TERRESTRIAL MAGNETISM 317 



at each point on the surface. This would not be true if currents 

 existed passing through the surface. For in order that the 

 potential shall be definite the work done in carrying unit pole 

 from one point, A, to another point, B, must be the same by all 

 paths, or, what is equivalent, the work done round any closed 

 path ABA must be zero. But if a current C passes through 

 the area enclosed by ABA, the work done round the path is 4?rC, 

 and this work will be different for different paths enclosing 

 different amounts of current. Hence the work from A to B is 

 indefinite and in this case there is not a potential.* 



Let us first consider the distribution of magnetism within the 

 surface. Let R be the radius of the earth, r the distance from the 

 earth^s centre of a point P outside the earth, I the latitude, and X 

 the longitude of the point of intersection of r with the surface. 

 Then it can be shown (Gauss, I.e.) that the potential at P may be 

 put in the form of a converging series : 



v U 



v= B or 



where B , B r B 2 , &c., are certain functions of / and X known as 

 spherical harmonics. The general form of each of these functions 

 is known. f If the order is designated by the suffix, that of the 

 nth order B n contains %n + 1 arbitrary constants. 



At a very great distance from the earth the potential tends to a 



value V = ^ , where M is the earth's magnetic moment and 



is the angle its magnetic axis makes with r. The first term, 



B R 



;-, must, therefore, vanish, or B = ; and it is easily shown, 



too, that in the second term B x = a sin I + b cos / sin X + c cos /cos X, 

 since B^ 2 must be equal to M cos ; a, &, and c are constants 

 depending on the position of the magnetic axis. 



Next let us consider the distribution without the surface. It 

 can be proved that the potential within the surface may be put in 

 the form of a converging series : 



where again A , A v &c., are spherical harmonics, but with arbi- 

 trary constants other than those in B r B 2 , &c. V is the potential 

 at the earth's centre, since it is the value of V when r = 0. Being 



* It is now known that there are minute vertical currents in the air, but too 

 minute to be considered in a theory which does not pretend to give an exact 

 account. 



f Thomson and Tait's Natural Philosophy, vol. i. p. 187. 



