386 Permanent Magnetism [CH. xi 



On replacing //, by Idxdydz, we find for the energy of a system of 

 magnetised bodies 



the integration being taken throughout all magnetised matter. 



449. An alternative proof can be given of equations (380) and (381), 

 following the method of 106, in which we obtained the energy of a system 

 of electric charges. 



Out of the magnetic materials scattered at infinity, it will be possible to 

 construct n systems, each exactly similar as regards arrangement in space to 

 the final system, but of only one-nth the strength of the final system. If n 

 is made very great, it is easily seen that the work done in constructing a 



single system vanishes to the order of , so that in the limit when n is very 



great, the work done in constructing the series of n systems is infinitesimal. 

 Thus the energy of the final system may be regarded as the work done in 

 superposing this series of n systems. 



Let us suppose so many of the component systems to have been super- 

 posed, that the system in position is K times its final strength, where K 

 is a positive quantity less than unity. The potential of the field at any 

 point will be /cH. On bringing up a new system let us suppose that K is 

 increased to K, + die, so that the strength of the new system is d/c times that 

 of the final system. In bringing up the new system, we place a magnet of 

 die times the strength of a in a field of force of potential /cO, and so on with 

 the other magnets. Thus the work done is 



die . /cH (a) + dfc . /cl (b) + . . . , 

 and on integration of the work performed, we obtain 



a) 



agreeing with equation (380), and leading as before to equation (381). 



450. If the magnetic matter consists solely of normally magnetised 

 shells, we may replace equation (381) by 



where ds denotes thickness and dS an element of area of a shell. Replacing 

 Ids by <f), so that <f> is the strength of a shell, we have 



dn 



