Page:DeSitterGravitation.djvu/16

geocentric or in heliocentric time. This, of course, is a purely formal change of unit, and has nothing to do with the number of days in the year.

The difference between heliocentric and geocentric time, apart from this change of unit, is purely periodic. We find

$t=\tau \left(1+{\tfrac {1}{2}}{\frac {\lambda ^{2}}{a}}\right)+{\frac {\lambda ^{2}}{an}}e\ \sin \ u,$

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which is the formula given by Minkowski, l.c., page 111. The value of the coefficient of $\sin u$ is, for the Earth, $0^{\mathrm {s} }.00083$.

10. For the integration of the equations (22), or the corresponding equations for the law II., the easiest way seems to be to use the Keplerian motion as a first approximation, and to compute "perturbations" by the method of variation of elements, introducing the radial, transversal, and orthogonal perturbing forces. Taking the orbital plane for plane of ($x,y$), these forces are found to be

for law I.:	$\mathrm {S} =\mathrm {S} _{1}+\mathrm {S} _{2}+\mu '\{-3\mathrm {S} _{1}-(2-{\tfrac {3}{2}}\mu '')\mathrm {S} _{2}+\mathrm {S} _{3}\},$
	$\mathrm {T} =(1-2\mu ')\mathrm {T} _{1},$
	$\mathrm {W} =0,$
for law II.:	$\mathrm {S} ={\tfrac {1}{2}}\mathrm {S} _{1}+\mathrm {S} _{2}+\mu '\{-3\mathrm {S} _{1}-(2-{\tfrac {3}{2}}\mu '')\mathrm {S} _{2}+\mathrm {S} _{3}\},$
	$\mathrm {T} =(1-2\mu ')\mathrm {T} _{1},$
	$\mathrm {W} =0,$

where

$\mathrm {S} _{1}=k^{2}\mathrm {M} {\frac {\phi ^{2}}{r^{2}}},\quad \mathrm {S} _{2}=k^{2}\mathrm {M} {\frac {\epsilon ^{2}}{r^{2}}},\quad \mathrm {S} _{3}=2{\frac {\lambda ^{2}}{r^{3}}},$ $\mathrm {T} _{1}=k^{2}\mathrm {M} {\frac {x\eta -y\xi }{r^{3}}}\epsilon .$

In the perturbing forces we can use the undisturbed or Keplerian motion. We find, then, taking the axis of $x$ towards the (unperturbed) perihelion,—

${\begin{aligned}\mathrm {S} _{1}&={\frac {\lambda ^{2}}{r^{3}}}(1+e\ \cos \ u)={\frac {\lambda ^{2}}{r^{2}}}\cdot {\frac {1+e^{2}+2e\ \cos \ v}{p}},\\\mathrm {S} _{2}&={\frac {\lambda ^{2}}{r^{4}}}ae^{2}\sin ^{2}u={\frac {\lambda ^{2}}{r^{2}}}\cdot {\frac {e^{2}\sin ^{2}v}{p}},\\\mathrm {S} _{3}&={\frac {\lambda ^{2}}{r^{4}}}ae{\sqrt {1-e^{2}}}\sin \ u={\frac {\lambda ^{2}}{r^{3}}}e\ \sin v.\end{aligned}}$

In the well-known equations for the variation of elements, we introduce as independent variable, instead of the time, either the excentric or the true anomaly, or the radius-vector, by—

$dt={\frac {r}{an}}du={\frac {r^{2}}{a^{2}n{\sqrt {1-e^{2}}}}}dv={\frac {\operatorname {cosec} v}{ane{\sqrt {1-e^{2}}}}}dr.$

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