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

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or the potential at any point outside the body is

 ${\displaystyle U={\frac {1}{a}}B_{0}\left({\frac {a}{r}}+{\frac {a^{2}}{r^{2}}}S_{1}Y_{1}+\dots +{\frac {a^{k+1}}{r^{k+1}}}S_{k}Y_{k}\right);}$ (80)

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

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

or

 ${\displaystyle 4\pi a\sigma =B_{0}\left(1+S_{2}Y_{2}+\dots +(k-1)S_{k}Y_{k}\right).}$ (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 ${\displaystyle k}$, the ratio of the difference of the superficial densities at any two points to their sum will be ${\displaystyle k-1}$ times the ratio of the difference of the radii of the same two points to their sum.