408 Dr. H. Bateman 



on some 



These equations are of a type familiar in hydrodynamics : 

 the first equation implies that X has the same value at the two 

 consecutive points (%,y,z, i), (# + v x St,y + VyBt, z + v z St, t + 8t); 

 consequently, i£ we regard v as the velocity of a particle of 

 electricity, the first two equations imply that a line of electric 

 force always consists of the same particles of electricity *. 



When we use vector notation the fundamental equations (2) 

 take the form 



rot H = - (^ + pv), div E = p, 



and the relation (4) signifies that the vectors E and H are 

 connected by the relation (EH) = 0: they are consequently 

 perpendicular to one another. A field in which this condition 

 is satisfied may be called special. 



§ 2. The Lines of Magnetic Force. — The theory of moving 

 lines of magnetic force may be developed along similar lines. 

 Let the equations of a line of magnetic force be 



Y(x, y, z, t) = const., T(#, y, z, t) = const., 



where V and T are uniform functions. Regarding V and T 

 as the rectangular coordinates of a point in a plane, we assume 

 that the flux of magnetic force through a closed curve C in 

 the a?, y, z space is represented by the area d(Y, T) of the 

 corresponding closed curve in the plane of V, T. The factor 

 is in this case taken to be unity because the flux of magnetic 

 force across the boundary of a closed surface is zero if there 

 is no free magnetism. Writing 



d(Y, T) = h x d(y, z) + h y d(z } x) + h z d(x, y) 



+ ce x d(x, t) + ce y d(y\ t) + ce z d(z, t) , 

 we have, as before, 



, _ arv,T) _.i5(v,T) i 



h *~ -b(y,z)> e '~'c a(*,t)'J- • • • (5) 



These equations give 



rote=-i|^, divh = 0, (eh)=0. . . (6) 

 c ot 



Fundamental Hypothesis. — We shall now assume that the 



* The flux across a closed circuit made up of particles of electricity 

 remains constant during the motion of these particles. 



