Page:DeSitterGravitation.djvu/24

where $p$ is the constant of general precession in longitude. We have thus—

$\Delta \Upsilon =-{\tfrac {1}{2}}q_{1}\cos ^{2}\mathrm {B} \ \sin \ 2\mathrm {L} +q_{1}\ \cos ^{2}\mathrm {B} \ \cos \ 2\mathrm {L} .pt.$

If we take for the axis of the transformation the direction of the Sun’s motion relatively to the fixed stars, and for $q$ the velocity of this motion, divided by the velocity of light, so that in the system ($x',y',z',t')$ the mean velocity of the stars is zero, we have—

$\mathrm {L} =270^{\circ },\quad \mathrm {B} =+55^{\circ },\quad q=0.000070.$

We find—

$\Delta \Upsilon =-0''.0000044{\text{ per century.}}$

(42)

This is, on the basis of the principle of relativity, the theoretical difference between the constant of procession as determined from the fixed stars (system $x',y',z',t'$) and from the motions in the solar system (system $x,y,z,t$).

15. If we take the orbital plane in the system ($x,y,z,t$) as the plane of ($x,y$), and choose the axis of $x$ perpendicular to the axis of the transformation, we have—

$\alpha =0,\qquad r_{q}=\beta y,$

and the formulæ for transformation to simultaneous relative coordinates in the system ($x',y',z',t'$) become—

$x'=x,\quad y'=y\left(1-\beta ^{2}q_{1}\right),\quad z'=-\beta \gamma q_{1}y,\quad ct'={\frac {ct-\beta qy}{\sqrt {1-q^{2}}}}.$

The problem is now, if $x,y$ are given as functions of $t$, to find the expression of $x',y',z'$ as functions of $t'$. Knowing that the orbit remains plane, we can eliminate $z'$ by combining with our Lorentz-transformation a rotation round the axis of $x'$ by an angle given by—

${\begin{aligned}\mathrm {K} \ \sin \ \mathrm {I} &=\ -\ \gamma \beta q_{1},\\\mathrm {K} \ \cos \ \mathrm {I} &=1-\beta ^{2}q_{1},\end{aligned}}$

from which

$\mathrm {K} ^{2}=1-2q_{1}+2q_{1}^{2}=1+2q_{1}{\sqrt {1-q^{2}.}}$

The new plane of $x'_{1},\ y'_{1}$ is thus the transformed orbital plane, and we have—

$x'_{1}=x'=x,\quad y'_{1}=y'\ \sec \ \mathrm {I} =\mathrm {K} y.$

If, to take a simple example, the orbit in ($x,y,z,t$) were a circle—

$x=a\ \cos \ nt,\qquad y=a\ \sin \ nt,$