Page:ThomsonMagnetic1889.djvu/2

2 -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

$\frac{d\xi_{1}}{dx}+\frac{d\eta_{1}}{dy}+\frac{d\zeta_{1}}{dz}=0,$|undefined

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

$\begin{array}{l} \dfrac{d\xi_{1}}{dx}+\dfrac{d\eta_{1}}{dy}+\dfrac{d\zeta_{1}}{dz}=\dfrac{d}{dx}\left(f\dfrac{du}{dx}+g\dfrac{du}{dy}+h\dfrac{du}{dz}\right)\\ \\ \qquad+\dfrac{d}{dy}\left(f\dfrac{dv}{dx}+g\dfrac{dv}{dy}+h\dfrac{dv}{dz}\right)+\dfrac{d}{dz}\left(f\dfrac{dw}{dx}+g\dfrac{dw}{dy}+h\dfrac{dw}{dz}\right), \end{array}$|undefined

we see that the currents will be closed if we add on to the components $$\xi_{1},\eta_{1},\zeta_{1}$$ the components $\xi_{0},\zeta_{0},\rho_{0}$ξ₀, η₀, ζ₀ [sic], 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