Page:Lorentz Grav1900.djvu/4

 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

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

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

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

(5) becomes

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