﻿Electromagnetic Lines and Tubes. 715 



the "internal composition" of the vector R. An electro- 

 static tube in three-dimensional space is characterized by 

 Only two quantities at each point: its direction, and the 

 electric force e which is inversely proportional to its 

 sectional area at the point. An electromagnetic tube in 

 hvperspace has not two but three such characteristic 

 quantities. Not only will the same tube in two different 

 places differ in direction, and in the magnitude of the vector 

 function of position R, which latter is inversely proport ional 

 to the sectional area of the tube, but R =• VE 2 + ti 2 will 

 differ in the relative proportions of the collinear E and H 

 of which it is composed. The composition of R is adequately 

 expressed by a para meter « such that 



H (11) 



s/(E 2 + H 2 ) 



sin a measures at any point the relative amount of magnetic 

 force in R which cannot possibly be transformed away. We 

 have further from (6) 



. 2EH 2 (eh) 



sin la. = = — - — - . 



E 2 + H 2 R 2 



Thus also sin 2 a gives a direct measure at each point of the 

 departure of the field from orthogonality of e and h, as 

 expressed by the value relative to R 2 of the scalar-product 

 invariant 2 (eh). 



Differentiating (11) we have 



EdH-H(?E = R*J*, 



the substitution of which in (8 a'-d') gives 





3J* Uy 



d-- 



')■ 









1 = 



r 

 = 0, 





+ o/ 



■)- 



:0. 



J 



(12) 



These equations disclose that the variation of « in each of 

 the four coordinate directions is determined by the extent to 



