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Mar. 1911. use planeto-centric time, the motion is Keplerian. Thus, e.g., the integrals of areas are

Or, introducing heliocentric time, to second orders—

In (22), on the other hand, it is not advantageous to introduce the proper-time of one of the bodies, since we should thereby lose the symmetry gained by the introduction of relative velocities and coordinates. In the second-order terms We can introduce the ordinary Keplerian motion. We find then, taking the orbital plane for plane of ($$x, y$$), for the two laws equally—

where

and v is the true anomaly.

Similarly we find for the vis-viva integral from (24)—

or in heliocentric time—

From (22), on the other hand, we find—

For the law II. we must add to (28) and (29) the term—