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Comparing (13) and (11) we find that (23) is equivalent to

${\frac {d^{2}{\boldsymbol {x}}}{d\tau ^{2}}}+k^{2}\mathrm {M} {\frac {\boldsymbol {x}}{{\boldsymbol {r}}^{3}}}=0.$

(24)

This is the equation given by Minkowski (l.c., p. 110). It could, of course, have been derived by taking $\mu =0$ from the beginning. In that case the acceleration of $m_{2}$ is zero, and we can take $m_{2}$ as the origin of our system of reference. Then $\xi _{2}=\eta _{2}=\zeta _{2}=0$, $\mathrm {B} _{1}={\boldsymbol {r}},\ \mathrm {x} _{1}={\boldsymbol {x}}$, and (24) is derived at once from (14).

In (24) $\tau$ is the proper-time of the planet, or planeto-centric time; in (23) $t$ is the time of the system of reference, which has been chosen with its origin in the mean position of the centre of gravity. Practically $t$ will coincide with the proper-time of the Sun, or heliocentric time.

The equations for the other two coordinates are, of course, quite similar.

The equations for the law II. are derived from those for the law I. by multiplying the right-hand members by $\mathrm {C}$. Introducing the relative velocities we find, neglecting fourth orders,

$\mathrm {C} ={\frac {1}{\sqrt {1-{\boldsymbol {\phi }}^{2}}}}.$

The only difference between the laws II. and I. is therefore that in (22) and (23,) the term ${{\boldsymbol {x}}\phi }^{2}$ is changed to ${\tfrac {1}{2}}{{\boldsymbol {x}}\phi }^{2}$. The simple form (24) is no longer possible with the law II. We have, however, in accordance with (12), for this law—

${\frac {d}{dt}}{\frac {d{\boldsymbol {x}}}{d\tau }}+k^{2}{\frac {\mathrm {M} }{{\boldsymbol {r}}^{3}}}=0.$

(25)

The only previous investigations on the same subject which have come to my knowledge, beyond the quoted papers of Poincaré and Minkowski, are those by Mr. Wilkens ("Zur Elektronentheorie," V.J.S. 1904, p. 209) and by Mr. Wacker ("Ueber Gravitation und Elektromagnetismus," Inaugural Dissertation. (Tübingen), 1909). Mr. Wilkens finds secular terms in all elements. Some of these have amounts which could not long remain undetected. Thus, e.g., his secular perturbation of the mean distance corresponds, for the Earth, to a shortening of the year by 19 seconds per century. His formulæ, however, are not in conformity with the principle of relativity, and his results are not confirmed here.

Mr. Wacker gives the equations (25). His equations are thus in conformity with the principle of relativity, and he adopts the law II., and neglects the mass of the planet.

8. The equation (24) is the ordinary equation of Keplerian motion. Consequently, if we neglect the mass of the planet and