446-448] Energy of a Magnetic Field 385 



Continuing this process we find that the total work done, W, is given by 



a (d) + H 6 (d) + n c (d) + etc. 



If, however, the magnets had been brought up in the reverse order, we 

 should have had 



w= n 6 (a) + n c (a) + n d (> + ... + n n (a) 



+ etc. 

 so that by addition of these two values for W, we have 



n & (a) + n c (a) + fl d ( tt ) + ... + n n (a) 



+ etc. 



The first line is equal to H (a) except for the absence of the term H a (a), 

 and so on for the other lines. Thus we have 



a) - SH a (a) .......... .............. (378). 



The quantity H a (a), the potential energy of the magnet a in its own 

 field of force, is purely a constant of the magnet a, being entirely independent 

 of the properties or positions of the other magnets 6, c, d, Thus in 

 equation (378), we may regard the term 2H a (a) as a constant, and may 

 replace the equation by 



W =iSfl (a) + constant ..................... (379). 



448. If we take the magnets a, b, c, ... n to be the ultimate magnetic 

 particles, the values of n a (a), n & (b), ... etc. all vanish, and their sum also 

 vanishes. Thus equation (379) assumes the form 



F=2n(a) (380), 



where the standard configuration from which W is measured is one in which 

 the ultimate particles are scattered at infinity. The value of O (a) for a 

 single particle is (cf. 420), 



an an an 



'- h m ~ \-n 

 ox .dy dz 



j. 25 



