272 ELEMENTARY LESSONS ON [CHAP. v. 



But the potential at P of the magnet whose pole is ///, will be 



v m f - J 

 V f-1 r. 2 / 



r 2 / I I \ 



fo - [L- - ) 



t COS /3 V ? 1 r 2 / 



but - - - which we may write - ^ - 



because r and r^ may be made as nearly equal as we please. 

 And since r - r 2 t cos /3 



r 1 / t cos p \ 

 v MI - I 2 1 



or the potential due to the element of the shel the strength 

 of the shell x the solid-angle subtended by the element of the 

 shell. Hence, if V be the sum of all the values of v for all the 

 different elements, and if w be the whole solid-angle (the sum 

 of all the small solid-angles such as c6), 



V P = * 



or, the potential due to a magnetic shell at a point is equal to 

 the strength of the shell multiplied by the solid -angle subtended 

 by the whole of the shell at that point. 



Hence wz represents the work that would have to be 

 done on or by a unit -pole, to bring it up from an 

 infinite distance to the point P, where the shell subtends 

 the solid-angle w. At a point Q where the solid-angle 

 subtended by the shell is different, the potential will be 

 different, the difference of potential between P and Q 



If a magnet-pole whose strength is m were brought 

 up to P, m times the work would have to be done, or 

 the mutual potential would be = mw. 



316. Potential of a Magnet-pole on a Shell. 

 It is evident that if the shell of strength i is to be 

 placed where it subtends a solid-angle w at the pole ;;/, 

 it would require the expenditure of the same amount of 

 work to bring up the shell from an infinite distance 

 on the one hand, as to bring up the magnet-pole from 



