11] FIGURE OF THE EARTH. 203 



Similarly the attraction of the zone generated by revolution of P'p' may 

 be shewn equal to the same quantity, and 



Attraction of the two zones = --- . 8PP'. 



r 2 



Now if we sum the attractions of all the elements that make up the 

 entire shell, since PP' varies from 2a when MP passes through to zero 

 when MP becomes a tangent, we have 



. . r. i i i 11 27TO. 47ra" M 



Attraction or spherical shell = - . 2a = - - = , 



/** /V* ,ji- 



since the mass is supposed to be measured by its surface. 

 Again, the potential at M of the zone generated by Pp 



PN . OK.OP.MP 



and the potential of the zone generated by P'p' 



OK.OP.MP' 



n - 



Together, the potential of the two zones is 



OK.OP(MP + MP') OP. OK 



~M07PK7MK~ * MO.PK ' M ' 



or, as above, = - . SPP', 



r 



and the potential of the whole shell = -- . 2a = 



r r 



In the same way we may prove that the attraction at an internal 

 point is zero and the potential constant*. 



2. Attraction of a sphere, supposing the law of attraction to be any 

 whatever. 



Let be the centre of the sphere, a its radius, M an attracted point, 

 OM=r, and let the attraction at distance p be denoted by f(p). 



Let a sphere described with centre M and any radius MQ = p cut the 

 attracting sphere in the spherical segment which is generated by revolution 

 of the arc QR about MO. Let Pp be an element of the arc RQ and 

 imagine an indefinitely thin shell of the attracting matter included between 



* Compare Newton's Principia, Book i. Prop. LXXI. 



262 



