268 ELECTROSTATICS. [PT. II. CH. VI. 



Making use of the equations (10) and the condition p rs =p sr , 

 the energy W=^ 8 e s V 8 



becomes 

 (15) W = %p u eS 



or W is a homogeneous quadratic function of the charges of the 

 n conductors, the coefficients being the coefficients of potential p. 

 This form will be denoted by W e . If we differentiate partially by 

 any charge e s we get 

 dW 



or the potential of any conductor is the partial derivative of the 

 energy of the system as a quadratic function of the charges, by the 

 corresponding charge. 



139. Properties of the Coefficients. As the energy of an 

 electrified system is intrinsically positive, the values of the co- 

 efficients q and p must be such that the functions W v and W e 

 shall be positive for all possible values of the Vs and e's. We may 

 deduce certain properties of the coefficients from the elementary 

 properties of the tubes of force and equipotential surfaces. Let 

 one conductor K s receive a positive unit of charge, all the others 

 being uncharged, its potential is then_p ss , and the energy 



W e = ^p ss e 2 = ^p ss , 



and since this must be positive p ss is positive, or: Any coefficient 

 of potential with double suffix is positive. 



Any conductor K r completely enclosed by K s has the same 

 potential, so that for these two p rs = p ss . Any conductor JT r out- 

 side of K s has a potential of the same sign but of less absolute 

 value. For the charge of a conductor is proportional to the excess 

 of unit tubes issuing from it over that entering. An uncharged 

 conductor accordingly has as many leaving as entering. Accordingly 

 all tubes have one end on K 8 and the other at infinity. (Fig. 57), 

 and the potential of K r> p rs is consequently intermediate between 

 that of K s and that at infinity, 



/Y) > v\ > 



Pss Prs ^ v> 



All coefficients of potential are positive, and those with double 

 suffixes are not greater than those with single suffixes. 



