Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/645

perpendiculaires entre elles, on aura celles-ci

${\begin{aligned}{\frac {\mathrm {S} x}{(\mathrm {TS} )^{3}}}+&{\frac {\mathrm {T} _{1}(x-x_{1})}{(\mathrm {TT_{1}} )^{3}}}+{\frac {\mathrm {T} _{2}(x-x_{2})}{(\mathrm {TT_{2}} )^{3}}}+\ldots ,\\{\frac {\mathrm {S} y}{(\mathrm {TS} )^{3}}}+&{\frac {\mathrm {T} _{1}(y-y_{1})}{(\mathrm {TT_{1}} )^{3}}}\,+{\frac {\mathrm {T} _{2}(y-y_{2})}{(\mathrm {TT_{2}} )^{3}}}\,+\ldots ,\\{\frac {\mathrm {S} z}{(\mathrm {TS} )^{3}}}+&{\frac {\mathrm {T} _{1}(z-z_{1})}{(\mathrm {TT_{1}} )^{3}}}\,+{\frac {\mathrm {T} _{2}(z-z_{2})}{(\mathrm {TT_{2}} )^{3}}}\,+\ldots .\end{aligned}}$

Mais le corps $\mathrm {S}$ est attiré de même par les corps $\mathrm {T,T_{1},T_{2}} ,\ldots ,$ avec les forces

$\mathrm {{\frac {T}{(TS)^{2}}},\quad {\frac {T_{1}}{(T_{1}S)^{2}}},\quad {\frac {T_{2}}{(T_{2}S)^{2}}}} ,\ldots ,$

lesquelles, étant décomposées suivant les mêmes directions, adonnent celles-ci

${\begin{aligned}-{\frac {\mathrm {T} x}{(\mathrm {TS} )^{3}}}-&{\frac {\mathrm {T} _{1}x_{1}}{(\mathrm {T_{1}S} )^{3}}}-{\frac {\mathrm {T} _{2}x_{2}}{(\mathrm {T_{2}S} )^{3}}}-\ldots ,\\-{\frac {\mathrm {T} y}{(\mathrm {TS} )^{3}}}-&{\frac {\mathrm {T} _{1}y_{1}}{(\mathrm {T_{1}S} )^{3}}}-{\frac {\mathrm {T} _{2}y_{2}}{(\mathrm {T_{2}S} )^{3}}}-\ldots ,\\-{\frac {\mathrm {T} z}{(\mathrm {TS} )^{3}}}-&{\frac {\mathrm {T} _{1}z_{1}}{(\mathrm {T_{1}S} )^{3}}}-{\frac {\mathrm {T} _{2}z_{2}}{(\mathrm {T_{2}S} )^{3}}}-\ldots .\end{aligned}}$

Retranchant donc ces forces des précédentes, on aura les véritables forces qui font décrire au corps $\mathrm {T}$ son orbite autour du corps $\mathrm {S} ,$ regardé comme immobile, c’est-à-dire, les valeurs des quantités $\mathrm {X,Y,Z} \,;$ on aura donc

${\begin{aligned}\mathrm {X} =\mathrm {\left[{\frac {S+T}{(TS)^{3}}}+{\frac {T_{1}}{(TT_{1})^{3}}}+{\frac {T_{2}}{(TT_{2})^{3}}}+\ldots \right]} x&+\mathrm {T} _{1}\mathrm {\left[{\frac {1}{(T_{1}S)^{3}}}-{\frac {1}{(TT_{1})^{3}}}\right]} x_{1}\\&+\mathrm {T} _{2}\mathrm {\left[{\frac {1}{(T_{2}S)^{3}}}-{\frac {1}{(TT_{2})^{3}}}\right]} x_{2}+\ldots \\\mathrm {Y} =\mathrm {\left[{\frac {S+T}{(TS)^{3}}}+{\frac {T_{1}}{(TT_{1})^{3}}}+{\frac {T_{2}}{(TT_{2})^{3}}}+\ldots \right]} y&+\mathrm {T} _{1}\mathrm {\left[{\frac {1}{(T_{1}S)^{3}}}-{\frac {1}{(TT_{1})^{3}}}\right]} y_{1}\\&+\mathrm {T} _{2}\mathrm {\left[{\frac {1}{(T_{2}S)^{3}}}-{\frac {1}{(TT_{2})^{3}}}\right]} y_{2}+\ldots \\\mathrm {Z} =\mathrm {\left[{\frac {S+T}{(TS)^{3}}}+{\frac {T_{1}}{(TT_{1})^{3}}}+{\frac {T_{2}}{(TT_{2})^{3}}}+\ldots \right]} z&+\mathrm {T} _{1}\mathrm {\left[{\frac {1}{(T_{1}S)^{3}}}-{\frac {1}{(TT_{1})^{3}}}\right]} z_{1}\\&+\mathrm {T} _{2}\mathrm {\left[{\frac {1}{(T_{2}S)^{3}}}-{\frac {1}{(TT_{2})^{3}}}\right]} z_{2}+\ldots ,\\\end{aligned}}$

et par conséquent

${\begin{aligned}\mathrm {Y} z-\mathrm {Z} y=&\mathrm {T} _{1}\mathrm {\left[{\frac {1}{(T_{1}S)^{3}}}-{\frac {1}{(TT_{1})^{3}}}\right]} (y_{1}z-yz_{1})\\+&\mathrm {T} _{2}\mathrm {\left[{\frac {1}{(T_{2}S)^{3}}}-{\frac {1}{(TT_{2})^{3}}}\right]} (y_{2}z-yz_{2})+\ldots ,\\\mathrm {X} z-\mathrm {Z} x=&\mathrm {T} _{1}\mathrm {\left[{\frac {1}{(T_{1}S)^{3}}}-{\frac {1}{(TT_{1})^{3}}}\right]} (x_{1}z-xz_{1})\\+&\mathrm {T} _{2}\mathrm {\left[{\frac {1}{(T_{2}S)^{3}}}-{\frac {1}{(TT_{2})^{3}}}\right]} (x_{2}z-xz_{2})+\ldots .\end{aligned}}$