328 CONSTITUTION OF MAGNETS. 



the plane Fds for that of yz, and the direction of the force F as 

 axis of y (Fig. 81). Let a be the angle which the element ds, 

 calculated in the positive direction, makes with the force F. The 

 projection of the parallelogram abb^ on the plane of the xz per- 

 pendicular to the force F is a new parallelogram ab'b'^. We may 

 consider this latter as having for base ab' = ds sin a, and for height ac 

 that is to say, the abscissa of the point a lt or the projection e of the 

 displacement aa l on the axis ax perpendicular to the plane fds. We 

 have then 



(25) dq = ds sin a x e. 



The corresponding work is 



?Fds sin axe. 



This work is the same as if the element ds were subjected to the 



action of a force 



sin a 



parallel to the axis of x that is, perpendicular to the plane Fds. 

 We are thus led to this important theorem : 



The action of a magnetic field on a shell is equivalent to that of a 

 system of forces applied at the different elements of the edge. 



The force, which we must suppose applied at each element, is 

 perpendicular to the plane which passes through the element, and to 

 the direction of the field, and is on the left of an observer placed in 

 the element along the positive direction, and looking at the direction 

 of the force F. 



344. If we consider these as real actions, we may enunciate 

 the following theorem : 



The action of a magnetic field on an element of the edge of a 

 shell) is equal to the product of the magnetic strength of the shell, by the 

 force of the field, the length of the element, and the sine of the included 

 angle in other words, by the surface dA. = Yds sin a of the parallelo- 

 gram constructed on the force F and the element ds. 



We have then simply 



As a particular case, if the magnetic system is reduced to a single 

 mass m at a distance r from the element ds, the force F is equal 



to , and the elementary action becomes 



r* 



d<b = < ds sin a . 



