﻿Electromagnetic Lines and Tubes. 711 



forms a complete equivalent of the original set (2) : 







E 



E 



9E 



a««, , a*. 



S H |S + (EM- HO (™= + ™*) = 0, . . 00 



3H 





E ?* - H |^ + i(E» + HO ( ^ 4 %g) = 0, . . (c') 

 oy ay ■ \3;c aw 



e|^_h|5 +!XE2 + ho(^+^) = 0. . . (<f) 



L(8) 



§ 3. A Flux Theorem for each of four Electromagnetic 

 Tubes. 



To interpret the equations (8) we observe that through 

 any point of the four-dimensional field not one only but 

 four electromagnetic lines may be constructed, each of which 

 is uniquely determined when the initial hyperplane in which 

 the field is specified is given. Starting from this hyperplane 

 the four axes' directions required to produce collinearity at a 

 given point are uniquely fixed. We can thus proceed to 

 a neighbouring point lying on any of these axes, and rotate 

 the system to give collinearity at this point, and by con- 

 tinuing this process obtain a continuous curved line in 

 hypers pace which at the origin coincides with the given 

 axis. AVe will call these curved lines the a?, y, z, /-electro- 

 magnetic lines respectively; bounded by a set of each of 

 them an x } y } z ) /-electromagnetic tube can be constructed 

 in the usual way. 



Consider now an infinitely thin a?- tube, which we may 

 take as having a rectangular cross-section at the origin. 



