Page:DeSitterGravitation.djvu/12

The coefficients $\mathrm {P,Q,R,P',Q',R'}$ are periodic functions of $t$ devoid of constant terms.^[1] Consequently $c\xi _{0},\ c\eta _{0}$ are also periodic, and $c\zeta _{0}$ is constant. The point the components of whose velocity id are the non-periodic parts of $c\xi _{0},\ c\eta _{0},\ c\zeta _{0}$ therefore has no acceleration. If we perform a Lorentz-transformation to a new system having this point as its origin, then the mean values of $\xi _{0},\ \eta _{0}$ and $\zeta _{0}$ are zero. Thus, if we neglect the periodic terms (which are, moreover, of the second order, and would introduce into (20) only terms of the fourth order, and those multiplied by $\mu '$), we have—

$\xi _{1}=(1-\mu )\xi ,\quad \xi _{2}=-\mu \xi ,\quad {\text{etc.}}$

By subtracting the second equation (20) from the first we find then—

${\frac {d^{2}{\boldsymbol {x}}}{dt^{2}}}={\frac {k^{2}\mathrm {M} }{{\boldsymbol {r}}^{3}}}\left\{-{\boldsymbol {x}}+{{\boldsymbol {r}}\epsilon \xi }+{{\boldsymbol {x}}\phi ^{2}}\right\}+{\frac {k^{2}\mathrm {M} }{{\boldsymbol {r}}^{3}}}\mu '\left\{2\lambda ^{2}{\boldsymbol {\frac {x}{r}}}-2{{\boldsymbol {r}}\xi }\epsilon -3{\boldsymbol {x}}\phi ^{2}+{\tfrac {3}{2}}\mu ''{\boldsymbol {x}}\epsilon ^{2}\right\},$

(22)

where

$\mu '=\mu (1-\mu )={\frac {m_{1}m_{2}}{\mathrm {M} ^{2}}},\ \mu ''=1-2\mu ={\frac {m_{2}-m_{1}}{\mathrm {M} }}.$

If the mass of the planet is neglected we have $\mu '=0$, and we find—

${\frac {d^{2}{\boldsymbol {x}}}{dt^{2}}}+k^{2}\mathrm {M} {\frac {\boldsymbol {x}}{{\boldsymbol {r}}^{3}}}={\frac {k^{2}\mathrm {M} }{{\boldsymbol {r}}^{3}}}\left\{{{\boldsymbol {r}}\epsilon \xi }+{\boldsymbol {x}}\phi ^{2}\right\}.$

(23)

1. If the true anomaly $v$ is introduced as independent variable instead of $t$, I find—

   ${\begin{aligned}&\quad {\frac {dc\xi _{0}}{dv}}=\mathrm {P} _{1}+\mathrm {Q} _{1}.c\xi _{0}+\mathrm {R} _{1}.c\eta _{0},{\text{ etc.}}\\\mathrm {P} _{1}&={\frac {k{\sqrt {\mathrm {M} }}}{p^{3/2}}}\lambda ^{2}\mu '\mu ''\{e\ \cos \ 2v+e^{2}\ \cos \ v-{\tfrac {3}{2}}e^{2}\ \sin ^{2}\ v\ \cos \ v\},\\\mathrm {P} '_{1}&={\frac {k{\sqrt {\mathrm {M} }}}{p^{3/2}}}\lambda ^{2}\mu '\mu ''\{e\ \cos \ 2v+2e^{2}\ \cos \ v-{\tfrac {3}{2}}e^{2}\ \sin ^{3}\ v\},\\\mathrm {Q} _{1}&=-{\frac {\lambda ^{2}\mu '}{p}}\left\{2\ \sin \ 2v-3e\ \cos ^{2}\ v\ \sin \ v\right\},\\\mathrm {R} _{1}&=\mathrm {Q} '_{1}={\frac {\lambda ^{2}\mu '}{p}}\left\{2\ \cos \ 2v+2e\ \cos \ v-3e\ \sin ^{2}\ v\ \cos \ v\right\},\\\mathrm {R} '_{1}&={\frac {\lambda ^{2}\mu '}{p}}\left\{2\ \sin \ 2v+4e\ \sin \ v-3e\ \sin ^{3}\ v\right\}.\end{aligned}}$

   These are not developments in powers of $e$, but are rigorous for all values of $e$. It may be mentioned that, if the first term within the brackets in (21), which arises from the non-stationary character of the motion, were neglected, $\mathrm {P} _{1}$ would have a constant term, and we should find an acceleration of the centre of gravity.