We keep the axes $(\xi, \eta)$ permanently fixed; the angle~$\theta$ which gives the direction of the sun (the old axis of~$x$) will change uniformly, and in the long run take all values with equal frequency independently of the moon's position in its orbit. We can only hope to observe the secular effects of the small forces $\Xi$,~$\Eta$, accumulated through a long period of time. Accordingly, averaging the trigonometrical functions, the secular terms are \[ \left. \begin{aligned} \Xi &= -3\frac{m}{a^{2}}\, v\, \frac{d\eta}{dt} = -2\omega\, \frac{d\eta}{dt}\Add{,} \\ \Eta &= \Neg 3\frac{m}{a^{2}}\, v\, \frac{d\xi}{dt} = \Neg2\omega\, \frac{d\xi}{dt}\Add{,} \end{aligned} \right\} \Tag{(44.5)} \] where \[ \omega = \tfrac{3}{2}mv/a^{2}. \Tag{(44.6)} \] If $(F_{\xi}, F_{\eta})$ is the Newtonian force, the equations of motion including these secular perturbing forces will be \[ \frac{d^{2}\xi}{dt^{2}} + 2\omega\, \frac{d\eta}{dt} = F_{\xi},\quad \frac{d^{2}\eta}{dt^{2}} - 2\omega\, \frac{d\xi}{dt} = F_{\eta}. \Tag{(44.7)} \] It is easily seen that $\omega$~is a very small quantity, so that $\omega^{2}$~is negligible. The equations~\Eq{(44.7)} are then recognised as the Newtonian equations referred to axes rotating with angular velocity~$-\omega$. Thus if we take the Newtonian orbit and give it an angular velocity~$+\omega$, the result will be the solution of~\Eq{(44.7)}. The leading correction to the lunar theory obtained from Einstein's equations is a precessional effect, indicating that the classical results refer to a frame of reference advancing with angular velocity~$\omega$ compared with the general inertial frame of the solar system. From this cause the moon's node and perigee will advance with velocity~$\omega$. If $\Omega$~is the earth's angular velocity \[ \frac{\omega}{\Omega} = \frac{3}{2}\, \frac{m}{a} = \tfrac{3}{2} · 10^{-8}. \] \PageSep{99} Hence the advance of perigee and node in a century is \[ 3\pi · 10^{-6} \text{ radians} = 1''.94. \] We may notice the very simple theoretical relation that Einstein's corrections cause an advance of the moon's perigee which is \emph{one half} the advance of the earth's perihelion. Neither the lunar theory nor the observations are as yet carried quite far enough to take account of this small effect; but it is only a little below the limit of detection. The result agrees with de~Sitter's value except in the second decimal place which is only approximate. There are well-known irregular fluctuations in the moon's longitude which attain rather large values; but it is generally considered that these are not of a type which can be explained by any amendment of gravitational theory and their origin must be looked for in other directions. At any rate Einstein's theory throws no light on them.