11] 



FIGURE OF THE EARTH. 



207 



remainder, divide it into shells with centre M. The attraction of a shell 

 with inner and outer radii MK, Mk, say p, p + Sp, is 



where KN is perpendicular to MO ; that is 



in which r stands for MO. 



This expression summed from p (r* + a" + 2rc)* to p = r + a. 



5. Attraction of a solid of revolution upon any point in its axis. 



Let us find the attraction of a circular lamina on a point situated 

 anywhere on a straight line through its centre perpendicular to its plane. 



Let M be the point, the centre of the circle, MO a, AO r. Let 

 Pp be an element of AO, MP = p, and let the attraction at distance p 

 be f(p). 



Join MP, Mp ; draw Pq perpendicular to Mp. 



The area generated by revolution of Pp about is 



2TT.OP. Pp. 



But by similar triangles Pp : pq = MP : PO ; 

 therefore OP.Pp = MP . pq, 



and the area in question is 2nMP . pq. 



The attraction of this elementary area upon M is in the direction MO 

 and is equal to 



and the attraction of the whole circular lamina is the integral of this 

 taken from p = a to p = (r 2 + a 2 )*. 



