Page:Lorentz Grav1900.djvu/2

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what follows, before passing to considerations of a different order (§ 5), I shall explain the reasons for which this theory of rapid vibrations as a cause of gravitation can not be accepted.

§ 2. Let an ion carrying a charge $e_{1}$, and having a certain mass, be situated at the point P(x,y,z); it may be subject or not to an elastic force, proportional to the displacement and driving it back to P, as soon as it has left this position. Next, let the aether be traversed by electromagnetic vibrations, the dielectric displacement being denoted by ${\mathfrak {d}}$, and the magnetic force by ${\mathfrak {H}}$, then the ion will be acted on by a force

$4\pi V^{2}e{\mathfrak {d}}$,

whose direction changes continually, and whose components are

${\mathsf {X}}=4\pi V^{2}e{\mathfrak {d}}_{\mathsf {x}},\ {\mathsf {Y}}=4\pi V^{2}e{\mathfrak {d}}_{\mathsf {y}},\ {\mathsf {Z}}=4\pi V^{2}e{\mathfrak {d}}_{\mathsf {z}}$,

(1)

In these formulae V means the velocity of light.

By the action of the force (1) the ion will be made to vibrate about its original position P, the displacement (x, y,z) being determined by well known differential equations.

For the sake of simplicity we shall confine ourselves to simple harmonic vibrations with frequency n. All our formulae will then contain the factor $\cos nt$ or $\sin nt$, and the forced vibrations of the ion may be represented by expressions of the form

$\left.{\begin{array}{l}{\mathsf {x}}=ae{\mathfrak {d}}_{\mathsf {x}}-be{\mathfrak {\dot {d}}}_{\mathsf {x}},\\{\mathsf {y}}=ae{\mathfrak {d}}_{\mathsf {y}}-be{\mathfrak {\dot {d}}}_{\mathsf {y}},\\{\mathsf {z}}=ae{\mathfrak {d}}_{\mathsf {z}}-be{\mathfrak {\dot {d}}}_{\mathsf {z}},\end{array}}\right\}$

(2)

with certain constant coefficients a and b. The terms with ${\mathfrak {\dot {d}}}_{\mathsf {x}},{\mathfrak {\dot {d}}}_{\mathsf {y}}$ and ${\mathfrak {\dot {d}}}_{\mathsf {z}}$ have been introduced in order to indicate that the phase of the forced vibration differs from that of the force (X,Y,Z); this will be the case as soon as there is a resistance, proportional to the velocity, and the coefficient b may then be shown to be positive. One cause of a resistance lies in the reaction of the aether, called forth by the radiation of which the vibrating ion itself becomes the centre, a reaction which determines at the same time an apparent increase of the mass of the particle. We shall suppose however that we have kept in view this reaction in establishing the equations of motion, and in assigning their values to the coefficients a and b.