Page:MortonCharge.djvu/5

(x y ζ) on the surface, are

$-{\frac {d\phi }{dx}},\ -{\frac {d\phi }{dy}},\ -{\frac {d\phi }{d\zeta }}$.

This force acts in the normal to the surface, and is proportional to the surface-density at (x y ζ), which we shall call σ'. Therefore

$\left({\frac {d\phi }{dx}},\ {\frac {d\phi }{dy}},\ {\frac {d\phi }{d\zeta }}\right)=-A\sigma '\left({\frac {dF}{dx}},\ {\frac {dF}{dy}},\ {\frac {dF}{d\zeta }}\right)/{\sqrt {\left({\frac {dF}{dx}}\right)^{2}+\left({\frac {dF}{dy}}\right)^{2}+\left({\frac {dF}{d\zeta }}\right)^{2}}}$

But

${\frac {d}{d\zeta }}=k{\frac {d}{dz}}$;

therefore, denoting differentiation with respect to x y z by subscripts 1 2 3,

$(\phi _{1},\ \phi _{2},\ \phi _{3})=-A\sigma '(F_{1},\ F_{2},\ F_{3})/{\sqrt {F_{1}^{2}+F_{2}^{2}+F_{3}^{2}}}$.

Now let σ be the surface-density at (x y z) on the moving conductor F(x y z)=C, then equating σ to the normal component of (f g h),

$\sigma ={\frac {fF_{1}+gF_{2}+hF_{3}}{\sqrt {F_{1}^{2}+F_{2}^{2}+F_{3}^{2}}}}$,

or putting in the values we have found for (f g h) in terms of φ

$\sigma =-{\frac {\phi _{1}F_{1}+\phi _{2}F_{2}+k^{2}\phi _{3}F_{3}}{\sqrt {F_{1}^{2}+F_{2}^{2}+F_{3}^{2}}}},$ $=A\sigma '{\sqrt {\frac {F_{1}^{2}+F_{2}^{2}+k^{2}F_{3}^{2}}{F_{1}^{2}+F_{2}^{2}+F_{3}^{2}}}}$.

Now the perpendicular from the origin on the tangent plane to F(x y z)=C at the point (x y z) is

$p={\frac {xF_{1}+yF_{2}+zF_{3}}{\sqrt {F_{1}^{2}+F_{2}^{2}+F_{3}^{2}}}}$,

and the perpendicular from the origin on the tangent plane to F(x, y, kζ)=C at (x y ζ) is

$p'={\frac {x{\frac {dF}{dx}}+y{\frac {dF}{dy}}+\zeta {\frac {dF}{d\zeta }}}{\sqrt {\left({\frac {dF}{dx}}\right)^{2}+\left({\frac {dF}{dy}}\right)^{2}+\left({\frac {dF}{d\zeta }}\right)^{2}}}}$