned}&\left[(1+m)(2-2m-g){\rm {B_{1}^{(0)}-A_{1}^{(1)}}}\right]\sin(gv-cv-\theta +\varpi )\\+&\left[(1-m)(2-2m-g){\rm {B_{1}^{(0)}+A_{1}^{(1)}}}\right]\sin(gv+cv-\theta -\varpi )\\\end{aligned}}\right\}$