Page:Lorentz Grav1900.djvu/15

${\begin{array}{ll}\Delta a=&0\\\\\Delta e=&0,018\ \delta ^{2}+1,38\ \delta \delta '\\\\\Delta \varphi =&0,95\ \delta ^{2}+0,28\ \delta \delta '\\\\\Delta \theta =&7,60\ \delta ^{2}-4,26\ \delta \delta '\\\\\Delta \varpi =-&0,09\ \delta ^{2}+1,95\ \delta \delta '\\\\\Delta \varkappa '=-&6,82\ \delta ^{2}-1,93\ \delta \delta '\end{array}}$

Now, $\delta '=1,6\ \times \ 10^{-4}$ and, if we put $\delta =5,3\ \times \ 10^{-5}$, we get

$\Delta e=117\ \times \ 10^{-10}$, $\Delta \varphi =51\ \times \ 10^{-10},$ $\Delta \theta =-137\ \times \ 10^{-10},\ \Delta \varpi =162\ \times \ 10^{-10},\ \Delta \varkappa '=355\ \times \ 10^{-10}$.

The changes that take place in a century are found from these numbers, if we multiply them by 415, and, if the variations of φ, θ, $\varpi$, and $\varkappa '$ are to be expressed in seconds, we have to introduce the factor $2,06\times 10^{5}$. The result is, that the changes in φ, θ, $\varpi$, and $\varkappa '$ amount to a few seconds, and that in e to 0,000005.

Hence we conclude that our modification of Newton's law cannot account for the observed inequality in the longitude of the perihelion — as Weber's law can to some extent do — but that, if we do not pretend to explain this inequality by an alteration of the law of attraction, there is nothing against the proposed formulae. Of course it will be necessary to apply them to other heavenly bodies, though it seems scarcely probable that there will be found any case in which the additional terms have an appreciable influence.

The special form of these terms may perhaps be modified. Yet, what has been said is sufficient to show that gravitation may be attributed to actions which are propagated with no greater velocity than that of light.

As is well known, Laplace has been the first to discuss this question of the velocity of propagation of universal attraction, and later astronomers have often treated the same problem. Let a body B be attracted by a body A, moving with the velocity p. Then, if the action is propagated with a finite velocity V, the influence which reaches B at time t, will have been emitted by A at an anterior moment, say t—τ. Let A₁ be the position of the acting body at this moment, A₂ that at time t. It is an easy matter to calculate the distance between these positions. Now, if the action at time