554 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. 



257. Equations for Bodies in Motion. All the equations 

 of this chapter have been deduced on the supposition that all the 

 media were at rest. In deducing their extension to the case of 

 media in motion we shall follow the method of Hertz, as given in 

 the last and crowning paper of his remarkable researches*. 



We shall suppose the media to be moving at every point with 

 velocities v whose components at any point are a, /3, 7. The 

 medium is not supposed necessarily to be moving like a rigid 

 body it may be deformed in any manner. At the surface of 

 separation between two media, although the velocity may be 

 discontinuous its normal component must be continuous, in order 

 to preclude the occurrence of vacant spaces. The fundamental 

 assumption made by Hertz is that as the medium moves or is 

 distorted, the lines of force are carried by the medium so as to pass 

 through the same material points. That is, this would be the 

 effect of the motion if it were the only influence at work to change 

 the field. Besides this, we have the usual effects that appear in 

 bodies at rest. 



Let 3, ), 3 %> $R> $1, represent the field at any point at rest 

 with respect to the coordinate-axes. The total change in $ at a 

 point in motion will depend on several causes, the first being the 

 change that is instantaneously taking place at the fixed point 



through which the material point happens to be passing. This we 



d 



shall denote by ^- . Secondly the point is displaced to new parts 

 ot 



of the field where the forces are different. The sum of these two 

 parts we shall call 



, 



dt~~dt 8a; dt dydt dz dt~ 8* 8# By 7 a* 



If a small element normal to the X-axis of area dS were 

 displaced parallel to itself, the flux through it would vary as just 

 stated. But if the element rotates, it takes in new amounts of 



flux* At the start the flux through it was $dS, but as it turns, 



* ^ 



it acquires a projection normal to the F-axis at the rate ^-, con- 



sequently its flux in the positive direction decreases from this 



* "Ueber die Grundgleichungen der Elektrodynamik fiir bewegte Korper." 

 Wied. Ann. 41, p. 369, 1890 ; Trans, p. 241. 



