384 Permanent Magnetism [OH. xi 



If e is the angle between the two elements ds, ds', the direction of these 

 elements being taken to be that in which the integration takes place, then 



dx dx dy dy dz dz' 



T~ T-> + j -JTT + -j- T~> = cos 6 > 

 ds ds ds ds ds ds 



[[COS 6 , , , 



so that n = 1 1 - dsds . 



JJ r 



From the rule as to directions given on p. 376, it will be clear that if the 

 integration is taken in the same direction round both circuits, then the 

 direction in which the n lines cross the circuit will be that of the direction 

 of magnetisation of the shell. 



Clearly n is symmetrical as regards the two circuits s and ', so that we 

 have the important result : 



The number of tubes of induction crossing the circuit s from a shell of unit 

 strength bounded by the circuit s' is equal to the number of tubes of induction 

 crossing the circuit s' from a shell of unit strength bounded by the circuit s. 



Here we have arrived at a purely geometrical proof of the theorem 

 already obtained from dynamical principles in 424. 



ENERGY OF A MAGNETIC FIELD. 



447. Let a, b } c, ... n be a system of magnetised bodies, the magnetisation 

 of each being permanent, and let us suppose that the total magnetic field 

 arises solely from these bodies. Let us suppose that the potential H at any 

 point is regarded as the sum of the potentials due to the separate magnets. 

 Denoting these by fi , fl&, .*. fl n , we shall have 



f} = n a + n 6 + ...+n n . 



Let us denote the potential energy of magnet a, when placed in the field 

 of force of potential II by H (a) ; if placed in the field of force arising from 

 magnet b alone, by fl&(a), etc. 



Let us imagine that we construct the magnetic field by bringing up the 

 magnets a, b, c, ... n in this order, from infinity to their final positions. 



We do no work in bringing magnet a into position, for there are no 

 forces against which work can be done. After the operation of placing a in 

 position, the potential of the field is H a . The operation of bringing magnet 

 a from infinity has of course been simply that of moving a field of force of 

 potential fl a from infinity, where this same field of force had previously 

 existed. 



On bringing up magnet b, the work done is that of placing magnet b in 

 a field of force of potential O a . The work done is accordingly fl a (6). 



The work done in bringing up magnet c is that of placing magnet c in a 

 field of force of potential fl a + fl&. It is therefore H a (c) + H& (c). 



