the transverse magnetic energy. There is no longitudinal magnetic energy, since α = α' = 0. We denote by A', B', C' the corresponding quantities in the ideal system. We first find:
In addition, we can observe that the actual field depends only on x = εt, y, and x, and write:
hence
In Lorentz's hypothesis we have B' = 2A', and A ' (being inversely proportional to the radius of the electron) is a constant independent of the velocity of the real electron; we get for the total energy:
and for the action (per unit time):
Now calculate the electromagnetic momentum; we find:
But there must be some relation between the energy E = A + B + C, the action per unit time H = A + B - C, and the momentum D. The first of these relations is:
the second is
hence
(2)
The second of equations (2) is always satisfied; but the first is so only if
that is to say if the volume of the ideal electron is equal to that of the real electron; or if the volume of the electron is constant; that's the hypothesis of Langevin.