180 NEWTON S PRHSTCIPIA. 



ference. Then the earth, being considered homogeneous, 

 the attraction of the spheroid on any point Z in OP, 

 resolved along this radius, will be proportional to its 

 distance O Z from the centre. The same is true for the 

 centrifugal force. Hence altogether the resolved part of 

 gravity at any point Z in O P resolved along the radius 

 varies as O Z. Let it be represented by co z, where z = OZ, 

 and co may vary with the position of P. The whole weight 

 of the column is therefore 



co/ z d z = -J- co r 2 ; 



J* 



and this must be the same for all radii ; hence co varies 

 inversely as r 2 , and therefore the attraction varies as 



2= , when Z lies on the surface. Hence the force 



of gravity towards the centre must vary on the sur 

 face of the same planet reciprocally as the distance of the 

 bodies from the centre of the earth. As the earth is 

 very nearly spherical, this must be also very nearly true 

 for the whole force of gravity. Let Gr then be the force 

 of gravity at the equator, g that at the point P, whose 

 latitude is X and radius r, then 



= I* - A / 1 __ 



= G- { 1 + e sin 2 A} nearly. 



We shall show in another chapter that the force of 

 gravity is proportional to the length of the seconds pen 

 dulum ; hence if / and L be the lengths in the latitude A 

 and at the equator, 



Z=L{1 + gsin 2 A}. 



The value of I was known in the latitude of Paris, 

 whence that of L can be found, and thence the length in 



