Let us suppose that a point-shaped particle, having the electrical charge e (to be called henceforth the electron) moves in the electromagnetic field; we assume the following about its law of motion.
If the electron be at rest at any definite epoch, then in the next “particle of time,” the motion takes place according to the equations
d²x d²y d²z
m ----- = eX, m ----- = eY, m ----- = eZ
dt² dt² dt²
Where (x, y, z) are the co-ordinates of the electron, and m is its mass.
Let the electron possess the velocity v at a certain epoch of time. Let us now investigate the laws according to which the electron will move in the ‘particle of time’ immediately following this epoch.
Without influencing the generality of treatment, we can and we will assume that, at the moment we are considering, the electron is at the origin of co-ordinates, and moves with the velocity v along the X-axis of the system. It is clear that at this moment (t = 0) the electron is at rest relative to the system k, which moves parallel to the X-axis with the constant velocity v.
From the suppositions made above, in combination with the principle of relativity, it is clear that regarded from the system k, the electron moves according to the equations
d²ξ d²η d²ζ