96 Systems of Conductors [CH. iv 



110. It is worth noticing that the coefficients of potential, capacity and 

 induction can be expressed as differential coefficients of the energy ; thus 



and so on. 



The last two equations give independent proofs of the relations 



Prs=Psr, q r s = q sr . 



PROPERTIES OF THE COEFFICIENTS. 



111. A certain number of properties can be deduced at once from the 

 fact that the energy must always be positive. For instance since the value 

 of W given by equation (38) is positive for all values of E lt E z , ... E n , it 

 follows at once that 



Pn,pz<L,pM, are positive, 



that Pup^ Pi*? is positive, that 



is positive 



and so on. Similarly from equation (39), it follows that 



qn, q&> #33, ... are positive, 

 and there are other relations similar to those above. 



112. More valuable properties can, however, be obtained from a con 

 sideration of the distribution of the lines of force in the field. 



Let us first consider the field when 



The potentials are Vi^pn* V> 2 =p l2 ,etc. 



Since conductors 2, 3, ... are uncharged, their potentials must be inter- 

 mediate between the highest and lowest potentials in the field. Thus the 

 potential of 1 must be either the highest or the lowest in the field, the other 

 extreme potential being at infinity. It is impossible for the potential of 1 

 to be the lowest in the field ; for if it were, lines of force would enter in at 

 every point, and its charge would be negative. Thus the highest potential 

 in the field must be that of conductor 1, and the other potentials must all 



