Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/308

276 Since $$p$$ is a homogeneous function of the degree $$- 1$$ in $$r$$ and $$a$$,

Since $$r$$ and $$a$$ are constant during the surface-integration,

But if $$P$$ is the potential due to a sheet of imaginary matter of surface-density $$\phi$$, and $$\Omega$$, the magnetic potential of the current-sheet, may he expressed in terms of $$P$$ in the form

671.] We may determine $$F$$, the $$x$$-component of the vector-potential, from the expression given in Art. 416, where $$\xi$$, $$\eta$$, $$\zeta$$ are the coordinates of the element $$dS$$, and $$l$$, $$m$$, $$n$$ are the direction-cosines of the normal.

Since the sheet is a sphere, the direction-cosines of the normal are $l=\frac{\xi}{a}$,$m=\frac{\eta}{a}$,$n=\frac{\zeta}{a}$.Butand

multiplying by $$\phi\, dS$$, and integrating over the surface of the sphere, we find