The coordinates of the image of the pole formed at the time are
, , ,
and if is the distance of this image from the point (, , ),
.
To obtain the potential due to the trail of images we have to calculate
.
If we write
,
,
the value of in this expression being found by making .
Differentiating this expression with respect to , and putting , we obtain the magnetic potential due to the trail of images,
.
By differentiating this expression with respect to or , we obtain the components parallel to or respectively of the magnetic force at any point, and by putting , , and in these expressions, we obtain the following values of the components of the force acting on the moving pole itself,
,
.
665.] In these expressions we must remember that the motion is supposed to have been going on for an infinite time before the time considered. Hence we must not take a positive quantity, for in that case the pole must have passed through the sheet within a finite time.
If we make , and negative, , and
,
or the pole as it approaches the sheet is repelled from it.
If we make , we find ,