202 FIGURE OF THE EARTH. [ll 



Join MO meeting the surface in A, A'. Through M draw any line 



MP cutting the surface in P, P', and let Mp be drawn in the plane 



OMP making an indefinitely small angle with MP and cutting the surface 

 in p, p'. 



Also draw OK, Ok perpendicular to MP, Mp respectively ; PN, pn 



perpendicular to MO, Pq perpendicular to Mp and Pr perpendicular to pn ; 

 let Ok meet MP in h. 



The surface of the elementary zone of the shell generated by revolution 



of the arc Pp about MO is 



27T.PN. Pp; 



but by similar triangles Pp : Pr = OP : PN. 

 Therefore PN.Pp = OP. Pr = OP . Nn, 



and the surface of the elementary zone is 



277 . OP . Nn = 2-rra . Nn, 



and the surface of the whole sphere, obtained by summing all such elements, 



. 2a = 



The zone in question evidently attracts M in the direction MO, and 

 since each element of the zone attracts M in a direction making the same 

 angle with MO, we have, resolving the attractions in this direction and 

 measuring the mass of the zone by its surface, 



MN 

 Attraction of zone - 2ir . PN . Pp . p - . 



But by similar triangles 



PN : M P = OK : MO, MN : MP = M. K : MO. 



r f OK.MK Pp 



Iheretore Attraction = 2ir =- . W. 



Also by similar triangles 



Pp:Pq=OP: PK and Pq : hk = MP : MK ultimately. 



Hence Pp_OP.MP Pp QP . hk 



hk'PK.MK - 



and Attraction of zone = 2ir . -->,. 



But Ok* - OK* = PK'' -pk* ; 



therefore ultimately hk. OK=(PK-pk)PK. 



Hence the attraction of the zone is 





