Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 3.djvu/38

le n^o 22 du Livre II,

${\frac {r}{a}}=1+{\frac {1}{2}}e^{2}-e\cos(nt+\varepsilon -\varpi )-{\frac {1}{2}}e^{2}\cos(2nt+2\varepsilon -2\varpi ),$

$v=nt+\varepsilon -\Pi +2e\sin(nt+\varepsilon -\varpi )+{\frac {5}{4}}e^{2}\sin(2nt+2\varepsilon -2\varpi ),$

ce qui donne les valeurs de ${\frac {r'}{a'}}$ et de $v',$ en marquant d’un trait les quantités $n,\varepsilon$ et $e.$ On a ensuite, par le n^o 48 du même Livre, le produit de $\sum {\rm {A}}^{(i)}\cos[i(n't-nt+\varepsilon '-\varepsilon )]$ par le sinus ou le cosinus d’un angle quelconque $ft+{\rm {I}}$ égal à

${\rm {A}}^{(i)}{\begin{aligned}&\sin \\&\cos \end{aligned}}\left[i(n't-nt+\varepsilon '-\varepsilon )+ft+{\rm {I}}\right].$

De là il est facile de conclure

${\begin{aligned}&{\rm {M}}^{(0)}={\frac {m'}{8}}\left[i(4i-5){\rm {A}}^{(i)}+2(2i-1)a{\frac {\partial {\rm {A}}^{(i)}}{\partial a}}+a^{2}{\frac {\partial ^{2}{\rm {A}}^{(i)}}{\partial a^{2}}}\right],\\&{\rm {M}}^{(1)}=-{\frac {m'}{4}}\times \\&\left[4i(i-1)^{2}{\rm {A}}^{(i-1)}+2(i-1)a{\frac {\partial {\rm {A}}^{(i-1)}}{\partial a}}-2(i-1)a'{\frac {\partial {\rm {A}}^{(i-1)}}{\partial a'}}-aa'{\frac {\partial ^{2}{\rm {A}}^{(i-1)}}{\partial a\partial a'}}\right],\\&{\rm {M}}^{(2)}={\frac {m'}{8}}\left[(i-2)(4i-3){\rm {A}}^{(i-2)}-2(2i-3)a'{\frac {\partial {\rm {A}}^{(i-2)}}{\partial a'}}+a'^{2}{\frac {\partial ^{2}{\rm {A}}^{(i-2)}}{\partial a'^{2}}}\right],\\&{\rm {M}}^{(3)}=-{\frac {m'}{8}}aa'{\rm {B}}^{(i-1)},\end{aligned}}$

et dans le cas de $i=1,$

${\rm {M}}^{(3)}={\frac {m'}{4}}{\frac {a}{a'^{2}}}-{\frac {m'}{8}}aa'{\rm {B}}^{0}.$

Représentons ${\rm {N}}\cos \left[i(n't-nt+\varepsilon '-\varepsilon )+{\rm {L}}\right]$ par les termes suivants

${\begin{aligned}&{\rm {N}}^{(0)}\cos i(n't-nt+\varepsilon '-\varepsilon )\\+&{\rm {N}}^{(1)}ee'\cos \left[i(n't-nt+\varepsilon '-\varepsilon )+\varpi -\varpi '\right]\\&{\rm {N}}^{(2)}ee'\cos \left[i(n't-nt+\varepsilon '-\varepsilon )-\varpi +\varpi '\right]\,;\end{aligned}}$

nous aurons

${\begin{aligned}&{\rm {N}}^{(0)}=-{\frac {m'}{4}}\times \\&\left[\left(e^{2}+e'^{2}\right)\left(4i^{2}{\rm {A}}^{(i)}2a{\frac {\partial {\rm {A}}^{(i)}}{\partial a}}-a^{2}{\frac {\partial ^{2}{\rm {A}}^{(i)}}{\partial a^{2}}}\right)-{\frac {\gamma ^{2}}{2}}aa'\left({\rm {B}}^{(i-1)}+{\rm {B}}^{(i+1)}+\right)\right],\\&{\rm {N}}^{(1)}={\frac {m'}{4}}\times \\&\left[4i(i-1)^{2}{\rm {A}}^{(i-1)}-2(i-1)a{\frac {\partial {\rm {A}}^{(i-1)}}{\partial a}}-2(i-1)a'{\frac {\partial {\rm {A}}^{(i-1)}}{\partial a'}}+aa'{\frac {\partial ^{2}{\rm {A}}^{(i-1)}}{\partial a\partial a'}}\right],\\&{\rm {N}}^{(2)}={\frac {m'}{4}}\times \\&\left[4i(i+1)^{2}{\rm {A}}^{(i+1)}+2(i+1)a{\frac {\partial {\rm {A}}^{(i+1)}}{\partial a}}+2(i+1)a'{\frac {\partial {\rm {A}}^{(i+1)}}{\partial a'}}+aa'{\frac {\partial ^{2}{\rm {A}}^{(i+1)}}{\partial a\partial a'}}\right],\end{aligned}}$