Page:MortonCharge.djvu/6

$={\frac {xF_{1}+yF_{2}+k\zeta F_{3}}{\sqrt {F_{1}^{2}+F_{2}^{2}+k^{2}F_{3}^{2}}}}$ $={\frac {xF_{1}+yF_{2}+zF_{3}}{\sqrt {F_{1}^{2}+F_{2}^{2}+k^{2}F_{3}^{2}}}}$; $\therefore {\frac {\sigma }{\sigma '}}={\frac {Ap}{p'}}$.

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

${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {k^{2}\zeta ^{2}}{c^{2}}}=1$; $\therefore \ \phi =C\int _{\mu }^{\infty }{\frac {d\lambda }{\sqrt {(a^{2}+\lambda )(b^{2}+\lambda )\left({\frac {c^{2}}{k^{2}}}+\lambda \right)}}}$ $=Ck\int _{\mu }^{\infty }{\frac {d\lambda }{\sqrt {(a^{2}+\lambda )(b^{2}+\lambda )\left(c^{2}+k^{2}\lambda \right)}}}$,

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

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