Page:The Meaning of Relativity - Albert Einstein (1922).djvu/118

The calculation of the planetary motion depends upon equation (90). From the first of equations (108b) and (90) we get, for the indices 1, 2, 3,

${\frac {d}{ds}}\left(x_{\alpha }{\frac {dx_{\beta }}{ds}}-x_{\beta }{\frac {dx_{\alpha }}{ds}}\right)=0$

or, if we integrate, and express the result in polar co-ordinates,

$r^{2}{\frac {d\phi }{ds}}={\text{constant.}}$

(111)

From (90), for $\mu =4$, we get

$0={\frac {d^{2}l}{ds^{2}}}+{\frac {1}{f^{2}}}{\frac {df^{2}}{dx_{\alpha }}}{\frac {dx_{\alpha }}{ds}}={\frac {d^{2}l}{ds^{2}}}+{\frac {1}{f^{2}}}{\frac {df^{2}}{ds}}.$

From this, after multiplication by $f^{2}$ and integration, we have

$f^{2}{\frac {dl}{ds}}={\text{constant.}}$

(112)

In (109b), (111) and (112) we have three equations between the four variables $s$, $r$, $l$ and $\phi$, from which the motion of the planet may be calculated in the same way as in classical mechanics. The most important result we get from this is a secular rotation of the elliptic orbit of the planet in the same sense as the revolution of the planet, amounting in radians per revolution to

${\frac {24\pi ^{3}\alpha ^{2}}{(1-e^{2})c^{2}T^{2}}}$

(113)