Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/375

will be of the same kind as in the electric case; so that the induced magnetic force H′ is given by an equation of the form

$\mathbf {H} ^{\prime }=(1/c^{2})\mathbf {b} _{1}$,

where e denotes some constant, and b₁, which is analogous to the vector-potential in the electric case, is a circuital vector whose curl is the electric force E₁, of the variable magnetic system. The value of b₁, is therefore $(1/4\pi ){\text{ curl Pot }}\mathbf {E} _{2}$: so we have

$\mathbf {H} ^{\prime }=-{\frac {1}{4\pi c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}{\text{ curl Pot }}\mathbf {a} _{1}$.

This must be added to H₁. Writing H₂, for the sum, H + H′, we see that H₂ is the curl of a₂, where

$\mathbf {a} _{2}=\mathbf {a} _{1}-{\frac {1}{4\pi c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}{\text{Pot }}\mathbf {a} _{1}$;

and the electric force E₂, will then be $-\mathbf {\dot {a}} _{2}$. This system is not, however, final; for we must now perform the process again with these improved values of the electric and magnetic forces and the vector-potential; and so we obtain for the magnetic force the value curl a₃, and for the electric force the value $-\mathbf {\dot {a}} _{3}$, where

${\begin{matrix}\mathbf {a} _{3}&=&\mathbf {a} _{1}-{\frac {1}{4\pi c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}{\text{Pot }}\mathbf {a} _{2}&\ \\\ &=&\mathbf {a} _{1}-{\frac {1}{4\pi c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}{\text{Pot }}\mathbf {a} _{1}&+{\frac {1}{(4\pi c^{2})^{2}}}{\frac {\partial ^{4}}{\partial t^{4}}}{\text{Pot Pot }}\mathbf {a} _{1}\end{matrix}}$

This process must again be repeated indefinitely; so finally we obtain for the magnetic force H the value curl a, and for the electric force E the value $-\mathbf {\dot {a}}$, where

$\mathbf {a} =\mathbf {a} _{1}-{\frac {1}{4\pi c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}{\text{Pot }}\mathbf {a} _{1}+{\frac {1}{(4\pi c^{2})^{2}}}{\frac {\partial ^{4}}{\partial t^{4}}}{\text{Pot Pot }}\mathbf {a} _{1}$

$-\ {\frac {1}{(4\pi c^{2})^{3}}}{\frac {\partial ^{6}}{\partial t^{6}}}{\text{Pot Pot Pot }}\mathbf {a} _{1}+$…

2 A 2