Page:DeSitterGravitation.djvu/7

394 The values of $$\xi_2$$ and the higher differential coefficients must be taken for the time $$t_2$$, defined by (8). We consider $$1/c$$ as a small quantity of the first order, and we wish our formulæ to be exact to the second order inclusive.

We find easily

where

$$\epsilon_1$$ and $$\epsilon_2$$ are of the first order, $$\chi_2$$ is of the second order.

In the equations of motion there appear the invariants—

To express these in simultaneous relative coordinates we have—

6. The equations of motion can be given in three forms:—

and similarly for the other coordinates.

$$\mathrm{X}$$ is the force according to the ordinary definition, or "Newtonian" force; $$(\mathrm{X})$$ is called the "Minkowskian" force. The mass $$m$$ is a constant.

Differentiating the formula $$(\kappa)^{2}=1+(\phi)^{2}$$, we derive a fourth equation analogous to (11) or (12), viz.: