122 124] POLARIZED DISTRIBUTIONS. 235 



while the potential due to the doublet is 



a spherical harmonic of degree 2. 



124. Solenoidal and Lamellar Polarizations. The 



volume-density of polarized matter has been found, 120 (6), 

 to be equal to the convergence of the polarization. If the polari- 

 zation is solenoidal, the volume -density vanishes, and the polariza- 

 tion is equal to a surface distribution, as in the original assump- 

 tion of 120. We may then divide the body into tubes of 

 polarization, or polarized solenoids. Such a solenoid possesses the 

 property that if it be cut anywhere the two cut ends will bear 

 equal and opposite charges, their amounts being the same wherever 

 the cut be made. The potential due to a solenoid of infinitesimal 

 section depends only on the position of its ends, and a solenoid 

 may be considered as equivalent to a doublet of points at a finite 

 distance apart. Again the polarization may be lamellar, that is it 

 may be the vector differential parameter of a function < which 

 will be called the potential of polarization. We then have 



/o\ A d<t> D 3<t> n d<!> 



A ^^' B= tt' G= Tc- 



Outside the polarized body, since / = 0, </> must be constant, 

 and accordingly discontinuous at the surface. 



Inserting this in the value of V the potential becomes 



Applying Green's theorem we obtain 



(io) F= 



But since 1/r is harmonic except for r = 0, if the attracted 

 point is outside of the polarized body, V is given by the surface 

 integral, 



(ii) F = - 



