Page:Lorentz Grav1900.djvu/3

( 561 )

Then, in what follows, we need only consider the forces due to the state of the aether, in so far as it not directly produced by the ion itself.

Since the formulae (2) contain e as a factor, the coefficients a and b will be independent of the charge; their sign will be the same for a negative ion and for a positive one. Now, as soon as the ion has shifted from its position of equilibrium, new forces come into play. In the first place, the force $4\pi V^{2}e{\mathfrak {d}}$ will have changed a little, because, for the new position, ${\mathfrak {d}}$ will be somewhat different from what it was at the point P. We may express this by saying that, in addition to the force (1), there will be a new one with the components,

$4\pi V^{2}e\left({\mathsf {x}}{\frac {\partial {\mathfrak {d}}_{\mathsf {x}}}{\partial x}}+{\mathsf {y}}{\frac {\partial {\mathfrak {d}}_{\mathsf {x}}}{\partial y}}+{\mathsf {z}}{\frac {\partial {\mathfrak {d}}_{\mathsf {x}}}{\partial z}}\right)$ etc.

(3)

In the second place, in consequence of the velocity of vibration, there will be an electromagnetic force with the components

$e\left({\dot {\mathsf {y}}}{\mathfrak {H}}_{\mathsf {z}}-{\dot {\mathsf {z}}}{\mathfrak {H}}_{\mathsf {y}}\right)$, etc.

(4)

If, as we shall suppose, the displacement of the ion be very small, compared with the wave-length, the forces (3) and (4) are much smaller than the force (1); since they are periodic — with the frequency 2n, — they will give rise to new vibrations of the particle. We shall however omit the consideration of these slight vibrations, and examine only the mean values of the forces (3) and (4), calculated for a rather long lapse of time, or, what amounts to the same thing, for a full period ${\tfrac {2\pi }{n}}$.

§ 3. It is immediately clear that this mean force will be 0 if the ion is alone in a field in which the propagation of waves takes place equally in all directions. It will be otherwise, as soon as a second ion Q has been placed in the neighbourhood of P; then, in consequence of the vibrations emitted by Q after it has been itself put in motion, there may be a force on P, of course in the direction of the line QP. In computing the value of this force, one finds a great number of terms, which depend in different ways on the distance r. We shall retain those which are inversely proportional to r or r², but we shall neglect all terms varying inversely as the higher powers of r; indeed, the influence of these, compared with that of the first mentioned terms will be of the order ${\tfrac {\lambda }{r}}$ , if $\lambda$ is the