576, 577] Isotropic Dielectric 501 



Since the variations Sf , By, Sf are independent and may have any values 

 at all points in the field, their coefficients must vanish separately, and we 



must have 



a dZ 8F A 



r< + ;s -- -^- = 0, etc. 



G dy dz 



These are the equations which the principle of least action gives as the 

 equations of motion, and we see at once that they are simply the equations 

 of system (JB). 



Homogeneous medium. 



577. Let us next consider the solution of the systems of equations (A) 

 and (B) (of page 498) when p and K are constants throughout the medium, 

 and the medium contains no electric charges. From the first equation of 

 system (A), we have 



G 2 dP ~dy \G dt dz \Cdt 



and on substituting the values of ^ ~- and ^ -- from the last two equa- 



te at (j dt 



tions of system (B), this equation becomes 



dz fa 



___ 3Z\ 

 7/ 3^ 2 dx\dy dz ) ' 



Since the medium is supposed to be uncharged, we have 



02V" 



so that the last term may be replaced by + -^-y , and the equation becomes 



O 2 dt* 



By exactly similar analysis we can obtain the differential equation satis- 

 fied by F, Z, a, 0, and 7, and in each case this differential equation is found 

 to be identical with that satisfied by X. Thus the three components of 

 electric force and the three components of magnetic force all satisfy exactly 

 the same differential equation, namely 



-**> 



where a stands for Cf'SSjL This equation, for reasons which will be seen 

 from its solution, is known as the " equation of wave-propagation." 



