Page:Joseph Louis de Lagrange - Œuvres, Tome 3.djvu/170

cette équation sera renfermée dans l’équation différentielle (E), ce qui donnera, comme nous l’avons vu (6), les deux équations finies

${\frac {dq}{dx}}-{\frac {p}{u}}{\frac {dq}{du}}=0,\quad {\frac {dq}{da}}-{\frac {r}{u}}{\frac {dq}{du}}=0.$

Substituons au lieu de $r$ sa valeur $p\mathrm {X} +u^{2}{\frac {d\mathrm {X} }{dx}}$ (4), et comme $\mathrm {X}$ est une fonction de $x$ et de $a$ sans $u,$ et que $p$ en est une de $x$ et de $u$ sans $a,$ on aura d’abord

${\begin{aligned}{\frac {dr}{dx}}=&X{\frac {dp}{dx}}+{\frac {dX}{dx}}p+{\frac {d^{2}X}{dx^{2}}}u^{2},\\{\frac {dr}{du}}=&\mathrm {X} {\frac {dp}{du}}+2{\frac {d\mathrm {X} }{dx}}u,\end{aligned}}$

d’où l’on trouvera

$q=\mathrm {X} {\frac {dp}{dx}}-{\frac {d\mathrm {X} }{dx}}\left(p-u{\frac {dp}{du}}\right)+{\frac {d^{2}\mathrm {X} }{dx^{2}}}u^{2},$

ensuite

${\begin{aligned}{\frac {dq}{dx}}=&\mathrm {X} {\frac {d^{2}p}{dx^{2}}}+{\frac {d\mathrm {X} }{dx}}u{\frac {d^{2}p}{dxdu}}-{\frac {d^{2}\mathrm {X} }{dx^{2}}}\left(p-u{\frac {dp}{du}}\right)+{\frac {d^{3}\mathrm {X} }{dx^{3}}}u^{2},\\{\frac {dq}{du}}=&\mathrm {X} {\frac {d^{2}p}{dxdu}}+{\frac {d\mathrm {X} }{dx}}u{\frac {d^{2}p}{du^{2}}}+2{\frac {d^{2}\mathrm {X} }{dx^{2}}}u,\\{\frac {dq}{da}}=&{\frac {d\mathrm {X} }{da}}{\frac {dp}{dx}}-{\frac {d^{2}\mathrm {X} }{dadx}}\left(p-u{\frac {dp}{du}}\right)+{\frac {d^{3}\mathrm {X} }{dx^{2}da}}u^{2}.\end{aligned}}$

Ainsi l’on aura les trois équations

(F)

$\mathrm {X} {\frac {dp}{dx}}+{\frac {d\mathrm {X} }{dx}}\left(u{\frac {dp}{du}}-p\right)+{\frac {d^{2}\mathrm {X} }{dx^{2}}}u^{2}=0,$

(G)

$\left\{{\begin{aligned}\mathrm {X} {\frac {d^{2}p}{dx^{2}}}+{\frac {d\mathrm {X} }{dx}}u&{\frac {d^{2}p}{dxdu}}-{\frac {d^{2}\mathrm {X} }{dx^{2}}}\left(p-u{\frac {dp}{du}}\right)+{\frac {d^{3}\mathrm {X} }{dx^{3}}}u^{2},\\&-p\left({\frac {\mathrm {X} }{u}}{\frac {d^{2}p}{dxdu}}+{\frac {d\mathrm {X} }{dx}}{\frac {d^{2}p}{du^{2}}}+2{\frac {d^{2}\mathrm {X} }{dx^{2}}}\right)=0,\end{aligned}}\right.$

(H)

$\left\{{\begin{aligned}{\frac {d\mathrm {X} }{da}}&{\frac {dp}{dx}}+{\frac {d^{2}\mathrm {X} }{dxda}}\left(u{\frac {dp}{du}}-p\right)+{\frac {d^{3}\mathrm {X} }{dx^{2}da}}u^{2}\\&-\left(\mathrm {X} p+{\frac {\mathrm {X} }{x}}u^{2}\right)\left({\frac {\mathrm {X} }{u}}{\frac {d^{2}p}{dxdu}}+{\frac {d\mathrm {X} }{dx}}{\frac {d^{2}p}{du^{2}}}+2{\frac {d^{2}\mathrm {X} }{dx^{2}}}\right)=0.\end{aligned}}\right.$