Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/744

de ces coordonnées étant dans le centre commun de gravité ; nous avons d’abord, par la propriété du centre de gravité, les trois équations

${\begin{alignedat}{2}\mathrm {X} +&mx+&m'x'+&m''x''+\ldots =0,\\\mathrm {Y} +&my+&m'y'+&m''y''+\ldots =0,\\\mathrm {Z} +&mz+&m'z'+&m''z''+\ldots =0.\end{alignedat}}$

Ensuite, si l’on fait, pour abréger,

${\begin{aligned}\Omega &=m\ \,\left[{\frac {1}{\sqrt {(x\ \,-\mathrm {X} )^{2}+(y\ \,-\mathrm {Y} )^{2}+(z\ \,-\mathrm {Z} )^{2}}}}-{\frac {1+m\ \ }{r}}\right]\\&+m'\ \left[{\frac {1}{\sqrt {(x'\,-\mathrm {X} )^{2}+(y'\,-\mathrm {Y} )^{2}+(z'\,-\mathrm {Z} )^{2}}}}-{\frac {1+m'\,}{r'}}\right]\\&+m''\left[{\frac {1}{\sqrt {(x''-\mathrm {X} )^{2}+(y''-\mathrm {Y} )^{2}+(z''-\mathrm {Z} )^{2}}}}-{\frac {1+m''}{r''}}\right]\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&+mm'\ \ \left[{\frac {1}{\sqrt {(x'\ -x\ )^{2}+(y'\ -y\ )^{2}+(z'\ -z\ )^{2}}}}-{\frac {1}{r}}-{\frac {1}{r'\ }}\right]\\&+mm''\ \left[{\frac {1}{\sqrt {(x''-x\,\ )^{2}+(y''-y\ )^{2}+(z''-z\ )^{2}}}}-{\frac {1}{r\ }}-{\frac {1}{r''}}\right]\\&+m'm''\left[{\frac {1}{\sqrt {(x''-x')^{2}+(y''-y')^{2}+(z''-z')^{2}}}}-{\frac {1}{r'}}-{\frac {1}{r''}}\right]\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\end{aligned}}$

on aura les équations suivantes, dans lesquelles je fais, pour agréger, $m+m'+m''+\ldots =\mathrm {M} ,$

${\frac {d^{2}\mathrm {X} }{dt^{2}}}={\frac {d\Omega }{d\mathrm {X} }},\quad {\frac {d^{2}\mathrm {Y} }{dt^{2}}}={\frac {d\Omega }{d\mathrm {Y} }},\quad {\frac {d^{2}\mathrm {Z} }{dt^{2}}}={\frac {d\Omega }{d\mathrm {Z} }},$

${\begin{alignedat}{3}{\frac {d^{2}x\ }{dt^{2}}}+{\frac {1+\mathrm {M} }{r^{3}}}x\ =&{\frac {d\Omega }{mdx}},&{\frac {d^{2}y\ }{dt^{2}}}+{\frac {1+\mathrm {M} }{r^{3}}}y\ =&{\frac {d\Omega }{mdy}},&{\frac {d^{2}z\ }{dt^{2}}}+{\frac {1+\mathrm {M} }{r^{3}}}z\ =&{\frac {d\Omega }{mdz}},\\{\frac {d^{2}x'}{dt^{2}}}+{\frac {1+\mathrm {M} }{r'^{3}}}x'=&{\frac {d\Omega }{m'dx'}},\ \ &{\frac {d^{2}y'}{dt^{2}}}+{\frac {1+\mathrm {M} }{r'^{3}}}y'=&{\frac {d\Omega }{m'dy'}},\ \ &{\frac {d^{2}z'}{dt^{2}}}+{\frac {1+\mathrm {M} }{r'^{3}}}z'=&{\frac {d\Omega }{m'dz'}},\\\end{alignedat}}$

$\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .$