193-196] Mechanical Forces on Dielectrics 169 



The first term in this expression 



a d * v 







47T ( fo 9a;9* dy dydz ~fc W 



/9F 9 2 F , 9F9 2 F 9F9 2 F\) . 



~* * I a~~ 3~2~ + 3T~ s~T + ^~ ^~o~ r dxdydz 

 \ox dxoy dy dy 2 dz dydzj) 



zmR*} ^ + - (ynR* - zmR*} dS . . .(111). 



The second term in expression (110) for L may, in virtue of the relations 

 (109), be expressed in the form 



- ^ 2(j(ynR 2 - zmR*) dS, 



which is exactly cancelled by the first term in expression (111). We are 

 accordingly left with 



1 [[(, o/7 8F 8F ZV\ ( 9F 8F 



L = n(ynR 2 -zmR 2 )-2 (l-^- +-m^- + n-^-\ (y ^ -- *~- 

 8-TrJJ ( w V 9^' dy dz J V dz dyj 



= - {y (IP XZ + mP yz + nP zz ) - z (lP xy + mP yy + nP yz )} dS, 



verifying that the couples are also accounted for by the supposed system of 

 ether-stresses. 



195. Thus the stresses in the ether are identical with those already 

 found in Chapter vi, and these, as we have seen, may be supposed to 



D2 



consist of a tension - per unit area across the lines of force, and a 



07T 



R z 



pressure per unit area in directions perpendicular to the lines of force. 



MECHANICAL FORCES ON DIELECTRICS IN THE FIELD. 



196. Let us begin by considering a field in which there are no surface 

 charges, and no discontinuities in the structure of the dielectrics. We shall 

 afterwards be able to treat surface-charges and discontinuities as limiting 

 cases. 



Let us suppose that the mechanical forces on material bodies are 5, H, Z 

 per unit volume at any typical point a, y, z of this field. 



Let us displace the material bodies in the field in such a way that the 

 point x, y, z comes to the point x -f S#, y + By, z 4- &z. The work done in 

 the whole field will be 



ZSz)dxdydz (112), 



