372 Permanent Magnetism [OH. xi 



429. The force from distribution III can be made to vanish by taking 

 the length of the cavity to be very great in comparison with the linear 

 dimensions of its ends. For the ends, of the cavity may then be treated as 

 points, and the force exerted by either end upon a unit pole placed at the 

 centre of the cavity will be 



57 



and this will vanish if S is small compared with Z 2 . The resultant force will 

 therefore arise solely from distributions I and II. 



The force arising from distribution II may be regarded as the force 

 arising from a distribution of volume-density 



94 dB 8(7 

 v dy dz 



spread throughout the whole of the magnetised matter, regardless of the 

 existence of the cavity, together with a distribution of volume-density 



dx dy dz, 



spread through the space occupied by the cavity. The force from this 

 latter distribution vanishes in the limit when the size of the cavity is 

 infinitesimal, so that the force from distribution II may be regarded as 

 that from a volume-density 



spread through all the original magnetised matter. 



We have now arrived at a force which is independent of the shape, size 

 and position of the cavity, provided only that these satisfy the conditions 

 which have already been laid down. This force we define to be the magnetic 

 force, at the point under discussion, inside the magnetised body. 



430. In the notation of 416, the force which has just been defined is 

 due to a distribution of surface-density <r, and a distribution of volume- 

 density p throughout the whole magnetised matter. The potential of these 

 distributions is 



ff 



dS + dxdydz, 



or HQ if we regard this as defined by equation (348). Thus, with this 

 meaning assigned to f! Qt the components of force at a point Q inside a 

 magnetic body will be 



dx ' dy ' dz 



