Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/267

 Rh According to our hypothesis a, b, c are identical with μα, μβ, μγ respectively. We therefore obtain we may write equation (1), {{numb form | $$ \left. \begin{align} 4 \pi \mu u &= \frac{dJ}{dx} + \nabla^2 F. \\ 4 \pi \mu v &= \frac{dJ}{dy} + \nabla^2 G, \\ 4 \pi \mu w &= \frac{dJ}{dz} + \nabla^2 H. \end{align} \right\} $$|(4)|Similarly,}} If we write {{numb form | $$ \left. \begin{align} F' = \frac{1}{\mu} \iiint{\frac{u}{r} dx \, dy \, dz,} \\ G' = \frac{1}{\mu} \iiint{\frac{v}{r} dx \, dy \, dz,} \\ H' = \frac{1}{\mu} \iiint{\frac{r}{r} dx \, dy \, dz.} \end{align} \right\} $$|(5)}}where r is the distance of the given point from the element x y z, and the integrations are to be extended over all space, then {{numb form | $$ \left. \begin{align} F &= F' + \frac{d\chi}{dx}, \\ G &= G' + \frac{d\chi}{dy}, \\ H &= H' + \frac{d\chi}{dz}. \end{align} \right\} $$|(7)}} The quantity χ disappears from the equations (A), and it is not related to any physical phenomenon. If we suppose it to be zero everywhere, J will also be zero everywhere, and equations (5), omitting the accents, will give the true values of the components of $$\mathfrak{A}$$.