196] Mechanical Forces on Dielectrics 171 



The change in p is due to two causes. In the first place, the electrifica- 

 tion at x, y, z was originally at x Bx, y By, z Sz, so that Bp has as part 

 of its value 



Again, the element of volume dxdydz becomes changed by displacement 

 into an element 



d } ( d )f9 



5- (Bx)dx\- \dy-\- =- (By) dy\ \dz + =- (Bz)( 

 dx^ } ( c dy *),( dz ^ ' 



j j j /-i d&c dBy dBz\ 

 or dtfefydsr 1 + - + ^ + - 



V dx dy dz J 



so that, even if there were no motion of translation, an original charge 

 pdxdydz would after displacement occupy the volume given by expression 

 (115), and this would give an increase in p of amount 



/3 Bx dBy d Bz\ 

 \dx dy dz / ' 



Combining the two parts of Bp given by expressions (114) and (115), 

 we find 



The change in K is also due to two causes. In the first place the point 

 which in the displaced position is at x, y, z was originally at x Bx, y &y, 

 z $z. Hence as part of the value in $K we have 



dK, ZK * 3K * 



- ^ Sx ^ 8y - ^ Sz. 



dx dy dz 



Also, with the displacement, the density of the medium is changed, so 

 that its molecular structure is changed, and there is a corresponding change 

 in K. If we denote the density of the medium by T, and the increase in r 

 produced by the displacement by Sr, the increase in K due to this cause 

 will be 



and we know, as in equation (116), that 



., I'dSx 



ST T( -^ 



V dx 



We now have, as the total value of 



T -^ -~- 



V dx dy dz J 



dK , dK , dK ., 

 ^-cx^-oy^cz 

 dx dy dz 



dK/dBx dBy dBz\ 

 dr \dx dy dz J ' 



