Page:DeSitterGravitation.djvu/9

396 Similarly we have for the force acting on $$m_2$$ from $$m_1$$,

We will now introduce simultaneous coordinates. Let these be for time $$t$$—

In the equations of motion of $$m_1$$, i.e. in the expression (14), we must use the coordinates and velocities of $$m_2$$ for the time $$t_2$$ defined by

and we have

In (15) we must use the coordinates and velocities of $$m_1$$ for the time $$t_1$$ defined by

and we have

Further, we have for use in (14)—

and in (15)—

The expression for $$\mathrm C$$ is the same in both cases.

We find then,

This form of the equations is not unique. We can multiply by any power of $$\mathrm C$$, or make more complicated alterations, for which the reader is referred to Poincaré.

Multiplying by $$\mathrm C$$ we get—