Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 3.djvu/245

sa variation est

${\frac {12m'}{h^{2}a}}\left[1+{\frac {3}{4}}\gamma ^{2}\cos(2gv-2\theta )\right]$

$\times \int {\frac {u'^{3}dv}{u^{4}}}\left[{\frac {\delta u}{u}}\sin(2v-2v')+{\frac {1}{2}}\delta v'.\cos(2v-2v')\right]$

$-\left({\frac {\partial ^{2}\delta u}{\partial v^{2}}}+\delta u\right)\int {\frac {3m'u'^{3}dv}{h^{2}u^{4}}}\sin(2v-2v')$

$-{\frac {9m'}{h^{2}a}}\int {\frac {u'^{2}\delta u'}{u^{4}}}dv.\sin(2v-2v').$

Le développement de ces termes donne, en observant que $c$ est à très-peu près $1-{\frac {3}{2}}m^{2},$ et que $g$ est à très-peu près $1+{\frac {3}{4}}m^{2},$

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

$-{\frac {3m^{2}}{a_{\text{ı}}}}\left\{{\begin{aligned}&{\frac {7+(2-2m-c)^{2}}{4(1-m)}}{\rm {A}}_{1}^{(1)}\\\\&\ \ -\left[4(1-m)^{2}-1\right]\left({\frac {1+m}{2-2m-c}}+{\frac {1-m}{2-2m+c}}\right){\rm {A}}_{2}^{(0)}\\\\&\ \ -{\rm {A}}_{1}^{(6)}e'^{2}+7{\rm {A}}_{1}^{(7)}e'^{2}\end{aligned}}\right\}$

$\times e\left(1-{\frac {5}{2}}e'^{2}\right)\cos(cv-\varpi )$

$+\left\{{\begin{aligned}&{\frac {6m}{a_{\text{ı}}}}\left[4{\rm {A_{2}^{(0)}+A_{2}^{(3)}-A_{2}^{(4)}-10A_{1}^{(1)}}}e^{2}+{\frac {5}{2}}\left({\rm {A_{1}^{(7)}-A_{1}^{(6)}}}e^{2}\right)\right]\\\\&\ \ -{\frac {3m^{2}}{4a_{\text{ı}}}}\left[4(1-m)^{2}-1\right]{\rm {A}}_{2}^{(0)}\left({\frac {7}{2-3m}}-{\frac {1}{2-m}}\right)\\\\&\ \ -{\frac {3m^{2}}{4a_{\text{ı}}(1-m)}}\left\{\left[(2-m)^{2}-1\right]{\rm {A}}_{2}^{(3)}+\left[(2-3m)^{2}-1\right]{\rm {A}}_{2}^{(4)}\right\}\\\\&\ \ -{\frac {3m^{2}}{4a_{\text{ı}}}}\left({\frac {11}{2}}{\rm {C_{2}^{(6)}+C_{2}^{(9)}-C_{2}^{(10)}}}\right)\end{aligned}}\right\}$

${\begin{aligned}&-{\frac {6m^{2}{\rm {A_{1}^{(8)}}}}{a_{\text{ı}}(2-3m-c)}}ee'\cos(2v-2mv-cv-c'mv+\varpi +\varpi ')\\\\&-{\frac {6m^{2}{\rm {A_{1}^{(9)}}}}{a_{\text{ı}}(2-m-c)}}ee'\cos(2v-2mv-cv+c'mv+\varpi -\varpi ')\\\\&-{\frac {6m^{2}}{a_{\text{ı}}(c-m)}}\left({\rm {A}}_{1}^{(6)}+{\frac {7}{2}}{\rm {A}}_{1}^{(1)}\right)ee'\cos(cv-c'mv-\varpi +\varpi ')\\\\&-{\frac {6m^{2}}{a_{\text{ı}}(c+m)}}\left({\rm {A}}_{1}^{(7)}-{\frac {1}{2}}{\rm {A}}_{1}^{(1)}\right)ee'\cos(cv+c'mv-\varpi -\varpi ')\\\\&+{\frac {6m^{2}{\rm {A_{2}^{(10)}}}}{a_{\text{ı}}(2c-2+2m)}}e^{2}\cos(2cv-2v+2mv-2\varpi )\\\\&+{\frac {6m^{2}{\rm {A_{2}^{(12)}}}}{a_{\text{ı}}(2g-2+2m)}}\gamma ^{2}\cos(2gv-2v+2mv-2\theta )\end{aligned}}$