110-114] 



Properties of the Coefficients 



be intermediate between this potential and the potential at infinity, and 

 must therefore all be positive. Thus p n , p l2 , p 13 , ... p ln are all positive and 

 the first is the greatest. 



Next let us put F 1 = l, F a = F 8 = ... = 0, 



so that the charges are q n , g 12 , q l3 , ... q ln . 



The highest potential in the field is that of conductor 1. Thus lines of 

 force leave but do not enter conductor 1. The lines may either go to the 

 other conductors or to infinity. No lines can leave the other conductors. 

 Thus the charge on 1 must be positive, and the charges on 2, 3, ... all negative, 

 i.e., qu is positive and q l2) q l3) ... are all negative. Moreover the total strength 

 of the tubes arriving at infinity is <?n + #i 2 + #13 + ... +qm, so that this must 

 be positive. 



113. To sum up, we have seen that 



(i) All the coefficients of potential ( p n , p l2 , . . . ) are positive, 

 (ii) All the coefficients of capacity (q n , q^ y ... ) are positive, 

 (iii) All the coefficients of induction (q l2 , qi S) ... ) are negative, 

 and we have obtained the relations 



(Pn - Pw) is positive, 

 (qu + 12 + + qm) is positive. 



In limiting cases it is of course possible for any of the quantities which 

 have been described as always positive or always negative, to vanish. 



VALUES OF THE COEFFICIENTS IN SPECIAL CASES. 

 Electric Screening. 



114. The first case in which we shall consider the values of the 

 coefficients is that in which one conductor, say 1, is completely surrounded 

 by a second conductor 2. 



FIG. 38. 



If E l = 0, the conductor 2 becomes a closed conductor with no charge 

 inside, so that the potential in its interior is constant, and therefore V l = F 2 . 

 Putting E l = 0, the relation F x = F 2 gives the equation 



j. 



