388 Permanent Magnetism [OH. xi 



the latter integral being taken over a sphere at infinity. Now at infinity H 

 is of the order of (cf. 67), while la + mb -f nc vanishes, and dS is of 



the order of r 2 , so that the surface integral vanishes on passing to the limit 

 r = oc . Also the volume integral vanishes since 



da db dc _ A 



8~ T ~ -- P o~~ ^> 

 x oy oz 



and hence the theorem is proved. 



Replacing a, 6, c by their values, as given by equations (359), we find that 

 equation (384) becomes 



(I [( 



(a 2 + /3 2 + 7 2 ) dxdydz + 4?r (4a + 5/3 + Oy) dxdyde = . . .(385). 



Both integrals are taken through all space, but since A = B = C = 

 except in magnetic matter, we can regard the latter integral as being taken 

 only over the space occupied by magnetic matter. This integral is therefore 

 equal, by equation (383), to 2TF, so that equation (385) becomes 



f-7 2 ) dxdydz (386), 



the integral being taken through all space. 



This expression is exactly analogous to that which has been obtained for 

 the energy of an electrostatic svstem, namely, 



TF = ~- ff((X* + F 2 + Z*) dxdydz. 



And, as in the case of an electrostatic system, equation (386) may be 

 interpreted as meaning that the energy may be regarded as spread through 



the medium at a rate - (a 2 + ft 2 -f 7 2 ) per unit volume. 



O7T 



TERRESTRIAL MAGNETISM. 



452. The magnetism of the earth is very irregularly distributed and is 

 constantly changing. The simplest and roughest approximation of all to the 

 state of the earth's magnetism is obtained by regarding it as a bar magnet, 

 possessing two poles near to its surface, the position of these in 1906 being 

 as follows : 



North Pole 70 30' N., 97 40' W. 



South Pole 73 39' S., 146 15' E. 



Another approximation, which is better in many ways although still 

 very rough, is obtained by regarding the earth as a uniformly magnetised 

 sphere. 



