# Page:Über die Möglichkeit einer elektromagnetischen Begründung der Mechanik.djvu/9

If the masses ${\displaystyle M}$ and ${\displaystyle m}$ would attract each other according to ${\displaystyle Weber}$'s law, one would have

${\displaystyle m{\frac {d^{2}r}{dt^{2}}}=-{\frac {\epsilon mM}{r}}\left\{1-{\frac {A^{2}}{2}}\left({\frac {dr}{dt}}\right)^{2}+rA^{2}{\frac {d^{2}r}{dt^{2}}}\right\}.}$

If we multiply with ${\displaystyle dr/dt}$ and then integrate, we have:

 ${\displaystyle {\frac {1}{2}}\left({\frac {dr}{dt}}\right)^{2}={\frac {\epsilon M}{r}}\left[1-{\frac {A^{2}}{2}}\left({\frac {dr}{dt}}\right)^{2}\right],}$

where the integration constant is so specified, that the body is at rest in infinite distance.

If we write this equation

 (10) ${\displaystyle {\frac {1}{2}}\left({\frac {dr}{dt}}\right)^{2}\left(1+A^{2}{\frac {\epsilon M}{r}}\right)={\frac {\epsilon M}{r}},}$

then it agrees with equation (9) up to a factor ${\displaystyle {\tfrac {16}{15}}}$ instead of 1. By consideration of the second approximation for inertia, we thus approximately obtain the same action between the two masses, as if the masses themselves would be invariable, but Weber's law would hold instead of Newton's law.

It's known, that Weber's law was applied with some success to the theory of molecular motion.

A precise test of these investigations, and an extension by applying it to other, fast moving celestial bodies with strong eccentric path, would lead us to a comparison of our results with experience. Though it is to be considered here, that new terms of same order are added by the motion in a curved path. For a body in elliptical path, the calculation would thus to be supplemented.

Only have with respect to cathode rays we have such great velocities required, so that the square of velocity multiplied with the reciprocal speed of light, is not getting too small.

The fastest, recently produced rays have ${\displaystyle {\tfrac {1}{3}}}$ of the speed of light. Here, the apparent increase of mass would be ca. 7 perc.; the slightest velocity is ${\displaystyle {\tfrac {1}{30}}}$ of the speed of light,[1] the corresponding increase of mass would amount

1. See P. Lenard, Sitzungsber. d. k. Akad. d. Wissensch. zu Wien 108. p. 1649. 1899.