260 ELECTROSTATICS. [PT. II. CH. VI. 



AF vanishing except in the bodies D, in which it is equal to 

 p.' Consequently the charge e on the inside of the conductor 



is equal and opposite to the whole charge of the bodies D. This 

 explains Experiment VI. 



If the conductor is not connected to earth, the same result as 

 far as the inner charge follows. There is, however, an outer 

 charge, but as there is no force in the substance of the conductor, 

 the distribution of this charge is unaffected by what is within. 

 This explains Experiment II. 



133. Tubes of Force. If we apply the above reasoning 



FIG. 56a. 



to the space inclosed by any tube of force, which must end either 

 at infinity or at conducting surfaces, we have, since A V = 0, 



-///"*-//,+//*-* ' 



or the ends of any tube of force cover equal and opposite charges, 

 the flux of force through the tube or the number of unit tubes 

 contained in it being 4?r times the absolute value of the charge. 



134. Theorems on Sign of Electrification. By means 

 of the properties of tubes of force and of the Potential Function, 

 we may deduce a number of theorems on the sign of the electrifi- 

 cation on the surfaces of conductors. These theorems are taken 

 from the excellent Lemons sur VElectricite et le Magnetisme, by 

 P. Duhem. 



We shall call a distribution in which the sign of the surface- 

 density is everywhere the same, monogenic. If the density varies 

 in sign we shall call the distribution amphigenic. We suppose 

 that all the conductors are external to each other, and that in 

 each case there are no conductors present except those mentioned. 



