Page:DeSitterGravitation.djvu/17

With the true anomaly as independent variable, the equations are then—

$da={\frac {2a^{2}r^{2}}{p}}\left[\mathrm {S} e\ \sin \ v+\mathrm {T} {\frac {p}{r}}\right]dv,$ ${\begin{aligned}de&=r^{2}{\Big [}\mathrm {S} \ \sin \ v+\mathrm {T} \ (\cos \ u+\cos \ v){\Big ]}dv,\\ed\varpi &=r^{2}{\Big [}-\mathrm {S} \ \cos \ v+\mathrm {T} (1+{\frac {r}{p}}\sin \ v){\Big ]}dv,\end{aligned}}$ $d\epsilon _{1}=-{\frac {2r^{3}}{a{\sqrt {1-e^{2}}}}}\mathrm {S} dv,$ $\delta \epsilon =\delta \epsilon _{1}+{\frac {e^{2}}{1+{\sqrt {1-e^{2}}}}}\delta \varpi .$

We find the following "perturbations":—

in $\delta a$:	from	$\mathrm {S} _{1}:-\ 2\lambda ^{2}{\frac {a^{2}}{r^{2}}}e\ \cos \ u$	(35)
	"	$\mathrm {S} _{2}:{\tfrac {2}{3}}\lambda ^{2}\left(1-e^{2}\right){\frac {a^{3}}{r^{3}}}-2\lambda ^{2}{\frac {a^{2}}{r^{2}}}e\ \cos \ u,$
	"	$\mathrm {S} _{3}:{\frac {4\lambda ^{2}}{\left(1-e^{2}\right)^{2}}}\left(e\ \cos \ v+{\tfrac {1}{4}}e^{2}\ \cos \ 2v\right),$
	"	$\mathrm {T} _{1}:-\ {\tfrac {2}{3}}\lambda ^{2}\left(1-e^{2}\right){\frac {a^{3}}{r^{3}}}$;
in $\delta e$:	from	$\mathrm {S} _{1},\ \mathrm {S} _{2},\ \mathrm {S} _{3}:{\tfrac {1}{2}}{\frac {1-e^{2}}{ae}}\times$ the corresponding terms in $\delta a$
	"	$\mathrm {T} _{1}:{\tfrac {1}{2}}{\frac {1-e^{2}}{ae}}.\delta a+{\frac {\lambda ^{2}}{a}}\cos \ v;$
in $e\delta \varpi$:	from	$\mathrm {S} _{1}:-\ {\frac {\lambda ^{2}}{p}}\left[\left(1+e^{2}\right)\sin \ v+{\tfrac {1}{2}}e\ \sin \ sv+ev\right],$
	"	$\mathrm {S} _{2}:-\ {\frac {\lambda ^{2}}{p}}.{\tfrac {1}{3}}e^{2}\sin ^{3}v,$
	"	$\mathrm {S} _{3}:-\ {\frac {\lambda ^{2}}{p}}\left[2\ \sin \ v-{\tfrac {1}{2}}e\ \sin \ 2v-ev\right],$
	"	$\mathrm {T} _{1}:\quad {\frac {\lambda ^{2}}{p}}\left[-{\tfrac {1}{2}}e\ \sin \ 2v+{\tfrac {1}{3}}e^{2}\sin ^{3}v+ev\right];$
in $\delta \epsilon _{1}$:	from	$\mathrm {S} _{1}:\quad 2{\frac {\lambda ^{2}}{a}}u-4{\frac {\lambda ^{2}}{a{\sqrt {1-e^{2}}}}}v,$
	"	$\mathrm {S} _{2}:\quad 2{\frac {\lambda ^{2}}{a}}u-2{\frac {\lambda ^{2}}{a{\sqrt {1-e^{2}}}}}(v-e\ \sin \ v),$
	"	$\mathrm {S} _{3}:-\ 4{\frac {\lambda ^{2}}{a{\sqrt {1-e^{2}}}}}v.$