573-575] Isotropic Dielectric 499 



Of these two systems of equations the former may be regarded as giving 

 the magnetic field in terms of the changes in the electric field, while the 

 latter gives the electric field in terms of the changes in the magnetic field. 

 We notice that, except for a difference of sign, the two systems of equations 

 are exactly symmetrical. Thus in an isotropic non-conducting medium 

 magnetic and electric phenomena play exactly similar parts. 



The two systems of equations may be regarded as expressing two facts for 

 which we have confirmation, although indirect, from experiment. System (A) 

 expresses, as we have seen, that the line-integral of magnetic force round a 

 circuit is equal to the rate of change (measured with proper sign) of the 

 surface integral of the polarisation, this rate of change being equal to 4-n- 

 times the total current through the circuit, while similarly system (B) ex- 

 presses that the line-integral of electric force round a circuit is equal to the 

 rate of change of the surface integral of the magnetic induction. These two 

 facts, however, are not independent of one another : the latter can be shewn 

 to follow from the former if we assume the whole mechanism of the system 

 to be dynamical in its nature. This might be suspected from what has 

 already been seen in 556, but we shall verify it before proceeding further. 



575. Assuming the whole field to form a dynamical system, the kinetic 

 and potential energies are given by 



W = -Jt- jjJK (X* + F 2 + Z*} dxdydz. 



The quantities a, /3, 7 must fundamentally be of the nature of velocities : 

 let us denote them by , 77, , so that f , 77, f are positional coordinates, and we 

 have 



giving the kinetic energy as a quadratic function of the velocities. The 

 motion can be obtained from the principle of least action, expressed by the 

 equation 



- W)dt=Q. 



We cannot, however, obtain the equations of motion until we know the 

 relation between the coordinates f, 77, f which enter in the kinetic energy, 

 and the coordinates X, Y, Z which enter in the potential energy. We shall 

 find that if we suppose this relation to be that expressed by equations (A), 

 then equations (B) will be obtained as the equations of motion. 



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