Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 3.djvu/281

l’équation différentielle en $u$ devient

${\begin{aligned}0={\frac {d^{2}u}{dv^{2}}}+u&+{\frac {3m^{2}}{a_{\text{ı}}}}\left[{\begin{aligned}&{\frac {7(2+11m+8m^{2})}{16}}-{\frac {7\left(10+19m+8m^{2}\right)}{8(2c-2+3m)}}\\\\&-{\frac {5{\rm {A}}_{1}^{(8)}}{2c-2+3m}}-{\frac {1}{2}}{\rm {A'}}_{1}^{(2)}\end{aligned}}\right]\\&\qquad \qquad \qquad \qquad \qquad \times e^{2}e'cos\left({\begin{aligned}&2cv-2v+2mv\\+&c'mv-2vp-vp'\end{aligned}}\right)\\\\&+{\frac {3m^{2}}{a_{\text{ı}}}}\left[{\begin{aligned}&{\frac {10+19m+8m^{2}}{8(2c-2+m)}}-{\frac {2+11m+8m^{2})}{16}}\\\\&-{\frac {5{\rm {A}}_{1}^{(9)}}{2c-2+m}}-{\frac {1}{2}}{\rm {A'}}_{1}^{(3)}\end{aligned}}\right]\\&\qquad \qquad \qquad \qquad \qquad \times e^{2}e'cos\left({\begin{aligned}&2cv-2v+2mv\\-&c'mv-2vp+vp'\end{aligned}}\right)\,;\end{aligned}}$

on aura donc

${\begin{aligned}{\rm {A'}}_{1}^{(2)}=&{\frac {-3m^{2}}{1-{\frac {3}{2}}m^{2}-(2c-2+3m)^{2}}}\\&\quad \times \left[{\frac {7(2+11m+8m^{2})}{16}}-{\frac {7\left(10+19m+8m^{2}\right)+40{\rm {A}}_{1}^{(8)}}{8(2c-2+3m)}}\right],\\\\{\rm {A_{1}^{'(3)}=}}&{\frac {-3m^{2}}{1-{\frac {3}{2}}m^{2}-(2c-2+m)^{2}}}\\&\quad \times \left[{\frac {10+19m+8m^{2}-40{\rm {A}}_{1}^{(9)}}{8(2c-2+m)}}-{\frac {2+11m+8m^{2})}{16}}\right].\end{aligned}}$

Si l’on désigne la partie correspondante de $nt+we$ par

${\begin{aligned}&{\rm {C}}_{1}^{'(2)}e^{2}e'cos(2cv-2v+2mv+c'mv-2vp-vp')\\+&{\rm {C}}_{1}^{'(3)}e^{2}e'cos(2cv-2v+2mv-c'mv-2vp+vp'),\end{aligned}}$

on aura, par le n^o 15,

${\rm {C_{1}^{'(2)}={\frac {\left\{{\begin{aligned}&{\frac {21m^{2}\left(10+19m+8m^{2}\right)+120m^{2}{\rm {A}}_{1}^{(8)}}{16(2c-2+3m)}}-{\frac {21m^{2}(2+3m)}{4(2-3m-c)}}\\\\&-{\frac {63m^{2}}{16(2-3m)}}-2{\rm {A_{1}^{'(2)}+3A_{1}^{(7)}-3A_{2}^{(4)}+3A_{1}^{(8)}A_{1}^{(1)}}}\end{aligned}}\right\}}{2c-2+3m}},}}$

${\rm {C_{1}^{'(3)}={\frac {\left\{{\begin{aligned}&{\frac {-3m^{2}\left(10+19m+8m^{2}\right)+120m^{2}{\rm {A}}_{1}^{(9)}}{16(2c-2+m)}}+{\frac {3m^{2}(2+m)}{4(2-m-c)}}\\\\&+{\frac {9m^{2}}{16(2-m)}}-2{\rm {A_{1}^{'(3)}+3A_{1}^{(6)}-3A_{2}^{(3)}+3A_{1}^{(9)}A_{1}^{(1)}}}\end{aligned}}\right\}}{2c-2+m}}.}}$

En réduisant ces formules en nombres, on trouve

${\begin{aligned}{\rm {A_{1}^{'(2)}=}}&\quad \,0{,}744932,\\{\rm {A_{1}^{'(3)}=}}&-0{,}0153320,\\{\rm {C_{1}^{'(2)}=}}&\quad \,0{,}563137,\\{\rm {C_{1}^{'(3)}=}}&-0{,}0235572,\end{aligned}}$