230 THEOKY OF NEWTONIAN FOKCES. [FT. I. CH. V. 



so that the vector g whose components are 



(9) = ^ + 47TA ) = F+47r, 3 = ^+47rC, 



has no divergence anywhere. Its normal component is continuous 

 at the surface of the body, for since by 82, 



" = 47T ^ C S ^^ + Fe OS ^^^ = ~ II C S (/ ^' 



(where the suffixes i and e denote values inside and outside, and 

 the opposite directions of the corresponding normals), we have 



( 10) g; cos (g;7i;) = Fi cos (FiUi) + 4-Tr/; cos (/^) 



= F e COS (-F> e ) = g e COS (g e W e ) = ge COS (ggW*). 



The vector g being everywhere solenoidal, its surface integral over 

 any closed surface vanishes, so that as many unit tubes enter as 

 leave the surface. Tubes leave the polarized body where a- is 

 positive, and enter it where a- is negative. They form closed 

 tubes, every one of which passes through the body. The vector g 

 is called by Maxwell the induction, and is characterized by the 

 solenoidal property. The line separating the region of positive a 

 from those of negative is linked with all the tubes of induction 

 belonging to the body. The induction is not in the same direction 

 as the force F unless the polarization / is. 



We obtain another physical conception of the induction by 

 considering the force in a cavity in the conductor. By hollowing 

 out a space in the body we remove a portion of the volume distri- 

 bution, but give rise to a new surface distribution. We shall 

 suppose the cavity so small that the volume-density of the part 

 removed may be considered constant. Now if we consider the 

 forces at corresponding points of geometrically similar distributions 

 of constant densities, we have for the action of the volume-density, 



fdr 



ffi 



and if we increase the dimensions in the ratio n, the element of 

 volume and the potential at a corresponding point are dr = n s dr, 



and the force 



_ 

 ds' d (ns) ds 



