Abraham found, in different notation (Göttinger Nachrichten, 1902, p. 37)
a is a constant. However, in the hypothesis of Abraham, we have θ = 1; then:
(5)
which defines the function φ.
This granted, imagine that the electron is subject to a binding, so there is a relation between r and φ; in the hypothesis of Lorentz this relation would be φr = const., in that of Langevin φ²r² = const. We assume in a more general way
b is a constant; hence:
What is the shape of the electron when the velocity become -εt, if we do not suppose the involvement of forces other than the binding forces? Its form will be defined by the equality:
(6)
or
or
If we want equilibrium to occur so that θ = k, it is necessary that , the logarithmic derivative of φ is equal to m.
If we develop and the right-hand side of (5) in powers of ε, equation (5) becomes: