Page:Lorentz Grav1900.djvu/12

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Instead of introducing two pairs of vectors (${\mathfrak {d,\ H}}$) and (${\mathfrak {d',H'}}$), both of which come into play in the electromagnetic actions, as well as in the phenomenon of gravitation, we might have assumed one pair for the electromagnetic field and one for universal attraction.

For these latter vectors, say ${\mathfrak {d,\ H}}$, we should then have established the equations (I), $\varrho$ being the density of ponderable matter, and for the force acting on unit mass, we should have put

$-\eta \left\{4\pi V^{2}{\mathfrak {d}}+{\mathfrak {\left[v.\ H\right]}}\right\}$,

where $\eta$ is a certain positive coefficient.

§ 8. Every theory of gravitation has to deal with the problem of the influence, exerted on this force by the motion of the heavenly bodies. The solution is easily deduced from our equations; it takes the same form as the corresponding solution for the electromagnetic actions between charged particles.^[1]

I shall only treat the case of a body A, revolving around a central body M, this latter having a given constant velocity p. Let r be the line MA, taken in the direction from M towards A, x, y, z the relative coordinates of A with respect to M, w the velocity of A's motion relatively to M, $\vartheta$ the angle between w and p, finally $p_{r}$ the component of p in the direction of r.

Then, besides the attraction

${\frac {k}{r^{2}}}$,

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which would exist if the bodies were both at rest, A will be subject to the following actions.

1^st. A force

$k\cdot {\frac {p^{2}}{2V^{2}}}\cdot {\frac {1}{r^{2}}}$

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in the direction of r.

2^nd. A force whose components are

$-{\frac {k}{2V^{2}}}{\frac {\partial }{\partial x}}\left({\frac {p_{\mathsf {r}}^{2}}{r}}\right),\quad -{\frac {k}{2V^{2}}}{\frac {\partial }{\partial y}}\left({\frac {p_{\mathsf {r}}^{2}}{r}}\right),\quad -{\frac {k}{2V^{2}}}{\frac {\partial }{\partial z}}\left({\frac {p_{\mathsf {r}}^{2}}{r}}\right).$.

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1. See the second of the above mentioned papers.