Page:Lorentz Grav1900.djvu/5

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of r. The mean value of ${\mathfrak {d}}^{2}$ for a full period is ${\tfrac {1}{2}}{\tfrac {c^{2}}{r^{2}}}$ and differentiating this with regard to x or to r, we should get $r^{3}$ in the denominator.

The terms in (6) which correspond to the part

$2\left({\mathfrak {d}}_{1{\mathsf {x}}}{\mathfrak {d}}_{2{\mathsf {x}}}+{\mathfrak {d}}_{1{\mathsf {y}}}{\mathfrak {d}}_{2{\mathsf {y}}}+{\mathfrak {d}}_{1{\mathsf {z}}}{\mathfrak {d}}_{2{\mathsf {z}}}\right)$

in ${\mathfrak {d}}^{2}$, may likewise be neglected. Indeed, if these terms are to contain no factors such as $\cos \ 2\pi k{\tfrac {r}{\lambda }}$ or $\sin \ 2\pi k{\tfrac {r}{\lambda }}$, there must be between ${\mathfrak {d}}_{1}$ and ${\mathfrak {d}}_{2}$, either no phase-difference at all, or a difference which is independent of r. This condition can only be fulfilled, if a system of waves, proceeding in the direction of QP, is combined with the vibrations excited by Q, in so far as this ion is put in motion by that system itself. Then, the two vectors ${\mathfrak {d}}_{1}$ and ${\mathfrak {d}}_{2}$ will have a common direction perpendicular to QP, say that of the axis of y, and they will be of the form

${\mathfrak {d}}_{1{\mathsf {y}}}=q\ cos\ n\ \left(t-{\frac {x}{V}}+\varepsilon _{1}\right)$ ${\mathfrak {d}}_{2{\mathsf {y}}}={\frac {c}{r}}\ cos\ n\ \left(t-{\frac {x}{V}}+\varepsilon _{2}\right)$

The mean value of ${\mathfrak {d}}_{1{\mathsf {y}}}{\mathfrak {d}}_{2{\mathsf {y}}}$ is

${\frac {1}{2}}{\frac {qc}{r}}\cos n\ \left(\varepsilon _{1}-\varepsilon _{2}\right)$,

and its differential coefficient with regard to x has $r^{2}$ in the denominator. It ought therefore to be retained, were it not for the extremely small intensity of the systems of waves which give rise to such a result. In fact, by the restriction imposed on them as to their direction, these waves form no more than a very minute part of the whole motion.

§ 4. So, it is only the terms in (2), with the coefficient b, with which we are concerned. The corresponding forces are

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

(7)

and

$-e^{2}b\left({\ddot {{\mathfrak {d}}_{\mathsf {y}}}}\ {\mathfrak {H}}_{\mathsf {z}}-{\ddot {{\mathfrak {d}}_{\mathsf {z}}}}\ {\mathfrak {H}}_{\mathsf {y}}\right)$.

(8)