If we now express these equations in terms of and , where
, ,
(8)
they become
a = y co r 2 - / , (9)
Equation (10) is satisfied if we assume any arbitrary function of and . and make
,
.
(11)
(12)
Substituting these values in equation (9), it becomes
.
(13)
Dividing by , and restoring the coordinates and , this becomes
.
(14)
This is the fundamental equation of the theory, and expresses the relation between the function, , and the component, , of the magnetic force resolved normal to the disk.
Let< 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 .
At the positive surface of the disk
.
(15)
Hence the first member of equation (14) becomes
.
(16)
But since satisfies Laplace s equation at all points external to the disk,
,
(17)
and equation (14) becomes
.
(18)
Again, since is the potential due to the distribution , the potential due to the distribution , or , will be . From this we obtain for the magnetic potential due to the currents in the disk,