ENERGY OF A MAGNETIC SHELL. 325 



The quantity in brackets (XA + Y/^ + Zv) represents the projection 

 F n of the force of the field on the perpendicular to the shell ; the 

 product F n ^S is the flow of force of the field corresponding to the 

 element dS ; this flow is positive when it traverses the shell from the 

 negative to the positive face, and negative when in the opposite 

 direction. Hence the integral of the second member simply ex- 

 presses the value of the flow limited to the edge, and therefore 

 is independent of the form of the surface to which it is attached. 

 Let Q be the value of this flow, the expression for the potential 

 energy of the shell is 



(22) W=-$Q. 



Consequently, the potential energy of a shell is equal to the product, 

 with the contrary sign, of the power of the shell by the floiv of force 

 which penetrates its negative surface. 



340. This result may be directly obtained. For the energy 

 of a mass m, in the field of a simple magnetic shell, is expressed by 



But the product ma* is the flow of force which starts from the 

 point in the angle w, and which therefore traverses the shell entering 

 by the positive surface. The flow dQ which enters by the negative 

 surface has the same value with the contrary sign - mu. We have 

 thus 



But the energy of a magnetic system in the field of the shell is 

 the sum of the energies of different masses; it is therefore the 

 product, taken with the opposite sign, of the magnetic strength < of 

 the shell by the sum of the flows of force which traverse it that is 

 to say, by the flow of force which starts from the system and enters 

 the shell by the negative face. 



341. If this system is a second shell S', the flow of force Q is 

 proportional to the magnetic strength <' of this second shell, and we 

 may write Q = M3*', the coefficient M being the flow of force which 

 the former shell would receive, if the power of the second were 

 equal to unity. The energy of the first shell, in the field of the 

 second, is therefore 



(23) W=-<M>'M. 



