1884.] equations of the electromagnetic field. 125 



Moreover the expression for the electromotive force which arises 

 from electromagnetic induction is the same on the two theories, 

 being according to Maxwell given by equations of the type 



p __dF ± d_/dN_dM\ . 



dt dt\dy dz J 



and according to Helmholtz in a magnetizable medium 



dU d (dN dM\ 

 F -~* dt + dt\dy~ dz) (J) ' 



The two equations are identical and may be written 



P=- d l 



dt 



Before however we can solve the problem completely we must 

 know the relation which exists between the values of F, G, and H, 

 or U, V, W and the current at any distant point. 



Let it, v, w be the components of the total electric current at 

 any point. Then since F, O, H depend on the action of the distant 

 current it is clear that there must be some relation between them 

 and the values of u, v, w. If we are considering the whole 

 electro-kinetic momentum round a closed curve we know the form 

 of the equation which expresses the connexion — for if i be the 

 current at any point distant r from the point considered and e the 

 angle between do- an element in the direction of i and that of an 

 element ds of this closed curve, then 



F x dx + Gf x dy + H x dz = /u, I — - — dsda ... 



Helmholtz starts from this and by means of some transformations 

 arrives at the equations 



(or in our notation) etc. 



7^^(1-^)^1-4^ (10), 



etc., 



d<$ 



k being a certain constant and -y defined by the equation 



du dv . dw 1 2 d<$> _ n n .. . 



dx dy dz 4<ir dt 



