Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/247

and

$\left(r{\frac {d^{2}r}{dt^{2}}}\right)_{1}=u^{2}{\frac {d^{2}r}{ds^{2}}}+2uu^{\prime }{\frac {d^{2}r}{ds\ ds^{\prime }}}+{u^{\prime }}^{2}{\frac {d^{2}r}{{ds^{\prime }}^{2}}}$.

By aid of these and the similar equations with the suffixes _{2, 3, 4} the equation for the ponderomotive force may be transformed into the equation

$F={\frac {\lambda \lambda ^{\prime }\ ds\ ds^{\prime }}{r^{2}}}{\begin{cases}\left(r{\frac {d^{2}r}{dt^{2}}}\right)_{1}\left(r{\frac {d^{2}r}{dt^{2}}}\right)_{2}\left(r{\frac {d^{2}r}{dt^{2}}}\right)_{3}+\left(r{\frac {d^{2}r}{dt^{2}}}\right)_{4}\\-{\frac {1}{2}}\left(r{\frac {dr}{dt}}\right)_{1}^{2}+{\frac {1}{2}}\left(r{\frac {dr}{dt}}\right)_{2}^{2}+{\frac {1}{2}}\left(r{\frac {dr}{dt}}\right)_{3}^{2}-{\frac {1}{2}}\left(r{\frac {dr}{dt}}\right)_{4}^{2}\end{cases}}$.

But this is the equation which we should have obtained had we set out from the following assumptions: that the ponderomotive force between two current-elements is the resultant of the force between the positive charge in ds and the positive charge in ds′, of the force between the positive charge in ds and the negative charge in ds′, etc.; and that any two electrified particles of charges e and e′, whose distance apart is r, repel each other with a force of magnitude

${\frac {ee^{\prime }}{r^{2}}}\left\{r{\frac {d^{2}r}{dt^{2}}}-{\frac {1}{2}}\left({\frac {dr}{dt}}\right)^{2}\right\}$.

Two such charges would, of course, also cxert on each other an electrostatic repulsion, whose magnitude in these units would be ee′c²/r², where c denotes a constant^[1] of the dimensions of a velocity, whose value is approximately 3 x 10¹⁰ cm./sec. So that on these assumptions the total repellent force would be

${\frac {ee^{\prime }c^{2}}{r^{2}}}\left(1+{\frac {r{\ddot {r}}}{c^{2}}}-{\frac {{\dot {r}}^{2}}{2c^{2}}}\right)$.

1. The units which have been adopted in the above investigation depend on the electrodynamic actions of currents; i.e. they are such that two unit currents flowing in parallel circular circuits at a certain distance apart exert unit ponderomotive force on each other. The quantity of electricity conveyed in unit time by such a unit current is adopted as the unit charge. This unit charge is not identical with the eleetrostatic unit charge, which is defined to be such that two suit charges at unit distance apart repel each other with unit ponderomotive force. Hence the necessity for introducing the factor c.

Q 2