﻿Electromagnetic Lines and Tubes. 713 



This equation expresses the fact that the flux of the 

 invariant function VE^ + H* over the cross-section y^ i z l of 

 an infinitely thin a?- tube is constant throughout the whole 

 tube ; since any tube may be formed by the juxtaposition of 

 infinitely thin elementary tubes, the statement is true for a 

 tube of finite cross-section also. In this case the corre- 

 sponding cross-section may be defined as any continuous 

 surface at any point of which the yz plane corresponding to 

 the .I'-line passing through the point differs only infinitesi- 

 mally from being tangential *. 



On the rectangle y\Z x as base not only an «r-tube but also 

 an /-tube may be constructed, since the /- as well as tho 

 .r-lines are at right angles to yz. The same figure 2 with 

 00' representing the /-axis gives 



and equation (8 6) reduces to 



^ l (</W + W.t /l z 1 ) = 0, . . . (9b) 



so that the flux theorem applies to the /-tube also. 



Any infinitely thin tube constructed of either y- or ^-lines 

 will at the origin be perpendicular to the plant xl. Take 

 the cross-section at the origin to be rectangular, formed of 

 infinitely short x- and /-lines, .r 3 and l Y . We find as before 



dflj^ia*, 'bOyi = i a/i 

 ~dx xi ~dy ' "dl h ~dy ' 



d0*£_IJ^i B^z _ 1 d/i 



-dx x, "dz ' -d! ~ I, *z ' 



* In the general field the #-lines are twisted in the yz plane (v. infra) 

 and a closed surface everywhere perpendicular to them cannot be 

 uniquely constructed; i.e., if we go from the origin always perpen- 

 dicular to the r-lines distances, first y Y and then Zi, we do not arrive at 

 the same point of the final a>line a^ will be reached by going the same 

 distances in the opposite order, mathematically the conditions of uncon- 

 ditional integrability will not be satisfied for the y l and z x displacements. 

 For the purpose of reckoning the flux this feature of the tubes is 

 immaterial, all the surfaces formed by joining up the points obtained by 

 displacements in any order will over an infinitesimal region only differ 

 from each other in area by second order quantities. The curvatures of 

 the .r-lines, and the second order displacements which they have under- 

 gone at O', Y and Z out of the hyperplane xyz, are also of negligible 

 effect. 



