224 Methods for the Solution of Special Problems [CH. vin 



Putting V n = r n S n , the surface density of electrification on the sphere 

 is, by Coulomb's Law, 



- 



"rTT 



This result is indeed obvious from 258. on considering that the 

 surface electrification must give rise to the potential (180). 

 If n is different from zero, 



[[ 



where the integration is over any sphere, so that 



ff 



IIS n dS = (^=^0), 



and \\V n dS =0 (n^O) (181). 



Thus the total charge on the sphere 



and TJ was the potential of the original field at the centre of the sphere. 



262. Incidentally we may notice, as a consequence of (181), that the 

 mean value of a potential averaged over the surface of any sphere which 

 does not include any electric charge is equal to the potential at the 

 centre (cf. 50). 



If the sphere is introduced insulated, we superpose on to the field 

 already given, the field of a charge E spread uniformly over the surface 



Tjl 



of the sphere, and the potential of this field is . We obtain the par- 

 ticular case of an uncharged sphere by taking E V Q a, and the potential 

 of this field, namely K(~)> j us * annihilates the first term in expression 

 (180), to which it has to be added. 



It will easily be verified that, on taking the potential of the original 

 field to be V l = Fx, we arrive at the results already obtained in 217. 



