# Page:Thomson1881.djvu/4

${\displaystyle F={\frac {\mu ep}{4\pi }}\int \int \int {\frac {1}{PQ}}{\frac {d^{2}}{dx^{2}}}{\frac {1}{\rho }}dx\ dy\ dz}$;

Now ${\displaystyle {\frac {d^{2}}{dx^{2}}}{\frac {1}{\rho }}={\frac {Y_{2}}{\rho ^{3}}}}$, where Y2 is a surface harmonic of the second order. And when ρ > R,

${\displaystyle {\frac {1}{PQ}}={\frac {1}{\rho }}+{\frac {R}{\rho ^{2}}}Q_{1}+{\frac {R^{2}}{\rho ^{3}}}Q_{2}+\dots }$;

and when ρ < R,

${\displaystyle {\frac {1}{PQ}}={\frac {1}{R}}+{\frac {\rho }{R^{2}}}Q_{1}+{\frac {\rho ^{2}}{R^{3}}}Q_{2}+\dots }$;

where Q1, Q2, &c. are zonal harmonics of the first and second orders respectively referred to OP as axis.

Let Y'2 denote the value of Y2 along OP. Then, since ${\displaystyle \int Y_{n}Q_{m}ds}$, integrated over a sphere of unit radius, is zero when n and m are different, and ${\displaystyle {\frac {4\pi }{2n+1}}Y_{n}^{'}}$ when n=m, Y'n being the value of Yn at the pole of Qn, and since there is no electric displacement within the sphere,

 ${\displaystyle F={\frac {\mu ep}{4\pi }}\times {\frac {2\pi Y_{2}^{'}}{5}}\left\{\int _{R}^{\infty }{\frac {R^{2}}{\rho ^{4}}}d\rho +\int _{a}^{R}{\frac {\rho \ d\rho }{R^{3}}}\right\}}$ ${\displaystyle ={\frac {\mu ep}{5}}Y_{2}^{'}\left({\frac {5}{6R}}-{\frac {a^{2}}{2R^{3}}}\right)}$,

or, as it is more convenient to write it,

${\displaystyle ={\frac {\mu ep}{5}}\left({\frac {5R^{2}}{6}}-{\frac {a^{2}}{2}}\right){\frac {d^{2}}{dx^{2}}}{\frac {1}{R}}}$.

By symmetry, the corresponding values of G and H are

 ${\displaystyle G={\frac {\mu ep}{5}}\left({\frac {5R^{2}}{6}}-{\frac {a^{2}}{2}}\right){\frac {d^{2}}{dx\ dy}}{\frac {1}{R}}}$, ${\displaystyle H={\frac {\mu ep}{5}}\left({\frac {5R^{2}}{6}}-{\frac {a^{2}}{2}}\right){\frac {d^{2}}{dx\ dz}}{\frac {1}{R}}}$.

These values, however, do not satisfy the condition

${\displaystyle {\frac {dF}{dx}}+{\frac {dG}{dy}}+{\frac {dH}{dz}}=0}$.

If, however, we add to F the term ${\displaystyle {\frac {2\mu ep}{3R}}}$, this condition will be satisfied; while, since the term satisfies Laplace's equation, the other conditions will not be affected: thus we have finally