﻿Faraday-Tube Theory of Electro-Magnetism. 605 



The linear density will remain fjud, so that the velocity of 

 propagation along the tube will be \l c 2 — v 2 . Since the tube 

 itself is in motion with velocity v in a perpendicular direction, 

 the propagation of the disturbance in space will be with 



v 



velocity c in a direction making an angle sin -1 - with the 



tube. When v = c the disturbance will not be propagated at 

 all along the tube, which will lie in the wave-front ; and the 

 traction (E+/i,VqYqD) will vanish. 



9. To take into account a general velocity of the tube in 

 the direction of its length, let us restrict ourselves to plane- 

 polarized radiation. We shall take the A'-axis in the direction 

 of propagation, and the y-axis in that of the disturbance. 

 Since we are dealing only with transverse vibrations, the 

 velocity of the tubes in the direction of the ray will be 

 constant from point to point along a tube. Let u be this 

 cr-component of velocity. Also let (.i*, y) be the coordinates 

 of a point on some particular tube at time t, so that y is a 

 function of x and t. Then the whole ^/-component of velocity 

 of the point will be 



1=1= !+«!• ■•",.;■ C«> 



It is obvious that the shearing motion perpendicular to the 

 .r-axis of the tubes in their vibration will not affect the 

 number of tubes per unit area passing through a plane normal 

 to the #-axis. Thus the quantity d Xi the ^-component of 

 electric displacement, will be constant at a point on the 

 tube, or 







+ u$-\d x =0. 



Also, if d y 



is the y-coir 



ponent, we shall have 





dy 



oy 





d x 



- *' 



ad thus 



d y : 



-^d 



-■dx** 





d 2 : 





The momentum per unit length along the tube is 





\d { B = 



^VdjVqd 







=fi(q.d— d.qdj. 



(43) 



