POLARIZED SPHERE. 141 



For the outside, the layer in question may be replaced by two 

 homogeneous spheres, or by two masses of opposite signs equal 



to -7rR 3 /> concentrated at A and A', and the moment of which is 

 tar = 8 . - 7rR 3 p = - 7rR 3 o- = UO-Q, 



<J O 



u being the volume of the sphere. We shall have then, for the 

 distance r> in a direction at an angle o> with the axis, 



cos w x 



The surface of the elementary zone du being 



dS = 27rR 2 sin o> dfo>, 

 the corresponding mass is 



27rR 2 cr sin w cosco du = d. 7rR 2 cr sin 2 o>. 



The total mass M of each of the layers is then 



o-^S = Y d. 7rR 2 (r sin 2 <o = 7rR 2 (r . 



M 



This result might have been directly obtained by considering 

 that at each point of the layer the thickness along the axis being 

 equal to 8, the total volume is equal to the product of this constant 

 thickness by the projection of the hemisphere on a plane perpen- 

 dicular to AA'. 



The flow of force from the positive layer is 



Q = 4 7rM = (47rR 2 ) 7T(r = STTO-, 



0' 



S being the whole surface of the sphere. 



158. It will be useful to collate here all the preceding results, 

 and to express each of them by quantities as a function of the 



