RELATION OF CHARGES TO POTENTIALS. 53 



The same reasoning applies to all the other conductors; thus 

 taking them in decreasing magnitude of potential, A 3 , for instance, 

 receives lines of force from A l and A 2 , and these latter may be con- 

 sidered as proceeding indirectly from A r On each of the insulated 

 conductors the negative charge is less than the positive of A 15 provided 

 that none of them forms a closed surface completely surrounding the 

 conductor A r 



68. RELATION OF CHARGES TO POTENTIALS. Denoting by A 1? 

 A 2 , . . . A n the conductors, let M 1? M 2 , . . . M n be the respective 

 charges, and V lf V 2 , . . . V n the corresponding potentials. 



Let us first suppose all the conductors insulated, in the neutral 

 state, and at zero potential. If we give unit positive charge to one of 

 them, A 15 its potential becomes a n , and those of the other conductors 

 are respectively a 21 , a sl , . . . a nl . If instead of unit charge, the 

 charge M x be given to A 15 all the potentials would be multiplied by 

 M! ; they would be a n M 15 a 21 M 15 . . . a nl M r Let us suppose that 

 Aj is discharged, and that we give the charge M 2 to A 2 , the 

 potentials will become a 12 M 2 , a 22 M 2 ..... a w2 M 2 ; and so forth. 

 Now the final state, when all the conductors receive their respective 

 charges simultaneously, is that in which all the states, obtained thus 

 in succession, are superposed ; to express then the potential of each 

 conductor we shall have an equation of the form 



and, therefore, n similar equations for the whole system. 



From this the following theorem is deduced. 



In any electrical system in equilibrium, the potentials of the several 

 conductors may be expressed as a linear function of the charges. 



Among the n 2 coefficients of the equation (i), the coefficient a pl> 

 expresses the potential of the conductor A p when it is charged with 

 unit electricity, all the others being in the neutral state ; a coefficient 

 such as a qp denotes the potential, which a conductor such as A q 

 would acquire in the same time. 



It is easy to see that these latter coefficients satisfy the relation 



(2) o. pq = a qp . 



For, let us consider the two successive states in which each of the 

 conductors A p and A q is alone charged with unit electricity, all the 

 others being in the neutral state ; applying Gauss' theorem (62) the 

 relation (2) is at once obtained. 



