Page:Joseph Louis de Lagrange - Œuvres, Tome 4.djvu/415

on aura, aux quantités du second ordre près,

${\begin{aligned}&{\frac {1}{\sqrt {(x-x')^{2}+(y-y')^{2}+(z-z')^{2}}}}\\&={\frac {1}{\sqrt {(x-\mathrm {X} ')^{2}+(y-\mathrm {Y} ')^{2}+(z-\mathrm {Z} ')^{2}}}}\\&+\mathrm {\frac {M''}{M'+M''+M'''+\ldots }} \\&\quad \qquad \times {\frac {(x-\mathrm {X} ')(x''-x')+(y-\mathrm {Y} ')(y''-y')+(z-\mathrm {Z} ')(z''-z')}{\left[(x-\mathrm {X} ')^{2}+(y-\mathrm {Y} ')^{2}+(z-\mathrm {Z} ')^{2}\right]^{\frac {3}{2}}}}\\&+\mathrm {\frac {M'''}{M'+M''+M'''+\ldots }} \\&\quad \qquad \times {\frac {(x-\mathrm {X} ')(x'''-x')+(y-\mathrm {Y} ')(y'''-y')+(z-\mathrm {Z} ')(z'''-z')}{\left[(x-\mathrm {X} ')^{2}+(y-\mathrm {Y} ')^{2}+(z-\mathrm {Z} ')^{2}\right]^{\frac {3}{2}}}}\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\\\end{aligned}}$

${\begin{aligned}&{\frac {1}{\sqrt {(x-x'')^{2}+(y-y'')^{2}+(z-z'')^{2}}}}\\&={\frac {1}{\sqrt {(x-\mathrm {X} ')^{2}+(y-\mathrm {Y} ')^{2}+(z-\mathrm {Z} ')^{2}}}}\\&+\mathrm {\frac {M'}{M'+M''+M'''+\ldots }} \\&\quad \qquad \times {\frac {(x-\mathrm {X} ')(x'-x'')+(y-\mathrm {Y} ')(y'-y'')+(z-\mathrm {Z} ')(z'-z'')}{\left[(x-\mathrm {X} ')^{2}+(y-\mathrm {Y} ')^{2}+(z-\mathrm {Z} ')^{2}\right]^{\frac {3}{2}}}}\\&+\mathrm {\frac {M'''}{M'+M''+M'''+\ldots }} \\&\quad \qquad \times {\frac {(x-\mathrm {X} ')(x'''-x'')+(y-\mathrm {Y} ')(y'''-y'')+(z-\mathrm {Z} ')(z'''-z'')}{\left[(x-\mathrm {X} ')^{2}+(y-\mathrm {Y} ')^{2}+(z-\mathrm {Z} ')^{2}\right]^{\frac {3}{2}}}}\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\\\end{aligned}}$

${\begin{aligned}&{\frac {1}{\sqrt {(x-x''')^{2}+(y-y''')^{2}+(z-z''')^{2}}}}\\&={\frac {1}{\sqrt {(x-\mathrm {X} ')^{2}+(y-\mathrm {Y} ')^{2}+(z-\mathrm {Z} ')^{2}}}}\\&+\mathrm {\frac {M'}{M'+M''+M'''+\ldots }} \\&\quad \qquad \times {\frac {(x-\mathrm {X} ')(x'-x''')+(y-\mathrm {Y} ')(y'-y''')+(z-\mathrm {Z} ')(z'-z''')}{\left[(x-\mathrm {X} ')^{2}+(y-\mathrm {Y} ')^{2}+(z-\mathrm {Z} ')^{2}\right]^{\frac {3}{2}}}}\\&+\mathrm {\frac {M''}{M'+M''+M'''+\ldots }} \\&\quad \qquad \times {\frac {(x-\mathrm {X} ')(x''-x''')+(y-\mathrm {Y} ')(y''-y''')+(z-\mathrm {Z} ')(z''-z''')}{\left[(x-\mathrm {X} ')^{2}+(y-\mathrm {Y} ')^{2}+(z-\mathrm {Z} ')^{2}\right]^{\frac {3}{2}}}}\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\\\end{aligned}}$

et ainsi de suite.

Ces quantités étant multipliées respectivement par $\mathrm {M',M'',M'''} ,\ldots$ et ensuite ajoutées ensemble, on verra que tous les termes du premier ordre se détruiront mutuellement ; car il est visible qu’on a, en général, quelles que soient les quantités $\mathrm {M',M'',M'''} ,\ldots$ et $x',x'',x''',\ldots ,$

${\begin{aligned}0&=\mathrm {M'\ M''} (x''-x\ \ )+\mathrm {M'\ M'''} (x'''\,-x'\ \,)+\ldots \\&+\mathrm {M''\ M'} (x'\ -x''\,)+\mathrm {M''M'''} (x'''\,-x''\,)+\ldots \\&+\mathrm {M'''M'} (x'\ -x''')+\mathrm {M'''M''} \,(x''\ -x''')+\ldots \\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\end{aligned}}$