Page:DeSitterGravitation.djvu/21

The second lines must be halved for law II. We have put—

${\begin{aligned}&\lambda '={\frac {k{\sqrt {\mathrm {M} }}}{c}},\quad \mathrm {M} ={\text{mass of Earth + Moon,}}\\&\nu =n'/c,\quad n'={\text{mean motion of Sun.}}\\&l={\text{the Moon's mean anomaly.}}\end{aligned}}$

I find then for the two laws—

I. || II.
$S={\tfrac {1}{2}}{\frac {m}{\mathrm {M} }}{\frac {\lambda '\nu }{\rho {\sqrt {a}}}}(1+e\ \cos \ l),$	${\frac {1}{4}}{\frac {m}{\mathrm {M} }}{\frac {a\nu ^{2}}{\rho }}\left(1-e\ \cos \ l\right),$
$T=-{\tfrac {1}{2}}{\frac {m}{\mathrm {M} }}{\frac {\lambda '\nu }{\rho {\sqrt {a}}}}e\ \sin \ l,$	$0,$
$W=-{\tfrac {1}{2}}{\frac {m}{\mathrm {M} }}{\frac {\nu }{\rho }}\left({\frac {\lambda '}{\sqrt {a}}}-2a\nu \right)i\ \sin \ (l-\Omega )$	${\tfrac {1}{4}}{\frac {m}{\mathrm {M} }}{\frac {a\nu ^{2}}{\rho }}i\ \sin \ (l-\Omega ).$

There are no secular perturbations in $a$, $e$, and $i$. In the other elements we find—

I. || II.
$\delta \varpi =-{\frac {m}{\mathrm {M} }}{\frac {a^{3/2}\lambda '\nu }{\rho }}.nt,$	$-{\tfrac {1}{4}}{\frac {m}{\mathrm {M} }}{\frac {a^{3}\nu ^{2}}{\rho }}.nt,$
$\delta \epsilon _{1}=-{\frac {m}{\mathrm {M} }}{\frac {a^{3/2}\lambda '\nu }{\rho }}.nt,$	$-{\tfrac {1}{2}}{\frac {m}{\mathrm {M} }}{\frac {a^{3}\nu ^{2}}{\rho }}.nt,$
$\delta \Omega =-{\tfrac {1}{4}}{\frac {m}{\mathrm {M} }}{\frac {a^{2}\nu }{\rho }}\left({\frac {\lambda '}{\sqrt {a}}}-2a\nu \right).nt,$	$+{\tfrac {1}{8}}{\frac {m}{\mathrm {M} }}{\frac {a^{3}\nu ^{2}}{\rho }}.nt,$

The numerical values for one century are—

I. || II.
$\delta \varpi =-1''.272$	$-0''.024$
$\delta \Omega =-0.270$	$+0.011$

These quantities are too small to be detected from observation. Even the value of $\delta \varpi$ for law I. is well within the limits of uncertainty of the observed value.

13. We are thus left with the motion of the perihelion of Mercury as the only effect which reaches an appreciable amount. Unfortunately this same motion presents the well-known excess of observation on theory, which has been explained by Seeliger by the attraction of the masses forming the zodiacal light.^[1] Until

1. "Das Zodiakallicht und die empirischen Glieder in der Bewegung der innern Planeten," Münchener Sitzungsberichte, 1906, p. 595.