125, 126] POLARIZED DISTRIBUTIONS. 239 



126. Energy of Polarized Distributions. If a polarized 

 distribution is placed in a field of which the potential is F, then- 

 mutual energy is, by 117, 



which, by 120, is equal to 



( i ) W= - (JV {A cos (nx) + B cos (ny) + C cos (nz)} dS 



i dB 



+ ^ 



Integrating by Green's theorem, this becomes 



" 



The integrand is the negative of the geometric product of the 

 polarization and the force of the field. This result may be ob- 

 tained directly for a doublet as we obtained the potential in 

 122. 



If the polarization is lamellar, the energy of the distribu- 

 tion is 



dV 



For a polarized shell the volume integral disappears, and the 

 surface integral becomes 



Accordingly the energy of a polarized shell is equal to the 

 product of its strength by the flux of force through it in the 

 direction opposite to the polarization. 



If we wish to find the energy of the polarized distribution 

 itself, we must put for V in the above formulae the potential 

 due to the distribution itself, and multiply by the factor one- 

 half, as in 117. It is important to notice that the energy of 

 polarized distributions is defined as the work that they are capable 

 of doing if every particle is allowed to retire to infinity carrying 



