362 ELECTROSTATICS AND MAGNETISM. [PT. II. CH. IX. 



But using the equation for the apparent surface density (21) when 

 the true surface density is zero, 



(29) 



so that if we put 



(30) /-T 



giving / in the direction of F and g, we obtain the proper surface 

 density. From the components of /, 



> 



we obtain for the volume density due to the polarization, 120 (6) 

 fdA dB dC\ 



(32) p' =__ + _ + _ 



\dx dy dzj 



a* ag) a3\ i /aar 3F 



and since by ( 1 8) we have 



we must have 



(34) ? + ?+f = 0- 



dx dy dz 



or the induction is solenoidal. The induction accordingly possesses 

 the property of the vector called the induction in 121, and by the 

 equations (31) is equal to it if /^ is equal to unity. 



187. Examples. Point-charge in Medium bounded by 

 Plane Face. Suppose we have a point-charge e placed at P at 

 a distance a from a plane face separating two media of inductivities 

 /AJ, /* 2 , their extent being infinite. We may solve the problem of 

 induction by the method of images as in 152. Suppose that e 

 lies on the left of the dividing plane, and that at its geometrical 



