﻿Electromagnetic Lines and Tubes. 709 



constructing a line of force in an electrostatic field, and 

 the lines so traced out may be called "electromagnetic 

 lines." 



Close to the origin the line passing through it lies along 

 the #- direction satisfying equation (1) in the arbitrary 

 hyper plane in which the field is initially specified. This is 

 the case beouse the rotation of the axes in the plane zl does 

 not change the axes x and y. Thus the length dx of the line 

 lies in the original hyperplane. For its next infinitesimal 

 portion, however, the rotation to produce collinearity is in 

 general in a plane infinitesimally inclined to the original zl, 

 so that the line bends out of the hyperplane. It is thus in 

 general impossible to draw continuous electromagnetic lines 

 in a fixed hyperplane ; at all points, however, the tangents 

 to the lines passing through them may be drawn. 



§ 2. Transformation of the Electromagnetic Equations into 

 terms of the Lines. 



The axes xyzl having been chosen so that at a given 

 point 



e x = E, h x = H, e y — e z = h y = h z = 0, 



at a neighbouring point of the four-dimensional field e and h 

 will have the values 



~E + de x , de y , de 2 , H + dh x , dh y , dh z . 



AVe can determine the infinitesimal angles through which 

 the axes have to be rotated to produce collinearity again 

 by the following process. The rotations necessitated by 

 the variations de y , dh y are independent of de Xi de z , etc., and 

 we can treat them as existing separately. Suppose therefore 

 that the new field consists simply of E and H along x, and 

 de y and dh y along y. Rotate the axes through the angle d6 xy 

 in the plane of xy until the relation (1) is again obeyed. 

 Wo shall have for the components of e and h along the new 

 axes x ! y 



e x '= E, A X ' = H, 



e y - = — E dd xy + de y , h x y = — H dd sy +- dJi y , 



and, since they must satisfy the relation 



<?2'<V + h x Ji y = 0, 



