Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/301

presented by ${\frac {dM}{dt}}\delta t$. This quantity, which is the increment of $M$ in the time $\delta t$, may be regarded as itself representing a magnetic system.

If we suppose that at the time $t$ a positive image of the system ${\frac {dM}{dt}}\delta t$ is formed on the negative side of the sheet, the magnetic action at any point on the positive side of the sheet due to this image will be equivalent to that due to the currents in the sheet excited by the change in $M$ during the first instant after the change, and the image will continue to be equivalent to the currents in the sheet, if, as soon as it is formed, it begins to move in the negative direction of $z$ with the constant velocity $R$.

If we suppose that in every successive element of the time an image of this kind is formed, and that as soon as it is formed it begins to move away from the sheet with velocity $R$, we shall obtain the conception of a trail of images, the last of which is in process of formation, while all the rest are moving like a rigid body away from the sheet with velocity $R$.

663.] If $P^{\prime }$ denotes any function whatever arising from the action of the magnetic system, we may find $P$, the corresponding function arising from the currents in the sheet, by the following process, which is merely the symbolical expression for the theory of the trail of images.

Let $P_{\tau }$ denote the value of $P$ (the function arising from the currents in the sheet) at the point ($x$, $y$, $z+R\tau$), and at the time $t-\tau$, and let $P_{\tau }^{\prime }$ denote the value of $P^{\prime }$ (the function arising from the magnetic system) at the point ($x$, $y$, $-(z+R\tau )$), and at the time $t-\tau$. Then

and equation (21) becomes

and we obtain by integrating with respect to $\tau$ from $\tau =0$ to $\tau =\infty$,

as the value of the function $P$, whence we obtain all the properties of the current sheet by differentiation, as in equations (3), (9), &c.

664.] As an example of the process here indicated, let us take the case of a single magnetic pole of strength unity, moving with uniform velocity in a straight line.