Page:DeSitterGravitation.djvu/20

12. In the motion of the Moon we have first the same terms as in the planetary motion, but $\mathrm {M}$ in the expression for $\lambda ^{2}$ is now the sum of the masses of the Earth and Moon. The motion of the perigee in one century for the law II. is found from (36)—

$\delta \varpi =+0''.0101.$

But, in addition to this, the disturbing force of the Sun is changed. To find the effect of this, we must use the first equation of (20), first taking for $m_{1}$ the Moon and for $m_{2}$ the Sun, and then for $m_{1}$ the Earth and for $m_{2}$ the Sun. The difference of the two equations thus derived—without the first term, which gives the ordinary Newtonian perturbation—will furnish the new terms in the acceleration of the Moon with respect to the Earth.

We can neglect the masses of the Moon and the Earth, and the acceleration of the Sun. Then, if we take the origin of the system of reference in the Sun, we find for the additional terms—

${\frac {d^{2}x}{dt^{2}}}=k^{2}m\left\{{\frac {\Delta \epsilon _{1}\xi _{1}}{\Delta ^{3}}}+{\frac {x_{1}\phi _{1}^{2}}{\Delta ^{3}}}-{\frac {\rho \epsilon _{2}\xi _{2}}{\rho ^{3}}}-{\frac {x_{2}\phi _{2}^{2}}{\rho ^{3}}}\right\},$

where the suffix 1 refers to the Moon and 2 to the Earth, and where

$\Delta$	= distance	Sun — Moon,
$\rho$	= ⁠"	Sun — Earth,

and $m$ is the Sun's mass. For the law II. the terms involving $\phi _{1}^{2}$ and $\phi _{2}^{2}$ are halved.

A development in series cannot be avoided. We will retain only the terms of the first order in the parallax, and in the inclination and excentricities, and of these only those which can give secular perturbations, i.e. which do not contain the Sun's longitude. I find—

${\begin{aligned}{\frac {d^{2}x}{dt^{2}}}&=-{\frac {k^{2}m\nu }{\rho }}\left[{\tfrac {1}{2}}\left({\frac {\lambda '}{\sqrt {a}}}-a\nu \right)\cos \ l+{\tfrac {3}{4}}a\nu e+{\tfrac {1}{2}}\left({\frac {\lambda '}{\sqrt {a}}}-{\tfrac {1}{2}}a\nu \right)e\ \cos \ 2l\right]\\&+{\frac {k^{2}m\nu }{\rho }}\left[\left({\frac {\lambda '}{\sqrt {a}}}-{\tfrac {1}{2}}a\nu \right)\cos \ l+{\tfrac {3}{4}}a\nu e+\left({\frac {\lambda '}{\sqrt {a}}}-{\tfrac {1}{4}}a\nu \right)e\ \cos \ 2l\right],\\{\frac {d^{2}y}{dt^{2}}}&=-{\frac {k^{2}m\nu }{\rho }}\left[{\tfrac {1}{2}}\left({\frac {\lambda '}{\sqrt {a}}}-a\nu \right)\sin \ l+{\tfrac {1}{2}}\left({\frac {\lambda '}{\sqrt {a}}}-{\tfrac {1}{2}}a\nu \right)e\ \sin \ 2l\right]\\&+{\frac {k^{2}m\nu }{\rho }}\left[\left({\frac {\lambda '}{\sqrt {a}}}-{\tfrac {1}{2}}a\nu \right)\sin \ l+\left({\frac {\lambda '}{\sqrt {a}}}-{\tfrac {1}{4}}a\nu \right)e\ \sin \ 2l\right],\\{\frac {d^{2}z}{dt^{2}}}&={\tfrac {1}{2}}{\frac {k^{2}m\nu }{\rho }}\left({\frac {\lambda '}{\sqrt {a}}}-a\nu \right)\sin \ i\ \sin \left(l-\Omega \right)\\&-{\frac {k^{2}m\nu }{\rho }}\left({\frac {\lambda '}{\sqrt {a}}}-{\tfrac {3}{2}}a\nu \right)\sin \ i\ \sin \left(l-\Omega \right).\end{aligned}}$