Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/222

or the potential at any point outside the body is

$U={\frac {1}{a}}B_{0}\left({\frac {a}{r}}+{\frac {a^{2}}{r^{2}}}S_{1}Y_{1}+\dots +{\frac {a^{k+1}}{r^{k+1}}}S_{k}Y_{k}\right);$

(80)

and if $\sigma$ is the surface-density at any point

$4\pi \sigma =-{\frac {dU}{dr}},$

or

$4\pi a\sigma =B_{0}\left(1+S_{2}Y_{2}+\dots +(k-1)S_{k}Y_{k}\right).$

(81)

Hence, if the surface differs from that of a sphere by a thin stratum whose depth varies according to the values of a spherical harmonic of degree $k$, the ratio of the difference of the superficial densities at any two points to their sum will be $k-1$ times the ratio of the difference of the radii of the same two points to their sum.