Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/305

If we now express these equations in terms of $r$ and $\theta$, where

they become

a = y co r 2 - / , (9)

Equation (10) is satisfied if we assume any arbitrary function $\chi$ of $r$ and $\theta$. and make

$\phi ={\frac {d\chi }{d\theta }}$, $\psi =\sigma r{\frac {d\chi }{dr}}$.	(11)
	(12)

Substituting these values in equation (9), it becomes

Dividing by $\sigma r^{2}$, and restoring the coordinates $x$ and $y$, this becomes

This is the fundamental equation of the theory, and expresses the relation between the function, $\chi$, and the component, $\gamma$, of the magnetic force resolved normal to the disk.

Let< $Q$ be the potential, at any point on the positive side of the disk, due to imaginary matter distributed over the disk with the surface-density $\chi$.

At the positive surface of the disk

Hence the first member of equation (14) becomes

But since $Q$ satisfies Laplace s equation at all points external to the disk,

and equation (14) becomes

Again, since $Q$ is the potential due to the distribution $\chi$, the potential due to the distribution $\phi$, or ${\frac {d\chi }{d\theta }}$, will be ${\frac {dQ}{d\theta }}$. From this we obtain for the magnetic potential due to the currents in the disk,