It must be pointed out that these perturbing forces are Einstein's corrections to the law of central force~$m/r^{2}$, where $r$~is \emph{the coordinate used in our previous work}. Whether these forces represent the actual differences between Einstein's and Newton's laws depends on what Newton's~$r$ is supposed to signify. De~Sitter, making a slightly different choice of~$r$, obtains different expressions for $R$,~$\Phi$\footnotemark.\footnotetext {\Title{Monthly Notices}, vol.~76, p.~723, equations~(53).} One cannot say that one set of perturbing forces rather than the other represents the difference from the older theory, because the older theory was not sufficiently explicit. The classical lunar theory has been worked out on the basis of the law~$m/r^{2}$; the ambiguous quantity~$r$ occurs in the results, and according as we have assigned to it one meaning or another, so we shall have to apply different corrections to those results. But the final comparison with observation does not depend on the choice of the intermediary quantity~$r$. Take fixed rectangular axes referred to the ecliptic with the sun as origin, and let \\ \Indent $(a, 0, 0)$ be the coordinates of the earth at the instant considered, \\ \Indent $(x, y, z)$ the coordinates of the moon relative to the earth. Taking the earth's orbit to be circular and treating the mass of the moon as infinitesimal, the earth's velocity will be $(0, v, 0)$, where $v^{2} = m/a$. To find the difference of the forces $R$, $\Phi$, $Z$ on the moon and on the earth, we differentiate~\Eq{(44.23)} and set \[ \delta r = x,\quad \delta(u, v, w) = (dx/dt, dy/dt, dz/dt), \] and, after the differentiation, \[ r = a,\quad (u, v, w) = (0, v, 0). \] \PageSep{98} The result will give the perturbing forces on the moon's motion relative to the earth, viz.\ \[ \left. \begin{aligned} \delta R &= \begin{aligned}[t] X &= \frac{4mx}{a^{3}}\, v^{2} - \frac{6m^{2}x}{a^{4}} - \frac{4m}{a^{2}}\, v\, \frac{dy}{dt} \\ &= -\frac{2m^{2}x}{a^{4}} - \frac{4m}{a^{2}}\, v\, \frac{dy}{dt}\Add{,} \end{aligned} \\ \delta\Phi &= Y = \frac{2m}{a^{2}}\, v\, \frac{dx}{dt}\Add{,} \\ Z &= 0\Add{.} \end{aligned} \right\} \Tag{(44.3)} \] We shall omit the term $-2m^{2}x/a^{4}$ in~$X$. It can be verified that it gives no important observable effects. It produces only an apparent distortion of the orbit attributable to our use of non-isotropic coordinates (\SecRef{43}). Transforming to new axes $(\xi, \eta)$ rotated through an angle~$\theta$ with respect to $(x, y)$ the remaining forces become \[ \left. \begin{aligned} \Xi &= \frac{m}{a^{2}} v \left(-2\cos\theta \sin\theta\, \frac{d\xi}{dt} - (4\cos^{2}\theta + 2\sin^{2}\theta)\, \frac{d\eta}{dt}\right)\Add{,} \\ \Eta &= \frac{m}{a^{2}} v \left(\Neg2\cos\theta \sin\theta\, \frac{d\eta}{dt} + (4\sin^{2}\theta + 2\cos^{2}\theta)\, \frac{d\xi}{dt}\right)\Add{.} \end{aligned} \right\} \Tag{(44.4)} \]