Page:DeSitterGravitation.djvu/10

(II.)$\quad (\mathrm {X} )_{2}={\frac {k^{2}m_{1}m_{2}}{A_{2}^{3}{\sqrt {\left(1-\phi _{1}^{2}\right)\left(1-\phi _{2}^{2}\right)}}}}\left\{\left({\boldsymbol {x}}-\delta x_{1}\right)\left(1-\xi _{1}\xi _{2}-\eta _{1}\eta _{2}-\zeta _{1}\zeta _{2}\right)+{\boldsymbol {r}}\epsilon _{2}\xi _{1}\right\}.$

(17)

This last form (II.) is preferred by Lorentz (l.c., p. 1239), because the corresponding Newtonian force, $\mathrm {X} _{i}=(\mathrm {X} )_{i}{\sqrt {1-\phi _{i}^{2}}}$, does not contain the velocity of $m_{i}$. (I.) is the law adopted by Minkowski, presumably because it gives the simplest result for a planet of infinitesimal mass.

As has already been remarked, the values of $\xi _{2},\ \eta _{2},\ \zeta _{2},$ in $(\mathrm {X} )_{1}$, and those of $\xi _{1},\ \eta _{1},\ \zeta _{1},$ in $(\mathrm {X} )_{2}$ differ from the values at the time $t$ for which the simultaneous coordinates are taken. If we are content to neglect third orders, however, we can assume all velocities to correspond to the time $t$. If the motion were quasi-stationary, i.e. if the accelerations could be neglected, the velocities would be constant, and also $\delta x_{2}$ and $\delta x_{1}$ would disappear. The equations (16) and (17) would in that case be rigorous.

7. We will first consider the law (I.). Introducing the developments of $\mathrm {B} _{1}$ and $\mathrm {A} _{2}$ we find—

${\begin{aligned}(\mathrm {X} )_{1}&={\frac {k^{2}m_{1}m_{2}}{{\boldsymbol {r}}^{3}}}\left\{-{\boldsymbol {x}}-\delta x_{2}-{\boldsymbol {r}}\epsilon _{1}\xi _{2}+{\tfrac {3}{2}}{\boldsymbol {x}}\epsilon _{2}^{2}+3{\boldsymbol {\frac {x}{r}}}\chi _{2}\right\},\\(\mathrm {X} )_{2}&={\frac {k^{2}m_{1}m_{2}}{{\boldsymbol {r}}^{3}}}\left\{{\boldsymbol {x}}-\delta x_{1}+{\boldsymbol {r}}\epsilon _{2}\xi _{1}-{\tfrac {3}{2}}{\boldsymbol {x}}\epsilon _{1}^{2}+3{\boldsymbol {\frac {x}{r}}}\chi _{1}\right\}\end{aligned}}$

(18)

We thus have by (13)—

${\begin{aligned}m_{1}{\frac {d^{2}\mathrm {x} _{1}}{dt^{2}}}&={\frac {k^{2}m_{1}m_{2}}{{\boldsymbol {r}}^{3}}}\left\{-{\boldsymbol {x}}-\delta x_{2}+3{\boldsymbol {\frac {x}{r}}}\chi _{2}+{\tfrac {3}{2}}{\boldsymbol {x}}\epsilon _{2}^{2}-{\boldsymbol {r}}\epsilon _{1}\xi _{2}+{\boldsymbol {r}}\epsilon _{1}\xi _{1}+{\boldsymbol {x}}\phi _{1}^{2}\right\}\\m_{2}{\frac {d^{2}\mathrm {x} _{2}}{dt^{2}}}&={\frac {k^{2}m_{1}m_{2}}{{\boldsymbol {r}}^{3}}}\left\{{\boldsymbol {x}}-\delta x_{1}+3{\boldsymbol {\frac {x}{r}}}\chi _{1}-{\tfrac {3}{2}}{\boldsymbol {x}}\epsilon _{1}^{2}+{\boldsymbol {r}}\epsilon _{2}\xi _{1}-{\boldsymbol {r}}\epsilon _{2}\xi _{2}-{\boldsymbol {x}}\phi _{2}^{2}\right\}\end{aligned}}$

(19)

Now we have, to second orders,—

${\begin{aligned}\delta x_{2}&={\tfrac {1}{2}}\Delta _{2}^{2}{\frac {d^{2}x_{2}}{c^{2}dt^{2}}}={\tfrac {1}{2}}{\frac {k^{2}m_{1}}{c^{2}}}\cdot {\boldsymbol {\frac {x}{r}}}\\\delta x_{1}&={\tfrac {1}{2}}\Delta _{1}^{2}{\frac {d^{2}x_{1}}{c^{2}dt^{2}}}=-{\tfrac {1}{2}}{\frac {k^{2}m_{2}}{c^{2}}}\cdot {\boldsymbol {\frac {x}{r}}}\\\chi _{2}={\boldsymbol {\frac {x}{r}}}\delta x_{2}&+{\boldsymbol {\frac {y}{r}}}\delta y_{2}+{\boldsymbol {\frac {z}{r}}}\delta z_{2}={\tfrac {1}{2}}{\frac {k^{2}m_{1}}{c^{2}}}\end{aligned}}$ $\chi _{1}=-{\tfrac {1}{2}}{\frac {k^{2}m_{2}}{c^{2}}}.$

Further, if we put

$\mathrm {M} =m_{1}+m_{2},\quad m_{1}=\mu \mathrm {M} ,\quad m_{2}=(1-\mu )\mathrm {M} ,$ $\lambda ^{2}={\frac {k^{2}\mathrm {M} }{c^{2}}},$