Page:DeSitterGravitation.djvu/10

Mar. 1911.

This last form (II.) is preferred by Lorentz (l.c., p. 1239), because the corresponding Newtonian force, $$\mathrm{X}_{i}=(\mathrm{X})_{i}\sqrt{1-\phi_{i}^{2}}$$, does not contain the velocity of $$m_i$$. (I.) is the law adopted by Minkowski, presumably because it gives the simplest result for a planet of infinitesimal mass.

As has already been remarked, the values of $$\xi_{2},\ \eta_{2},\ \zeta_{2},$$ in $$(\mathrm{X})_1$$, and those of $$\xi_{1},\ \eta_{1},\ \zeta_{1},$$ in $$(\mathrm{X})_2$$ differ from the values at the time $$t$$ for which the simultaneous coordinates are taken. If we are content to neglect third orders, however, we can assume all velocities to correspond to the time $$t$$. If the motion were quasi-stationary, i.e. if the accelerations could be neglected, the velocities would be constant, and also $$\delta x_2$$ and $$\delta x_1$$ would disappear. The equations (16) and (17) would in that case be rigorous.

7. We will first consider the law (I.). Introducing the developments of $$\mathrm{B}_1$$ and $$\mathrm{A}_2$$ we find—

We thus have by (13)—

Now we have, to second orders,—

Further, if we put