Page:SearleEllipsoid.djvu/13

 that as far as terms in $$u^2 / v^2$$ the electric part of the energy is unaltered by the motion.

(C) Energy of a very slender Ellipsoid. When the ellipsoid is so slender that $$b^2 /a^2$$ may be neglected in comparison with unity we have

When $$u / v$$ is small, this becomes

$\mathrm{W}=\frac{q^{2}}{2\mathrm{K}a}\left\{ \left(1+\frac{u^{2}}{v^{2}}\right)\log\frac{2a}{b}+\frac{1}{2}\frac{u^{2}}{v^{2}}\right\}$.|undefined

(D) Energy of a Disk.

When $$a^2 < \alpha b^2$$ the ellipsoid is more oblate than Heaviside's, and $$l^2$$ becomes negative. In this case let us write

$r^{2}=b^{2}-\frac{a^{2}}{\alpha}$,|undefined

so that $$r$$ is the radius of the disk which is the "image" of the ellipsoid $$a, b$$. Then writing $$\sqrt{-1}=i$$ we have from (23)

$\mathrm{W}=\frac{q^{2}}{4\mathrm{K}ir\sqrt{\alpha}}\left(1-\frac{u^{2}a^{2}}{v^{2}r^{2}\alpha}\right)\log\frac{1+i\sqrt{a}r/a}{1-i\sqrt{\alpha}r/a}+\frac{q^{2}u^{2}a}{2Kv^{2}r^{2}\alpha}$.|undefined

But

$\frac{1}{i}\log\frac{1+xi}{1-xi}=2\left(x-\frac{x^{3}}{3}+\frac{x^{5}}{5}\dots\right)=2\tan^{-1}x$,|undefined

so that (23) becomes

When $$a = 0$$ we find for the energy of a disk of radius $$r$$ moving along its axis

In all these cases it will be found that when $$u = v$$ the energy becomes infinite, so that it would seem to be impossible to make a charged body move at a greater speed than that of light.