121, 122] POLARIZED DISTRIBUTIONS. 231 



while for surface distributions 



r ,_ T f/*S' ff 1<Mf _ BF 8F '_ 3F ' 

 'jjf J}'m r "M~ ~8T 



Accordingly as we decrease the linear dimensions indefinitely, the 

 force from the volume distribution decreases indefinitely, while the 

 force from the surface distribution remains finite. Consequently 

 in an infinitely small cavity the force does not depend on the 

 volume density of the part removed, but only on the surface 

 densities formed on the surface of the cavity. This will be at 



all points 



(T I cos (In), 



the normal being directed into the cavity. Suppose the cavity is in 

 the form of a cylinder with generators in the direction of the 

 polarization. Then the density on the sides is zero, and on the ends 

 / and /. If a is the radius of the cylinder, 26 its length, the 

 action of the ends on a point at the center of the cylinder is the 

 same as the action of two circular discs, of surface density / and 

 -/, which, by 81, is 



* -il. 



+ a 2 j 



This is a function only of a/6, as we have just shown that the action 

 is independent of the linear dimensions. If the radius is infinitely 

 small in comparison with the length the action vanishes. Accord- 

 ingly in such a cavity the force is that due to the action of the 

 rest of the body, or 



Y dV dV dV 



A= ^r, Jr = -^ , Zl ;r . 



ox dy oz 



If on the other hand the length of the cylinder is infinitesimal in 

 comparison with the radius, the force is 4?r/, so that the total force 

 in the cavity is 



or the induction is equal to the force in a thin crack perpendicular 

 to the lines of polarization. 



122. Potential due. to Polarized Distribution. If we 



introduce the expressions of the volume and surface densities in 

 terms of the polarization, we obtain for the potential due to a 

 polarized distribution 



