274 ATTRACTION OF BODIES. 



the sphere, r its distance from the centre of the sphere, or the 

 distance of the ring from that centre, du consequently the 

 thickness of the ring, TT the semicircle whose radius is unity, 

 and F the function of f representing the attracting force. 

 The whole attraction of the sphere, therefore, is the integral 

 taken from f = r u to f = r + u, and the expression be- 



comes - - + ffdf X fdfY with (r + M) (r w), 



substituted for/, when / results from this integration. Then 

 let F =, or the attraction be that of gravitation ; ihe ex- 



2-K.uduCr C d f Zir.udu / 



pression becomes -- I fdf X r, = - - X - 



r J J f r 



1 2 w . udu (r 4- u) (r u) 2 -rrudu 



_ \s _ / _ v> 



",, ~ ~"xv>~ ~A~ 



f r 2 r 



u = 2nu*du X -; and the coefficient of dr. taking the 

 r 



2 w y 3 du 

 differential with r as the variable, is + - - s ; consequently 



the attraction is inversely as the square of the distance of the 

 particle from the centre of the sphere, and is the same as if 

 the whole sphere were in the centre.* 



* Mec. Gel. liv. ii. ch. 2. The expression is here developed ; but it 

 coincides with the analysis in 11.' 



1 This Tract and the last are both taken from the 'Analytical View of 

 the Principia,' Lib. I. 



