46 GENERAL THEOREMS. 



62. GAUSS' THEOREM. Given two electrical systems, one con- 

 sisting of the masses m v m^, m B , ..... and producing a potential V, 

 the other of masses m' v m' 2 , m' 3 , ..... and producing a potential V, 

 we shall have the identical equation : 



that is to say, that the sum of the elementary masses of the first 

 system, multiplied respectively by the value of the potential of the 

 second system at the point which they occupy, is equal to the corres- 

 ponding sum relative to the masses of the second system ; the sum- 

 mations must be replaced by integrals if the masses occupy a finite 

 extent. 



This proposition is an identity ; to see this, we need only replace 

 the potentials by their values as functions of the masses, and of the 

 distances. It then appears that each member of the equation is 

 equal to the sum of the products obtained by multiplying each 

 mass of one system by a mass of the other, and dividing the product 

 by the distance separating them. 



It will be sufficient to remark that the sums J5T* m'V and ^ mV' 

 denote the work which must be expended to bring them respectively 

 in presence of each other, from an infinite distance to the positions 

 which they occupy, the two systems ^m and jgm' being supposed 

 rigid; and in either case the work is evidently the same. 



When the two systems we are considering are conductors in 

 equilibrium, A v A 2 , A 3 , the potential is constant on each conductor, 

 and if we denote their total masses by M 15 M 2 , M 3 ..... , and M' 1$ 

 M' 2 , M' 3 ..... , the equation becomes 



M l V 'l + M 2 V/ 2 + M 3 V ' 3 ..... = M/ 1 V 1 + M/ 2 V 2 + M/ 3 V 



63. The following theorems may be considered as corollaries of 

 that of Gauss, as has been shown by M. Bertrand : 



I. If a conductor A in the neutral state, whether insulated or not, 

 is exposed to the action of an electrical mass m placed successively at two 

 points P and P' of the dielectric, the potential due to the induced charge 

 at A, will be the same at the point P' in the first case, as at the point P 

 in the second. 



Let us observe, in the first place, that any point of the dielectric 

 may always be considered as the centre of an infinitely small con- 

 ducting sphere, for the charge which this sphere acquires is constantly 

 zero, and always produces at its centre a potential equal to zero. 



