Page:Philosophical magazine 21 series 4.djvu/311

applied to Electric Currents. 287 in unit of volume is

$E=C\mu\left(\alpha^{2}+\beta^{2}+\gamma^{2}\right),$

where C is a constant to be determined.

Let us take the case in which

Let

and let

then $$\phi_{1}$$ is the potential at any point due to the magnetic system $$m_1$$, and $$\phi_{2}$$ that due to the distribution of magnetism represented by $$m_2$$. The actual energy of all the vortices is

the integration being performed over all space.

This may be shown by integration by parts (see Green's 'Essay on Electricity,' p. 10) to be equal to

Or since it has been proved (Green's 'Essay,' p. 10) that

Now let the magnetic system $$m_1$$ remain at rest, and let $$m_2$$ be moved parallel to itself in the direction of $$x$$ through a space $$\delta x$$; then, since $$\phi_{1}$$ depends on $$m_{1}$$ only, it will remain as before, so that $$\phi_{1}m_{1}$$ will be constant; and since $$\phi_{2}$$ depends on $$m_2$$ only, the distribution of $$\phi_{2}$$ about $$m_2$$ will remain the same, so that $$\phi_{2}m_{2}$$ will be the same as before the change. The only part of E that will be altered is that depending on $$2\phi_{1}m_{2}$$, because $$\phi_{1}$$ becomes $$\phi_{1}+\tfrac{d\phi_{1}}{dx}\delta x$$ on account of the displacement. The variation of actual energy due to the displacement is therefore

But by equation (12), the work done by the mechanical forces on $$m_2$$ during the motion is

and since our hypothesis is a purely mechanical one, we must