152 Intelligence and Miscellaneous Articles. 



or, if the simultaneous modification of the charges and potentials is 

 infinitesimal, 



SMSV=SVSM (2') 



whence this theorem, a consequence of Gauss's principle explained 

 according to § I. : — 



In a system of fixed conductors, in which two distinct systems of 

 equilibrium are considered,, the sum of the products of the initial charge 

 of each conductor and the variation of its potential from one state to 

 the other is equal to the sum of the products of the initial potential and 

 the variation of the charge. 



III. When conductors, maintained at constant potentials, are left 

 to their mutual actions, the energy of the system tends towards a 

 maximum. 



Clerk Maxwell has demonstrated this theorem* by means of the 

 linear equations which exist between the potentials and the charges. 

 The following process is more direct and speedy. 



Let us suppose, at the beginning, each conductor A,, A 2 , . . . 

 insulated, and impress on the system an infinitely small deformation j 

 the charges M r M 2 , ... do not change ; there are for the respective 

 potentials the falls oV,, oV 2 , . . . ; the loss of energy, equal to the 

 external work accomplished, is 



aw=— isMffv". 



Now, the conductors being fixed, let us connect them to constant 

 batteries, in order to restore the potentials to their original values. 

 This restoration of the potentials cannot be effected without addi- 

 tional charges £M X , BI 2 , . . . , regulated by the relation (2'). The 

 positive variation of the initial energy will therefore be 



8'\Y=+i2MaV. 



We hence conclude that 



3\y + nV=0, 

 and 



a'W-oW=2M8V. 



Therefore (1) the work accomplished, during the displacement, 

 by the electrical forces is equal to the augmentation of energy of 

 the system ; (2) the energy furnished by the sources is equal to 

 twice one or the other of those quantities, and is expended exactly, 

 half in mechanical work, half in electrical work or potential energy. 



According to the equality (2), the preceding theorem applies to 

 a finite deformation; but, for its application to the theory of elec- 

 trometers, there is, as is known, only occasion to consider an ele- 

 mentary modification. — Comptes Rendus de V Academic des Sciences, 

 Jan. 9, 1882, t. xciv. pp. 74-76. 



• ' Electricity and Magnetism/ vol. i. p. 90. 



