ACTION OF A FIELD ON A MAGNETIC SHELL. 329 



Hence, 'the action of a magnetic pole on an element of the edge 

 of a shell is proportional to the magnetic strength of the shell, to the 

 sine of the angle which the element makes with right line joining the 

 pole to the element, and inversely as the square of the distance. 



345. We may further remark that dq represents the flow of 

 force cut by an element ds during the displacement aa 19 the flow 

 of force thus cut being counted positive or negative, according as the 

 displacement is to the left or right of the observer whose position has 

 been defined as above. From this follows the theorem : 



The work of magnetic forces during the displacement is equal to the 

 product of the shell by the sum of the flows of force cut by each of the 

 elements of the contour. 



346. Let us suppose that the external system reduces to a 

 magnetic mass equal to unity, and placed at the origin O of the 

 co-ordinates. Let C be the edge of the shell, and ds an element at 

 the distance r, at a point M whose co-ordinates are x, y, and z. 



Let A, fj,j and v be the cosines of the angles which the force d$ 

 makes with the axes. This force being perpendicular to the element 



ds and to the straight line OM along which is directed the action 

 proceeding from the point O, we have the ratios 



= , 

 \dx + pdy + vdz = ; 



from which we deduce 



ydz - zdy zdx xdz xdy -ydx rds sin a ' 

 a being the angle which the right line OM makes with the element 



* * 



As the force d$ is equal to ds sin a, its components d^ dy, and 



(zdx xdz) , 



= (xdy -ydx}. 



