166 Mr. G. H. Livens on the Mechanical 



so that the forces acting will be derived from the variation 

 of W ; the variation with respect to <j> leads to the electric 

 forces, and that with respect to the material configuration 

 leads to the mechanical ones. The problem is to determine 

 the mechanical forces when there is electric equilibrium, that 

 is, when the variation with respect to <j> yields a null result. 

 The form of W above expressed does not lead to this null 

 result. We can, however, by integration by parts derive 

 the form 



W= \dv\ r <l>dp, 



-H 



the essence of this transformation being that in the new 

 integral the distribution of the energy among the elements 

 of volume dv of the medium has been altered. This form 

 does not satisfy the above requirements either, but by 



•combining the two forms we obtain 



W= {dvL<j>- f E (ME)l, 



whose variation with respect to $ is null as required ; 

 although, as integration by parts is employed, the variation 

 is not null for each single element of mass. This integral is 

 then taken to represent the actual distribution of the 

 organized energy in the medium when in electric equilibrium 

 and not merely its total amount, and variation of it with 

 respect to the material configuration should thus give the 

 actual bodily distribution of mechanical forcive, not merely 

 its statical resultant on the hypothesis that the system is 

 absolutely rigid. Now. in finding the variation of W arising 

 from a virtual displacement hs of the polarized material, we 

 have to respect the condition that the free charge pdv is 

 merely displaced, so that by the equation of continuity 



6>+(V,/^) = 0, 



and also that each element of the material is moved on with 

 its own elastic constants, so that if V' denotes the vector of 

 space variation taken without reference to the variation of E, 



oT> + (&?,V')I> = 0, 



while things have been arranged so that a variation of <j> 

 produces no result, — but only, however, no aggregate result 

 on integration by parts. The transitions at interfaces are 

 supposed to be gradual, so that the volume integrals can all 

 be extended over lie entire field, without the necessity for 



