500 



Displacement Currents 



[OH. xvn 



576. Assuming that the magnetic coordinates f, ?;, f are connected with 

 the electric coordinates X, F, Z by equations (A), we have 



= _ = _ 



C dt ~dy dz ~ dt\dy dz 

 so that on integration we obtain 



except for a series of constants which may be avoided by assigning suitable 

 values to f, rj and f. Using equations (534), we have the potential energy 

 expressed as a function of f , 77 and f, and the kinetic energy expressed as a 

 function of j, 17 and , and may now proceed to find the equations of motion 

 by the principle of least action. 



We have 



- 



so that 



dxdydz 

 1 1 1 (S + 6877 + c) dxdydz, 



As in 545, we suppose the values of Sf, 877, 8f all to vanish at the 

 instants = and = r, so that the first term on the right hand disappears. 

 We have also 



C [[f^vft^ d&A irfi*% d ^\ rfiZv 9Sf\] , 



"fcj|jrw^S^ F (w"^ 



on substituting the values of jOJT, etc., from equations (534). The volume 

 integral may be transformed by Green's Theorem, and we obtain 



c 



Collecting terms, we find that 



fb dX dZ\z (c dY 



+ (n + -3- - o- ^ + 7^ + o -- 

 \C dz dxj \G dx 



5- 



dy J J 



..}dS. 



