RECIPROCAL ACTION OF TWO SHELLS. 



33* 



which, expressing the forces Z and Y as a function of the co-ordinates 

 x, y and z of the point M where the element ds is situate, gives 



(27) 



>' ds'( 



Ix-xdy 



zdx - xdz 



We may also consider the action of the edge C on the element 

 ds' t as the resultant of the direct actions which each of the ele- 

 ments ds would exert on the element ds'. The only condition 

 imposed on this elementary action is, that the integral of the 

 partial components extended to the edge C shall reproduce the 

 preceding expressions. 



348. In accordance with this, the simplest solution for the action 

 of ds on ds' is a force/, the components of which parallel to the axes 

 f-x>fy>fz are > representing by a the product 



(28) 



xdy-yd X= _ ( ,y_ 

 r 6 r 3 \x 



xdz - zdx 



j / 



= -ad ( - 



349. To each of the components of the elementary action an 

 exact differential of the co-ordinates x, y, and z may be added, since 

 the integrals extended to the contour C will give values of zero for 

 these terms. There is therefore an infinite number of expressions 

 by which the actions of the elements of two magnetic shells may be 

 expressed. 



Let X, Y, and Z be functions of the co-ordinates x, y, and z ; 

 the problem will be satisfied if we take as components of the action 



(29) 



