Properties of the Electric Field, 157 



The rate of increase in the number of tubes parallel to x is 

 thus 



which may be written as 



d , n d t* i \ ( d f dg dh\ 



df dg dh _ 

 dx dy dz ~~ ™ 



p being the density of the free electricity. 



Hence df/dt, the rate of increase in the number of tubes 

 parallel to a?, may be written as 



dt = Jy [gu - fv) ~Iz [wf - uh) - up; 

 and similarly 



^ , . df da dh ,, 



But since -^-4- up, ~ + vp, -=- +wp are the components of 



the current parallel to x, y, z, we have, if a, /3, 7 are the 

 components of the magnetic force, 



A (df x \ dy d/3 

 Hdt +Up ) = dy-dz-> 



dy 



. (dh \ d/3 da 



ilr {dt +W P) = cL*-d-y 



Hence we may regard the moving tubes of electrostatic 

 induction as producing a magnetic force whose components 

 a, /3, 7 parallel to the axes of x, y, z are given by 



a = 47T (hv — gw)) ^ 



f3 = ±7r(fic-h U ), [ (1) 



y = ±7r(gu-fv). J 



In other words, a moving tube of electrostatic induction 

 may be regarded as producing a magnetic force at right 



