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 is given by an equation of the form

where e denotes some constant, and, which is analogous to the vector-potential in the electric case, is a circuital vector whose curl is the electric force, of the variable magnetic system. The value of, is therefore $$(1/4\pi)\text{ curl Pot }\mathbf{E}_2$$: so we have

This must be added to. Writing, for the sum, , we see that is the curl of , where

and the electric force, 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, and for the electric force the value $$-\mathbf{\dot{a}}_3$$, where

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