Page:MortonCharge.djvu/8

But by a well-known transformation, when we take the integral through all space, we have

$\int (F{\frac {d^{2}\phi }{dx\ dz}}+G{\frac {d^{2}\phi }{dy\ dz}}+H{\frac {d^{2}\phi }{dz^{2}}})d\tau$

$=\int {\frac {d\phi }{dz}}({\frac {dF}{dx}}+{\frac {dG}{dy}}+{\frac {dH}{dz}})d\tau$

$=0$ since (F G H) is circuital.

$\therefore$ the expression for this part of the energy reduces to

$-u(1-k^{2})\int H{\frac {d^{2}\phi }{dz^{2}}}d\tau =-{\frac {u^{2}}{V^{2}}}\int H{\frac {d^{2}\phi }{dz^{2}}}d\tau$.

Therefore if the velocity u ceases to be negligible in comparison with V, we have a correction of the second order in the ratio ${\frac {u}{V}}$ in addition to the expression involving the convection-current simply. It also appears from the above that the force on the moving charge cannot, unless this term be neglected, be expressed in terms of the magnetic intensity at the charge, but will depend on the entire field.