Page:PoincareDynamiqueJuillet.djvu/27

That of Langevin:

$l=k^{-{\frac {1}{3}}},\quad k=\theta ,\quad klr=const..$

We then find:

$H={\frac {\varphi \left({\frac {\theta }{k}}\right)}{k^{2}r}}.$

Abraham found, in different notation (Göttinger Nachrichten, 1902, p. 37)

$H={\frac {a}{r}}{\frac {1-\epsilon ^{2}}{\epsilon }}\log {\frac {1+\epsilon }{1-\epsilon }},$

a is a constant. However, in the hypothesis of Abraham, we have θ = 1; then:

(5)	$\varphi \left({\frac {1}{k}}\right)=ak^{2}{\frac {1-\epsilon ^{2}}{\epsilon }}\log {\frac {1+\epsilon }{1-\epsilon }}={\frac {a}{\epsilon }}\log {\frac {1+\epsilon }{1-\epsilon }},$

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

$r=b\theta ^{m}\,$

b is a constant; hence:

$H={\frac {1}{bk^{2}}}\theta ^{-m}\varphi \left({\frac {\theta }{k}}\right).$

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)	${\frac {\partial H}{\partial \theta }}=0,$

or

$-m\theta ^{-m-1}\varphi +\theta ^{-m}k^{-1}\varphi ^{\prime }=0$

or

${\frac {\varphi ^{\prime }}{\varphi }}={\frac {mk}{\theta }}.$

If we want equilibrium to occur so that θ = k, it is necessary that ${\tfrac {\theta }{k}}=1$, the logarithmic derivative of φ is equal to m.

If we develop ${\frac {1}{k}}$ and the right-hand side of (5) in powers of ε, equation (5) becomes:

$\varphi \left(1-{\frac {\epsilon ^{2}}{2}}\right)=a\left(1-{\frac {\epsilon ^{2}}{3}}\right)$

neglecting higher powers of ε.