Page:DeSitterGravitation.djvu/14

use planeto-centric time, the motion is Keplerian. Thus, e.g., the integrals of areas are^[1]

$x{\frac {dy}{d\tau }}-y{\frac {dx}{d\tau }}=\mathrm {G} ,\ {\text{etc}}.$

Or, introducing heliocentric time, to second orders—

$x{\frac {dy}{dt}}-y{\frac {dx}{dt}}=\mathrm {G} \left\{1-{\tfrac {1}{2}}{\frac {\lambda ^{2}}{p}}\left(1+e^{2}+2e\ \cos \ v\right)\right\}$[2].

(26)

In (22), on the other hand, it is not advantageous to introduce the proper-time of one of the bodies, since we should thereby lose the symmetry gained by the introduction of relative velocities and coordinates. In the second-order terms We can introduce the ordinary Keplerian motion. We find then, taking the orbital plane for plane of ($x,y$), for the two laws equally—

$x{\frac {dy}{dt}}-y{\frac {dx}{dt}}={\text{const. }}-(1-2\mu ')\mathrm {G} _{0}{\frac {\lambda ^{2}}{p}}e\ \cos \ v,$

(27)

where

$\mathrm {G} _{0}=a^{2}n{\sqrt {1-e^{2}}}=k{\sqrt {\mathrm {M} }}{\sqrt {p}},$

and v is the true anomaly.

Similarly we find for the vis-viva integral from (24)—

$\left({\frac {dx}{d\tau }}\right)^{2}+\left({\frac {dy}{d\tau }}\right)^{2}=k^{2}\mathrm {M} \left({\frac {2}{r}}-{\frac {1}{a}}\right),$

or in heliocentric time—

$\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}={\frac {k^{2}\mathrm {M} \left({\frac {2}{r}}-{\frac {1}{a}}\right)}{1+\lambda ^{2}\left({\frac {2}{r}}-{\frac {1}{a}}\right)}}.$

(28)

From (22), on the other hand, we find—

${\begin{aligned}\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}&=2{\frac {k^{2}\mathrm {M} }{r}}+{\text{const.}}-4k^{2}\mathrm {M} {\frac {\lambda ^{2}}{a}}\left({\frac {a}{r^{2}}}-{\frac {1}{r}}\right)+\\&+2k^{2}\mathrm {M} \mu '{\frac {\lambda ^{2}}{a}}\left[{\frac {4a}{r^{2}}}-{\frac {5}{r}}+{\frac {1}{2}}\mu ''\left\{{\frac {ap}{r^{3}}}-3{\frac {a}{r^{2}}}+{\frac {3}{r}}\right\}\right].\end{aligned}}$

(29)

For the law II. we must add to (28) and (29) the term—

$+k^{2}\mathrm {M} {\frac {\lambda ^{2}}{a}}\left({\frac {a}{r^{2}}}-{\frac {1}{r}}\right)+{\text{const.}}$

1. As we shall now use only relative coordinates, the distinction between the different types of letters has become unnecessary, and is dropped.
2. Mr. Wacker has a similar formula, l. c., page 34.