Page:ThomsonMagnetic1889.djvu/2

placement-currents are due entirely to variations in the electric displacement $(f,g,h)$ caused by the motion of the sphere and the medium, the components $\xi _{1},\eta _{1},\zeta _{1}$ of these currents would be given by

${\begin{array}{c}\xi _{1}=u{\dfrac {df}{dx}}+v{\dfrac {df}{dy}}+\left(w-w_{0}\right){\dfrac {df}{dz}},\\\\\eta _{1}=u{\dfrac {dg}{dx}}+v{\dfrac {dg}{dy}}+\left(w-w_{0}\right){\dfrac {dg}{dz}},\\\\\zeta _{1}=u{\dfrac {dh}{dx}}+v{\dfrac {dh}{dy}}+\left(w-w_{0}\right){\dfrac {dh}{dz}}.\end{array}}$

These values, however, do not satisfy the equation

${\frac {d\xi _{1}}{dx}}+{\frac {d\eta _{1}}{dy}}+{\frac {d\zeta _{1}}{dz}}=0,$

unless the dielectric is moving uniformly; so that, if the circuits are to be closed, the motion of the medium must produce some other effect analogous to a current.

Since

${\begin{array}{l}{\dfrac {d\xi _{1}}{dx}}+{\dfrac {d\eta _{1}}{dy}}+{\dfrac {d\zeta _{1}}{dz}}={\dfrac {d}{dx}}\left(f{\dfrac {du}{dx}}+g{\dfrac {du}{dy}}+h{\dfrac {du}{dz}}\right)\\\\\qquad +{\dfrac {d}{dy}}\left(f{\dfrac {dv}{dx}}+g{\dfrac {dv}{dy}}+h{\dfrac {dv}{dz}}\right)+{\dfrac {d}{dz}}\left(f{\dfrac {dw}{dx}}+g{\dfrac {dw}{dy}}+h{\dfrac {dw}{dz}}\right),\end{array}}$

we see that the currents will be closed if we add on to the components $\xi _{1},\eta _{1},\zeta _{1}$ the components $\xi _{0},\zeta _{0},\rho _{0}$, where

${\begin{array}{c}-\xi _{0}=f{\dfrac {du}{dx}}+g{\dfrac {du}{dy}}+h{\dfrac {du}{dz}},\\\\-\eta _{0}=f{\dfrac {dv}{dx}}+g{\dfrac {dv}{dy}}+h{\dfrac {dv}{dz}},\\\\-\zeta _{0}=f{\dfrac {dw}{dx}}+g{\dfrac {dw}{dy}}+h{\dfrac {dw}{dz}}.\end{array}}$

The medium is assumed to be incompressible, so that

${\frac {du}{dx}}+{\frac {dv}{dy}}+{\frac {dw}{dz}}=0.$

Hence the components of the total effective currents are