Page:Lorentz Grav1900.djvu/4

( 562 )

wave-length, and we shall suppose this to be a very small fraction.

We shall also omit all terms containing such factors as $\cos \ 2\pi k{\tfrac {r}{\lambda }}$ or $\sin \ 2\pi k{\tfrac {r}{\lambda }}$ (k a moderate number). These reverse their signs by a very small change in r; they will therefore disappear from the resultant force, as soon as, instead of single particles P and Q, we come to consider systems of particles with dimensions many times greater than the wave-length.

From what has been said, we may deduce in the first place that, in applying the above formulae to the ion P, it is sufficient, to take for ${\mathfrak {d}}$ and ${\mathfrak {H}}$ the vectors that would exist if P were removed from the field. In each of these vectors two parts are to be distinguished. We shall denote by ${\mathfrak {d}}_{1}$ and ${\mathfrak {H}}_{1}$ the parts existing independently of Q, and by ${\mathfrak {d}}_{2}$ and ${\mathfrak {H}}_{2}$ the parts due to the vibrations of this ion.

Let Q be taken as origin of coordinates, QP as axis of x, and let us begin with the terms in (2) having the coefficient a.

To these corresponds a force on P, whose first component is

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

(5)

Since we have only to deal with the mean values for a full period, we may write for the last term

$-e^{2}a\left({\mathfrak {d}}_{y}{\mathfrak {\dot {H}}}_{z}-{\mathfrak {d}}{}_{z}{\mathfrak {\dot {H}}}_{y}\right)$,

and if, in this expression, ${\mathfrak {\dot {H}}}_{y}$ and ${\mathfrak {\dot {H}}}_{z}$ be replaced by

$4\pi V^{2}\left({\frac {\partial {\mathfrak {d}}_{\mathsf {z}}}{\partial x}}-{\frac {\partial {\mathfrak {d}}_{\mathsf {x}}}{\partial z}}\right)$ and $4\pi V^{2}\left({\frac {\partial {\mathfrak {d}}_{\mathsf {x}}}{\partial y}}-{\frac {\partial {\mathfrak {d}}_{\mathsf {y}}}{\partial x}}\right)$,

(5) becomes

$2\pi V^{2}e^{2}a{\frac {\partial \left({\mathfrak {b^{2}}}\right)}{\partial x}}$,

(6)

where ${\mathfrak {d}}$ is the numerical value of the dielectric displacement.

Now, ${\mathfrak {d}}^{2}$ will consist of three parts, the first being ${\mathfrak {d}}_{1}^{2}$, the second ${\mathfrak {d}}_{2}^{2}$ and the third depending on the combination of ${\mathfrak {d}}_{1}$ and ${\mathfrak {d}}_{2}$.

Evidently, the value of (6), corresponding to the first part, will be 0.

As to the second part, it is to be remarked that the dielectric displacement, produced by Q, is a periodic function of the time. At distant points the amplitude takes the form ${\tfrac {c}{r}}$, where c is independent