Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 11.djvu/326

312

THÉORIE DES SATELLITES DE JUPITER.

Si l’on fait, pour abréger,

$\mathrm {R} =$

${\begin{aligned}m'{\frac {xx'+yy'+zz'}{r'^{3}}}&+m''{\frac {xx''+yy''+zz''}{r''^{3}}}+m'''{\frac {xx'''+yy'''+zz'''}{r'''^{3}}}\\&+\mathrm {S} {\frac {x\mathrm {X} +y\mathrm {Y} +z\mathrm {Z} }{\mathrm {D} ^{3}}}-\lambda -(1+m)\mathrm {V} \\&-m'\ \ \left(x{\frac {\partial \mathrm {V} '}{\partial x'}}\ +y{\frac {\partial \mathrm {V} '}{\partial y'}}\ \ +z{\frac {\partial \mathrm {V} '}{\partial z'}}\ \right)\\&-m''\ \left(x{\frac {\partial \mathrm {V} ''}{\partial x''}}\,+y{\frac {\partial \mathrm {V} ''}{\partial y''}}\ +z{\frac {\partial \mathrm {V} ''}{\partial z''}}\,\right)\\&-m'''\left(x{\frac {\partial \mathrm {V} '''}{\partial x'''}}+y{\frac {\partial \mathrm {V} '''}{\partial y'''}}+z{\frac {\partial \mathrm {V} '''}{\partial z'''}}\right)\\&-\quad \mathrm {S} \left(x\mathrm {\frac {\partial U}{\partial X}} \ \ \ +y\mathrm {\frac {\partial U}{\partial Y}} \ \ \,+z\mathrm {\frac {\partial U}{\partial Z}} \ \ \,\right),\end{aligned}}$

les trois équations différentielles précédentes deviendront

(A)

$\left\{{\begin{aligned}0=&{\frac {d^{2}x}{dt^{2}}}+(1+m){\frac {x}{r^{3}}}+{\frac {\partial \mathrm {R} }{\partial x}},\\0=&{\frac {d^{2}y}{dt^{2}}}+(1+m){\frac {y}{r^{3}}}+{\frac {\partial \mathrm {R} }{\partial y}},\\0=&{\frac {d^{2}z}{dt^{2}}}+(1+m){\frac {z}{r^{3}}}+{\frac {\partial \mathrm {R} }{\partial z}}.\end{aligned}}\right.$

II.

Si l’on multiplie la première des équations (A) par $dx,$ la seconde par $dy,$ la troisième par $dz$ et qu’en suite on les ajoute, on aura

${\begin{aligned}0={\frac {dxd^{2}x+dyd^{2}y+dzd^{2}z}{dt^{2}}}&+{\frac {1+m}{r^{3}}}\left(xdx+ydy+zdz\right)\\&+dx{\frac {\partial \mathrm {R} }{\partial x}}+dy{\frac {\partial \mathrm {R} }{\partial y}}+dz{\frac {\partial \mathrm {R} }{\partial z}}.\end{aligned}}$

Or on a

$xdx+ydy+zdz=rdr\,;$

de plus, on peut mettre la quantité

$dx{\frac {\partial \mathrm {R} }{\partial x}}+dy{\frac {\partial \mathrm {R} }{\partial y}}+dz{\frac {\partial \mathrm {R} }{\partial z}}$