676 Mr. G. H. Livens on the Mathematical 



and, further, if the polarization is induced according to 

 a linear isotropic law, 



fjuB = — grad (/> + 47rI . 

 Thus finally, /I \ 4tt div I 



^- Z 1 1 jA 47rdl " 

 div ( - orad d> ) = - 



is the characteristic equation satisfied by the potential </>. 

 In homogeneous media this is equivalent to 



V 2 </> = 47rdivl . 



The surface condition at a bounding interface, corre- 

 sponding to this equation, involves the continuity of 



B »=K-B-* +4rf 4 



It used to be assumed that a complete analogy exists 

 between the electrostatic and magnetostatic fields, the 

 electric force, complete electric displacement, and electric 

 polarization corresponding to the magnetic force, magnetic 

 induction, and polarization respectively ; but it has already 

 been pointed out that no such complete analogy exists. 

 With the suggested modification in the conception of the 

 permeability the analogy again presents itself as a pos- 

 sibility, but it is by no means perfect, although it is more 

 complete than is otherwise possible. There is, of course, 

 no reason why the two sets of relations should be analogous, 

 as the two cases are quite distinct, and the similarity in the 

 expressions for the energy is obtained only when the one is 

 treated as potential energy and the other as kinetic energy. 

 From another point of view it appears that a most unfortunate 

 mistake bas been Committed in starting with magnetism in a 

 statical theory with a potential function. The more correct 

 procedure is to begin with the magnetic induction as the true 

 sethereal magnetic force and then to separate out from this 

 vector the mechanically effective part, the magnetic force 

 proper, which is derived from a potential function under the 

 appropriate circumstances. By proceeding in the other way 

 we have been led to assume, among other things already 

 indicated in sufficient detail, that the magnetic force is the 

 proper vector to use in such expressions as the complete 

 electromagnetic force on a moving charge element, and 

 when we have then found that in ferromagnetic media 



