168 Mr. Gr. H. Livens on the Mechanical 



4. On close inspection it will be seen that the analysis of 

 the preceding paragraph determines in reality something 

 quite different from that which is the ultimate object of 

 search. In calculating the mechanical forcive on the element 

 of the dielectric medium by the variational method, care- 

 must be exercised to determine the change in the energy of 

 the moving element of the medium, because it is this change- 

 which is brought about by the action of the mechanical 

 forces w T hen the constitution is maintained constant. But in 

 the argument of the preceding paragraph, the work done by 

 the bodily forces acting on the material element during its 

 displacement is equated to the change in the energy in the 

 element of volume originally occupied by the element before 

 its displacement, which is quite a different thing from the 

 change in the energy of the moving element itself. The 

 analysis is therefore fundamentally unsound and will require 

 considerable modification. To obtain its proper legitimate 

 form we must confine our attention to a finite portion of the 

 dielectric medium and follow it in its motion. We enclose 

 the portion by a surface / on its outer boundary and then 

 notice, with Larmor, that in the displacement a space is left 

 unoccupied on the one side of the surface and a new space 

 is occupied on the other ; the result is that the above 

 expression for variation of W, or at least that part of it 

 referring to the portion of the dielectric medium inside/, is- 

 now represented by 



BW= f^[-(/)(V,p^)-(D, (S*V)E) + (&V) \ E (D^E)1 



+ (df[pcf>-^(I)dE)l i Ss n , 



wherein the volume integral is taken throughout the portion 

 of the medium enclosed by the surface / and the surface 

 integral over its outer boundary or / itself ; Ss n represents 

 the outward normal component on /of the virtual displace- 

 ment given to the medium. The volume integral of the first 

 term in the first integral now transforms by integration by- 

 parts into the integral 



together with a surface integral over /equal and opposite in 

 sign to the first part of the surface integral already present 

 in 8W; and the integral of the last term in the first integral 

 is equal but opposite in sign to the remaining part of this 



