160 Prof. J. J. Thomson on the 



case where we have any number of tubes moving with any 

 velocities. 



Let us suppose we have the tubes fi,ffi, h x moving with the 

 velocities u l9 r l7 w ly the tubes / 2 , g 2 , h 2 moving with the velo- 

 cities u 2 , v 2 , w 2 , and so on. Then the rate of increase in the 

 number of tubes in the element is 



This may be written as 



— 2 (gu-fv) -j^ X (fw-hu) - tup. 



Hence we see, as before, that the collection of tubes may be 

 regarded as producing a magnetic force whose components 

 a, f3, 7 are given by 



a = 4-7T % {hv —gw), \ 



fi = ±irt (fw-hu), y (5) 



7 = 4tt2 (gu-fv). ) 



The kinetic energy T per unit volume due to the motion of 

 these tubes is 



or 



tor[_{t(hv-gv,)}*+ \%{fw-hu)Y+ \X igu-fi>)\*]. 



Thus the momentum parallel to x of the tube with suffix (1) 

 dTjdui 



= 4tt{# 1 2 (gu-fv) — hi 2 (fw — hu) } 



Thus the components U, V, W of the momenta parallel to 

 the axes of a, y, z respectively are given by the equations 



U = ytg-pth, ) 



Y = *th-y$f, y (6) 



W=02f-*tg. j 



Thus, when we have a number of tubes moving about, the 

 resultant momentum at any point is perpendicular to both the 

 resultant magnetic force and the resultant electric displace- 

 ment, and is equal to the product of these two quantities into 

 the sine of the angle between them. 



