Page:MortonCharge.djvu/6

Rh

If now F(x y z) = C is an ellipsoid, then we know that $$\sigma\propto p$$, therefore also $$\sigma'\propto p'$$, that is the arrangement of charge on the moving ellipsoid is the same as if it were at rest.

7. Applying the above to the ellipsoid (a b c), we find that φ as a function of (x y ζ) is the potential of a free distribution on the ellipsoid

where μ is given by

$$\frac{x^{2}}{a^{2}+\mu}+\frac{y^{2}}{b^{2}+\mu}+\frac{\zeta^{2}}{\frac{c^{2}}{k^{2}}+\mu}=1$$,

or

$$\frac{x^{2}}{a^{2}+\mu}+\frac{y^{2}}{b^{2}+\mu}+\frac{z^{2}}{c^{2}+k^{2}\mu}=1$$.

Deteimining the value of the constant C so that the density at a point shall be $$\frac{ep}{4\pi abc}$$, we get

$$\phi=\frac{e}{8\pi}\int_{\mu}^{\infty}\frac{d\lambda}{\sqrt{(a^{2}+\lambda)(b^{2}+\lambda)\left(c^{2}+k^{2}\lambda\right)}}$$.

Putting b=a, c=ka, we get

$$\phi=\frac{e}{4\pi\sqrt{k^{2}(x^{2}+y^{2})+z^{2}}}$$.