RECIPROCAL ACTION OF A POLE AND A CURRENT. 447 



so that the moment M 2 of the actions exerted by the pole on any 

 arc AB has the value 



(4) M^ = I(cosy a -cosy & ). 



If the circuit is closed, this moment is null, and as the direction 

 of the z axis has been arbitrarily chosen, we see that the action of a 

 pole on a closed current passes through the pole. Conversely, the action 

 of a closed current on a pole also passes through the pole. 



465. Instead of following the course taken, and of proving that 

 Biot and Savart's law is satisfied by an action which is inversely as 

 the square of the distance, we might have pursued a perhaps more 

 rigorous course, and have admitted as an experimental fact that the 

 action of a closed current on a pole passes through the pole. 



The moment of the action of a pole on the element ds in 

 reference to the z axis will then be 



and the moment relative to an arc AB 



/B fA 



M,= -I ry(r)<Jcosy = I ^ 2 /(r 

 JA JB 



Integrating this expression by parts, we get 



If the current is closed, the first term of the second member is 

 null. As the moment must be null, whatever be the shape of the 

 circuit traversed by the current, the second term must be identical 

 with zero ; that is to say, the product r 2 f(r) must be a constant, and 

 therefore the force must be inversely as the square of the distance. 



If the arc, without being closed, terminates at two points, A and 

 B of a right line, about which it can turn, the couple of rotation 

 will not in general be zero. 



Supposing, for instance, that the points A and B are situate on 

 the same right line which passes through the pole, and on the same 

 side of the pole, the moment of the forces in reference to this 

 axis is null, and the current will not acquire any moment of rotation 

 about the right line. 



