444 ELEMENTARY ACTIONS. 



The ratio of the actions d$ and d<$ of the pole on the elements 

 ds and ds' becomes then 



ds sin a 

 d( r'" r .r ds r a 



i r . rds' r a 

 ds sin a 



The actions of the corresponding elements being inversely as the 

 distances a and ', this will also be the case with the resultants. 

 This is the law resulting from experiment. The action of a pole on 

 an element of current is then expressed by 



dssina 



d<p = mkt - - . 



As all the forces are parallel and in the same direction, the action 

 of the pole on the unlimited rectilinear current is 



. (ds sin a ( 



= mkt mki 



J r* 



ds cos 8 



Measuring the length of the circuit from the point A, we have 



dB 



i0tan0, ds = a , 

 cos 2 # 



a? = r^ cos 2 \ 



from which follows 



ds cos 6 i 



r 2 = # C( 



and therefore 



IT 



/&' f + 2 2;^/ 



This force is by symmetry applied at the point A, and the action 

 of the rectilinear current on the pole is applied at the same point, 

 but in the opposite direction. 



