Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/445

encloses a large number of molecules, but which is small compared with measurable quantities; and this equation may be written

${\bar {\rho \mathbf {u} }}={\frac {1}{V}}{\frac {d}{dt}}(VP)$.

Now, if Ṗ refers to differentiation at a fixed point of space (as opposed to a differentiation which accompanies the moving body), we have

$(d/dt)\mathbf {P} =\mathbf {\dot {P}} +(\mathbf {w.} \nabla )\mathbf {P}$,

and $(d/dt)V=V{\text{ div }}\mathbf {w}$; so that

${\begin{matrix}{\bar {\rho \mathbf {u} }}&=&{\bar {\mathbf {P} }}+(\mathbf {w.} \nabla )\mathbf {P} +{\text{div }}\mathbf {w.P} \\\ &=&{\bar {\mathbf {P} }}+{\text{curl }}[\mathbf {P.w} ]+{\text{div }}\mathbf {P.w} +(\mathbf {P.} \nabla )\mathbf {w} ,\end{matrix}}$

and therefore

${\bar {\rho \mathbf {u} }}+{\bar {\rho \mathbf {w} }}=\mathbf {\dot {P}} +{\text{curl }}[\mathbf {P.w} ]$.

This equation determines the part of ${\bar {\rho \mathbf {v} }}$ which arises from the dielectric molecules.

The general equations of the aether thus become, when the averaging process is performed,

${\begin{matrix}{\text{div }}\mathbf {\bar {d}} &=&4\pi c^{2}\rho _{1}-4\pi c^{2}{\text{div }}\mathbf {P} ,\qquad {\text{div }}\mathbf {\bar {h}} =0,\\{\text{curl }}\mathbf {\bar {d}} &=&-\mathbf {\bar {\dot {h}}} ,\\{\text{curl }}\mathbf {\bar {h}} &=&(1/c^{2})\mathbf {\bar {\dot {d}}} +4\pi \left\{{\begin{matrix}{\text{convection-current}}+{\text{conduction-current}}\\+\mathbf {\dot {P}} +{\text{curl }}[\mathbf {P.w} ]\end{matrix}}\right\}.\end{matrix}}$.

In order to assimilate those to the ordinary electromagnetic equations, we must evidently write

${\begin{matrix}\mathbf {\bar {d}} =\mathbf {E} ,&{\text{the electric force;}}\\(1/4\pi c^{2})\mathbf {E} +\mathbf {P} =\mathbf {D} ,&{\text{the electric induction;}}\\\mathbf {\bar {h}} =\mathbf {H} ,&{\text{the electric vector.}}\end{matrix}}$

The equations then become (writing ρ for ρ₁, as there is no longer any need to use the subscript),

${\begin{matrix}{\text{div }}\mathbf {D} &=&\rho ,\qquad \qquad -&{\text{curl }}\mathbf {E} &=&\mathbf {\dot {H}} ,\\{\text{div }}\mathbf {H} &=&0,\qquad \qquad &{\text{curl }}\mathbf {H} &=&4\pi \mathbf {S} ,\end{matrix}}$

where

S = conduction-current + convection-current + Ḋ + curl [P.w].