204 FIGURE OF THE EARTH. [11 



two spherical surfaces with the common centre M, their radii being p and 



Then the elementary surface generated by Pp 



and the volume corresponding to this surface contained between the surfaces 

 of the shell =2irpSp.Nn. 



The attraction of this matter upon M 



= 27rp3p . Nn .f(p) MN/p = irf(p) Sp (MN* - Mrf) = v f(p) 8p . 8 (MN"~). 



Hence if QS be drawn perpendicular to MO, the attraction of the whole 

 shell whose internal and external radii are p and p + 8p is 



irf(p) 8p (MR 2 - MS 1 ) = Trf(p) Sp . QS**. 



Now in the triangle MQO, since QS is drawn perpendicular to MO, 



we have 



2 + r- a 2 



MS = ' 



hence 



2r 



/V> 2 + ?' 2 -' 

 2r 



and the attraction of the whole sphere 



17 



= L r/(p) [(r + ) 2 - P*] [ P 2 - (r - a) 2 ] dp. 



*' J r-a 



p This is equivalent to Newton's Prop. LXXIX. 



