Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/417

blie dans le n^o 1, on aura ces nouvelles équations

${\begin{aligned}{\frac {d^{2}\alpha }{dt^{2}}}=&(1+m)\left({\frac {d^{2}{\frac {1}{r}}}{dx^{2}}}\alpha +{\frac {d^{2}{\frac {1}{r}}}{dxdy}}\beta +{\frac {d^{2}{\frac {1}{r}}}{dxdz}}\gamma \right)+\mu {\frac {d{\frac {1}{r}}}{dx}},\\{\frac {d^{2}\beta }{dt^{2}}}=&(1+m)\left({\frac {d^{2}{\frac {1}{r}}}{dydx}}\alpha +{\frac {d^{2}{\frac {1}{r}}}{dy^{2}}}\beta +{\frac {d^{2}{\frac {1}{r}}}{dydz}}\gamma \right)+\mu {\frac {d{\frac {1}{r}}}{dy}},\\{\frac {d^{2}\gamma }{dt^{2}}}=&(1+m)\left({\frac {d^{2}{\frac {1}{r}}}{dzdx}}\alpha +{\frac {d^{2}{\frac {1}{r}}}{dzdy}}\beta +{\frac {d^{2}{\frac {1}{r}}}{dz^{2}}}\gamma \right)+\mu {\frac {d{\frac {1}{r}}}{dz}}.\end{aligned}}$

J’observe qu’on peut satisfaire à ces équations en faisant $\alpha =\mathrm {K} x,$ $\beta =\mathrm {K} y,$ $\gamma =\mathrm {K} z,$ $\mathrm {K}$ étant un coefficient constant ; car elles deviennent par là

${\begin{aligned}\mathrm {K} {\frac {d^{2}x}{dt^{2}}}=&(1+m)\left({\frac {d^{2}{\frac {1}{r}}}{dx^{2}}}x+{\frac {d^{2}{\frac {1}{r}}}{dxdy}}y+{\frac {d^{2}{\frac {1}{r}}}{dxdz}}z\right)\mathrm {K} +\mu {\frac {d{\frac {1}{r}}}{dx}},\\\mathrm {K} {\frac {d^{2}y}{dt^{2}}}=&(1+m)\left({\frac {d^{2}{\frac {1}{r}}}{dydx}}x+{\frac {d^{2}{\frac {1}{r}}}{dy^{2}}}y+{\frac {d^{2}{\frac {1}{r}}}{dydz}}z\right)\mathrm {K} +\mu {\frac {d{\frac {1}{r}}}{dy}},\\\mathrm {K} {\frac {d^{2}z}{dt^{2}}}=&(1+m)\left({\frac {d^{2}{\frac {1}{r}}}{dzdx}}x+{\frac {d^{2}{\frac {1}{r}}}{dzdy}}y+{\frac {d^{2}{\frac {1}{r}}}{dz^{2}}}z\right)\mathrm {K} +\mu {\frac {d{\frac {1}{r}}}{dz}}.\end{aligned}}$

Mais les équations de l’orbite non altérée (n^o 6) donnent

${\frac {d^{2}x}{dt^{2}}}=(1+m){\frac {d{\frac {1}{r}}}{dx}},\quad {\frac {d^{2}y}{dt^{2}}}=(1+m){\frac {d{\frac {1}{r}}}{dy}},\quad {\frac {d^{2}z}{dt^{2}}}=(1+m){\frac {d{\frac {1}{r}}}{dz}}.$

Ensuite je remarque que la quantité ${\frac {1}{r}}$ est une fonction homogène de $x,y,z$ de la dimension $-1,$ qu’ainsi les quantités ${\frac {d{\frac {1}{r}}}{dx}},{\frac {d{\frac {1}{r}}}{dy}},{\frac {d{\frac {1}{r}}}{dz}}$ sont aussi