Page:Thomson1881.djvu/3

hence

$\left.{\begin{array}{ll}{\frac {df}{dt}}&={\frac {ep}{4\pi }}{\frac {d^{2}}{dx^{2}}}{\frac {1}{\rho }}\\\\{\frac {dg}{dt}}&={\frac {ep}{4\pi }}{\frac {d^{2}}{dx\ dy}}{\frac {1}{\rho }}\\\\{\frac {dh}{dt}}&={\frac {ep}{4\pi }}{\frac {d^{2}}{dx\ dz}}{\frac {1}{\rho }}\end{array}}\right\}$

(1)

Using Maxwell's notation, let F, G, H be the components of the vector-potential at any point; then, by 'Electricity and Magnetism,' § 616,

$F=\mu \int \int \int {\frac {u}{\rho '}}dx\ dy\ dz$, $G=\mu \int \int \int {\frac {v}{\rho '}}dx\ dy\ dz$, $H=\mu \int \int \int {\frac {w}{\rho '}}dx\ dy\ dz$,

where u, v, w are the components of the electric current through the element dx dy dz, and ρ' is the distance of that element from the point at which the values of F, G, H are required, μ is the coefficient of magnetic permeability. In the case under consideration,

$F=\mu \int \int \int {\frac {\frac {df}{dt}}{\rho '}}dx\ dy\ dz$;

substituting for ${\frac {df}{dt}}$ its value from equation (1), we get

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

with similar expressions for G and H.

Let us proceed to calculate the value of F at a point P.

Let 0 be the centre of the sphere; then OQ=ρ, PQ=ρ', OP = R,