If $\Omega_{S}$, $\Omega_{E}$ are the Newtonian potentials of the sun and earth, the leading terms of (1), (2), (3) will be relatively of order of magnitude \[ \Omega_{S}^{2},\quad \Omega_{S}\Omega_{E},\quad \Omega_{E}. \] For the moon $\Omega_{S} = 750\, \Omega_{E}$. We may therefore confine attention to terms of type~(1). If these prove to be too small to be detected, the others will presumably be not worth pursuing. We were able to work out the planetary orbits from Einstein's law independently of the Newtonian theory; but in the problem of the moon's motion we must concentrate attention on the difference between Einstein's and Newton's formulae if we are to avoid repeating the whole labour of the classical lunar theory. In order to make this comparison we transform \Eq{(39.31)} and \Eq{(39.32)} so that $t$~is used as the independent variable. \begin{align*} \frac{d^{2}}{ds^{2}} &= \left(\frac{dt}{ds}\right)^{2} \frac{d^{2}}{dt^{2}} + \frac{dt}{ds}\, \frac{d}{dt} \left(\frac{dt}{ds}\right) \frac{d}{dt} \\ &= \left(\frac{dt}{ds}\right)^{2} \left(\frac{d^{2}}{dt^{2}} + \lambda'\, \frac{dr}{dt}\, \frac{d}{dt}\right) \qquad\text{by \Eq{(39.42)}.} \end{align*} Hence the equations \Eq{(39.31)} and \Eq{(39.32)} become \begin{gather*} \frac{d^{2}r}{dt^{2}} + \tfrac{3}{2}\lambda'\left(\frac{dr}{dt}\right)^{2} - re^{-\lambda} \left(\frac{d\phi}{dt}\right)^{2} + \tfrac{1}{2} e^{2\nu} \nu' = 0,\displaybreak[0] \\ \frac{d^{2}\phi}{dt^{2}} + \lambda'\, \frac{dr}{dt}\, \frac{d\phi}{dt} + \frac{2}{r}\, \frac{dr}{dt}\, \frac{d\phi}{dt} = 0. \end{gather*} Whence \[ \left. \begin{aligned} \frac{d^{2}r}{dt^{2}} - r\left(\frac{d\phi}{dt}\right)^{2} + \frac{m}{r^{2}} &= R\Add{,} \\ r\left(\frac{d^{2}\phi}{dt^{2}} + \frac{2}{r}\, \frac{dr}{dt}\, \frac{d\phi}{dt}\right) &= \Phi\Add{,} \end{aligned} \right\} \Tag{(44.1)} \] where \[ \left. \begin{aligned} R &= -\tfrac{3}{2}\lambda' u^{2} - \frac{2m}{r^{2}}\, v^{2} + \frac{2m^{2}}{r^{3}}\Add{,} \\ \Phi &= -\lambda' uv\Add{,} \end{aligned} \right\} \Tag{(44.21)} \] and \[ u = dr/dt,\quad v = r\, d\phi/dt. \] \PageSep{97} Equations \Eq{(44.1)} show that $R$ and~$\Phi$ are the radial and transverse perturbing forces which Einstein's theory adds to the classical dynamics. To a sufficient approximation $\lambda' = -2m/r^{2}$, so that \[ \left. \begin{aligned} R &= \frac{m}{r^{2}} (3u^{2} - 2v^{2}) + \frac{2m^{2}}{r^{3}}\Add{,} \\ \Phi &= \frac{m}{r^{2}} · 2uv\Add{.} \end{aligned} \right\} \Tag{(44.22)} \] In three-dimensional problems the perturbing forces become \[ \left. \begin{aligned} R &= \frac{m}{r^{2}} (3u^{2} - 2v^{2} - 2w^{2}) + \frac{2m^{2}}{r^{3}}\Add{,} \\ \Phi &= \frac{m}{r^{2}} · 2uv\Add{,} \\ Z &= \frac{m}{r^{2}} · 2uw\Add{.} \end{aligned} \right\} \Tag{(44.23)} \]