Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 1.djvu/337

valeurs entières, positives ou négatives, de $i,$ la seule valeur $i=0$ étant exceptée, parce que nous avons fait sortir hors de ce signe les termes dans lesquels $i=0\,;\ m'g$ est une constante ajoutée à l’intégrale $\int d{\rm {R.}}$

En faisant donc

${\begin{aligned}{\rm {C}}&={\frac {1}{2}}a^{3}{\frac {\partial ^{2}{\rm {A}}^{(0)}}{\partial a^{2}}}+3a^{2}{\frac {\partial {\rm {A}}^{(0)}}{\partial a}}+6ag,\\\\{\rm {D}}&={\frac {1}{2}}a^{2}a'{\frac {\partial ^{2}{\rm {A}}^{(1)}}{\partial a\partial a'}}+a^{2}{\frac {\partial {\rm {A}}^{(1)}}{\partial a}}+aa'{\frac {\partial {\rm {A}}^{(1)}}{\partial a'}}+2a{\rm {A}}^{(1)},\\\\{\rm {C}}^{(i)}&={\frac {1}{2}}a^{3}{\frac {\partial ^{2}{\rm {A}}^{(i)}}{\partial a^{2}}}+{\frac {2i+1}{2}}a^{2}{\frac {\partial {\rm {A}}^{(i)}}{\partial a}}+{\frac {i(n-n')-3n}{2\left[i(n-n')-n\right]}}\\&\qquad \qquad \qquad \qquad \qquad \times \left(a^{2}{\frac {\partial {\rm {A}}^{(i)}}{\partial a}}+{\frac {2n}{n-n'}}a{\rm {A}}^{(i)}\right)\\\\&\qquad \qquad \qquad \qquad \qquad +{\frac {(i-1)n}{i(n-n')-n}}\left(a^{2}{\frac {\partial {\rm {A}}^{(i)}}{\partial a}}+2ia{\rm {A}}^{(i)}\right),\\\\{\rm {D}}^{(i)}&={\frac {1}{2}}a^{2}a'{\frac {\partial ^{2}{\rm {A}}^{(i-1)}}{\partial a\partial a'}}-(i-1)a^{2}{\frac {\partial {\rm {A}}^{(i-1)}}{\partial a}}\\&\qquad \qquad \qquad +{\frac {(i-1)n}{i(n-n')-n}}\left(aa'{\frac {\partial {\rm {A}}^{(i-1)}}{\partial a'}}-2(i-1)a{\rm {A}}^{(i-1)}\right)\,;\end{aligned}}$

en prenant ensuite pour unité la somme des masses ${\rm {M}}+m,$ et observant que, par le n^o 20, ${\frac {{\rm {M}}+m,}{a^{3}}}=n^{2},$ l’équation (X’) deviendra

$0={\frac {d^{2}\delta u}{dt^{2}}}+n^{2}\delta u-2n^{2}m'ag-{\frac {n^{2}m'}{2}}a^{2}{\frac {\partial {\rm {A}}^{(0)}}{\partial a}}$

${\begin{aligned}&-{\frac {n^{2}m'}{2}}\Sigma \left(a^{2}{\frac {\partial {\rm {A}}^{(i)}}{\partial a}}+{\frac {2n}{n-n'}}a{\rm {A}}^{(i)}\right)\cos i(n't-nt+\varepsilon '-\varepsilon )\\&+n^{2}m'{\rm {C}}e\cos(nt+\varepsilon -\varpi )+n^{2}m'{\rm {D}}e'\cos(nt+\varepsilon -\varpi ')\\&+n^{2}m'\Sigma {\rm {C}}^{(i)}e\cos \left[i(n't-nt+\varepsilon '-\varepsilon )+nt+\varepsilon -\varpi \right]\\&+n^{2}m'\Sigma {\rm {D}}^{(i)}e'\cos \left[i(n't-nt+\varepsilon '-\varepsilon )+nt+\varepsilon -\varpi \right],\end{aligned}}$

et, en intégrant,

$\delta u=2m'ag+{\frac {m'}{2}}a^{2}{\frac {\partial {\rm {A}}^{(0)}}{\partial a}}$

$-{\frac {m'}{2}}n^{2}\Sigma {\frac {a^{2}{\frac {\partial {\rm {A}}^{(i)}}{\partial a}}+{\frac {2n}{n-n'}}a{\rm {A}}^{(i)}}{i^{2}(n-n')^{2}-n^{2}}}\cos i(n't-nt+\varepsilon '-\varepsilon )$

${\begin{aligned}&+m'f_{\text{ı}}e\cos(nt+\varepsilon -\varpi )+m'f'_{\text{ı}}e'\cos(nt+\varepsilon -\varpi ')\\&-{\frac {m'}{2}}{\rm {C}}nte\sin(nt+\varepsilon -\varpi )-{\frac {m'}{2}}{\rm {D}}nte'\sin(nt+\varepsilon -\varpi ')\\&+m'\Sigma {\frac {{\rm {C}}^{i}n^{2}}{\left[i(n-n')-n\right]^{2}-n^{2}}}e\cos \left[i(n't-nt+\varepsilon '-\varepsilon )+nt+\varepsilon -\varpi \right]\\&+m'\Sigma {\frac {{\rm {D}}^{i}n^{2}}{\left[i(n-n')-n\right]^{2}-n^{2}}}e'\cos \left[i(n't-nt+\varepsilon '-\varepsilon )+nt+\varepsilon -\varpi '\right],\end{aligned}}$