448-451] Energy of a Magnetic Field 387 



For uniform shells, </> may be taken outside the sign of integration, and 

 the equation becomes 



W ~2 dS = 



(cf. 423), where n is the number of lines of induction which cross the shell, 



This calculation measures the energy from a standard configuration in 

 which the magnetic materials are all scattered at infinity. To calculate 

 the energy measured from a standard configuration in which the shells have 

 already been constructed and are scattered at infinity as complete shells, we 

 use equation (378), namely 



W = 1 2 

 from which we obtain 



)O' P)O 



where ^ denotes the values -^ at the surface of any shell if the shell itself 

 dn dn 



is supposed annihilated. 



If all the shells are uniform, this may again be written 



(382), 



=-n ........................... , 



where n' is the number of tubes of force from the remaining shells, which 

 cross the shell of strength <. An example of this has already occurred in 

 424. 



ENERGY IN THE MEDIUM. 



451. We have seen that the energy of a magnetic field is given by 

 (cf. equation (381)), 



the integration being taken over all magnetic matter. As a preliminary to 

 transforming this into an integral taken through all space, we shall prove 

 that 



Q .................. (384), 



the integration being through all space. 



The integral on the left can be written as 



Man .an an\, 

 a dv +b ty +c te) dxd y dz > 



and this by Green's Theorem, may be transformed into 



( + i + S) ****** -// n ^ +mb+ MC > ds > 



252 



