164 Mr. Gr. H. Livens on the Mechanical 



mainly with the mechanical forces on the medium as a 

 whole, for it is the part of the total that remains when the 

 internal configuration determined by the polarization is main- 

 tained constant during the establishment of the system. 



The function W m may now be regarded as the potential 

 function of the required mechanical forces. The expression 

 for the forcive on the element on account of the polarization 

 in it then follows as a matter of course, and its linear 

 constituents are represented per unit volume by the com- 

 ponents of the vector 



-gradW m =grad(PE), 



the differential operations not, however, affecting P ; the 

 angular components are similarly determined and are 

 represented by the vector 



[PE]. 



When a potential o£ force exists the former vector is 

 equivalent to 



(PV)E, 



wherein V is the usual Hamiltonian vector operator. Thus 

 in the case of isotropic media, when the induction of the 

 polarization follows a linear law so that 



the forcive per unit volume is completely specified by the 

 vector 



^-pigradE 2 . 



The additional forcive which acts on the medium on 

 account of the distribution of free charge of density p 

 throughout it is represented by a force on the element of 

 volume at any point whose intensity per unit volume is 



P E, 



xmd this, combined with the above forcive arising on account 

 of the polarization, is completely represented in the most 

 general possible case by an applied stress system whose nine 

 components are of the types 



T IX =E X B X -^W, T JS =E I D„ T„=:E,D„ 



wherein 



D=^E + P 



47T 



is the total displacement vector of Maxwell's theory. 



