14 PARTICULAR CASES OF EQUILIBRIUM. 



to those which are thus produced by two homogeneous masses equal 

 in density and of opposite signs, one of which has moved through an 

 infinitely small distance. 



The medium may be considered to be polarized, and the axis of 

 electrical polarization is parallel to the direction along which the 

 displacement has taken place. 



In the present case the density of the layer at each point is pro- 

 portional to the corresponding thickness P'P of the part which is not 

 common to the two spheres. Denoting by cr this density on the line 

 of the centres, we shall have 



As the thickness of the layers along the line of the centres is con- 

 stant, the value of the density, at a point P at the end of radius 

 which makes the angle o> with this right line, is 



o- = (T O cos to = pS cos w. 



The action on a point M in the interior, is that of two 

 homogeneous spheres whose radii are AM and A'M, on a point 

 of their respective surface. Hence, for the sphere A, it is equal 



to TT/a.AM, and is directed along AM; for the sphere A', it is 

 3 



equal to -^Trp.MA', and is directed along MA'. The resultant is 



3 



therefore proportional to AA' and has the value 

 4 A, 4 * 4 



- 7T/0 . A A = 7T/06 = 7TO-Q ; 



j 5 5 



it is constant. Let us denote this force by F { , and reckon it posi- 

 tively from left to right, we shall have 



In the interior of the sphere, the equipotential surfaces are planes 

 perpendicular to the axis AA' and are equidistant ; the potential at a 

 point varies proportionally to the abscissa x of the point, and as it is 

 zero at the centre, we have 



