262 ELECTROSTATICS. [FT. II. CH. VI. 



On the line on the first conductor separating positive and negative 

 values of <r, the potential does not vary as we leave the surface in 

 the direction of the normal, or in other words the equipotential 

 surface of which the conductor forms a part has a sheet cutting 

 the conductor normally. This sheet is closed, containing the 

 other conductor with the monogenic charge. This sheet is dotted 

 in Fig. 57. The direction of the lines of force is shown by the 

 arrows. 



THEOREM IV. If the system consists of two bodies with 

 equal and opposite charges, the distribution on each is mono- 

 genic. For if not, it is evident on inspection of Fig. 57, that 

 if we draw a sufficiently large surface including both con- 

 ductors, the tubes of force will cross it everywhere in the 

 same direction (outward or inward). But the total outward 

 flux of force is equal to 4?r times the total charge within the 

 surface, which is zero, accordingly all the tubes of force must 

 issue from one conductor and end on the other. 



THEOREM V. A charge concentrated at a point produces a 

 monogenic charge on a conductor whose charge is of equal amount 

 and opposite sign. For this is a particular case of the preceding 

 theorem. 



THEOREM VI. A negative charge concentrated at a point 

 produces a monogenic distribution on a conductor with a positive 

 charge of greater absolute value. For the charged point may 

 be considered as the limit of a conductor with potential oo . 

 This is then the lowest value of the potential occurring. The 

 value at infinity, namely zero, is not the greatest value, for then 

 all the values occurring would be negative, but as we approach 

 infinity the value approached by the potential is M/r, 74 (7), 

 where M is the total charge of all the distributions, which is 

 here positive. Since positive values occur, the highest value 

 attained must be on the conductor, whose distribution is there- 

 fore monogenic. 



The remaining theorems are expressed in terms of known 

 potentials, instead of charges, of the conductors. 



THEOREM VII. If two conductors have potentials of the same 

 sign, the distribution is monogenic on the one whose potential 

 has the greatest absolute value, and the density has the same 

 sign as the potential. 



