Page:DeSitterGravitation.djvu/19

The two laws I. and II. are the only ones that have been actually proposed, but we can, without violating the principle of relativity, multiply the forces by any power of $\mathrm {C}$, and consequently any (positive or negative or even fractional) multiple of the quantities (37) will be in agreement with that principle.

11. The final result by the method of the preceding article must, for the law I. and for $\mu =0$, of course be the same as that derived in article 9, and contained in the formulæ (30), (32), (33). We have, since $\varpi _{0}=0$,—

$x=a_{0}\left(\cos \ u_{0}-e_{0}\right)+\delta x,$ $y=a_{0}{\sqrt {1-e_{0}^{2}}}\sin \ u_{0}+\delta y,b$ $u_{0}-e_{0}\sin \ u_{0}=n_{0}t,\quad a_{0}^{3}n_{0}^{2}=k^{2}\mathrm {M} ,$ ${\begin{aligned}\delta x&={\frac {x}{a}}\delta a+{\tfrac {3}{2}}{\frac {an\ \sin \ u}{r}}\int \delta adt-{\frac {a^{2}}{p}}\sin \ u{\Big [}\delta \epsilon _{1}+e\ \cos \ v\left(\delta \epsilon -\delta \varpi \right){\Big ]}\\&\qquad \qquad \qquad \qquad \qquad \qquad +{\frac {ae^{2}}{\sqrt {1-e^{2}}}}\sin \ u\delta \varpi -\left[{\frac {a^{2}\sin ^{2}u}{r}}+a\right]\delta e,\\\delta y&={\frac {y}{a}}\delta a-{\tfrac {3}{2}}{\frac {an{\sqrt {1-e^{2}}}\cos \ u}{r}}\int \delta adt\\&\qquad \qquad \qquad \qquad \qquad +{\frac {a^{2}}{p}}{\sqrt {1-e^{2}}}\cos \ u{\Big [}\delta \epsilon _{1}+e\ \cos \ v\left(\delta \epsilon -\delta \varpi \right){\Big ]}\\&\qquad \qquad \qquad \qquad -ae\delta \varpi +\left[{\frac {a^{2}{\sqrt {1-e^{2}}}}{r}}\cos \ u-{\frac {ae}{\sqrt {1-e^{2}}}}\right]\sin \ u\delta e,\end{aligned}}$

(38)

where $\delta a,\delta e,e\delta \varpi ,\delta \epsilon _{1}$ must be collected from (35), taking the values for $S_{1}+S_{2}+T_{1}$, and

$e(\delta \epsilon -\delta \varpi )=e\delta \epsilon _{1}-{\sqrt {1-e^{2}}}.e\delta \varpi .$

On the other hand, from article 9, we have—

$\delta x=-a\ \sin \ u\Delta u+(\cos \ u-e)\Delta a-a\Delta e,$

$\delta y=a{\sqrt {1-e^{2}}}\cos \ u\Delta u+a{\sqrt {1-e^{2}}}\sin \ u\Delta a-{\frac {ae}{\sqrt {1-e^{2}}}}\sin \ u\Delta e,$

(39)

where

$r\Delta u=-\left({\tfrac {3}{2}}\Delta a+{\tfrac {1}{2}}\lambda ^{2}\right)(u-e\ \sin \ u)+\left(a\Delta e-\lambda ^{2}e\right)\sin \ u,$

and we have put

$a=a_{0}+\Delta a,\qquad e=e_{0}+\Delta e.$

By equating the values of $\delta x$ and $\delta y$ given by (38) and (39) we must then find for $\Delta a$ and $\Delta e$ constant values. To the first power of $e$ I find—

$\Delta a=\lambda ^{2},\qquad \Delta e={\tfrac {3}{2}}{\frac {\lambda ^{2}}{a}}.e.$