Page:Lorentz Grav1900.djvu/13

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3^rd. A force

$-{\frac {k}{V^{2}}}p\cdot {\frac {1}{r^{2}}}{\frac {dr}{dt}},$,

(17)

parallel to the velocity p.

4^th. A force

${\frac {k}{V^{2}}}{\frac {1}{r^{2}}}p\ w\ \cos \ \vartheta$,

(18)

in the direction of r.

Of these, (15) and (16) depend only on the common velocity p, ( 17) and (18) on the contrary, on p and w conjointly.

It is further to be remarked that the additional forces (15) — (18) are all of the second order with respect to the small quantities

${\frac {p}{V}}$ and ${\frac {w}{V}}$.

In so far, the law expressed by the above formulae presents a certain analog)- with the laws proposed by Weber, Riemann and Clausius for the electromagnetic actions, and applied by some astronomers to the motions of the planets. Like the formulae of Clausius, our equations contain the absolute velocities, i. e. the velocities, relatively to the aether.

There is no doubt but that, in the present state of science, if we wish to try for gravitation a similar law as for electromagnetic forces, the law contained in (15) — (18) is to be preferred to the three other just mentioned laws.

§ 9. The forces (15) — (18) will give rise to small inequalities in the elements of a planetary orbit ; in computing these, we have to take for p the velocity of the Sun's motion through space. I have calculated the secular variations, using the formulae communicated by Tisserand in his Mécanique céleste.

Let	a be the mean distance to the sun,
	e the eccentricity,
	φ the inclination to the ecliptic,
	θ the longitude of the ascending node,
	$\varpi$ the longitude of perihelion,
	$\varkappa '$ the mean anomaly at time t=0, in this sense that, if n be the mean motion, as determined by a, the mean anomaly at time t is given by