ATTRACTION OF BODIES. 269 



1 A 



integral will be found to be - x 7; - X (b* + /#) 



4 (a + b)* 3 n 



1 - n 5 - n 



(2a + 



and 



, 



the constant C is -. T - x - -- \- 



* 



. 

 4 (a + 6)* \o n I n 



b 3 ~ n J. This in every case gives an easy and a finite 



O """"* / / 



expression, excepting the three 'cases of n = 1, n = 3, and 

 n = 5, in which cases it is to be found by logarithms, or by 

 hyperbolic areas. To find the attraction of the whole sphere, 



when x = 2 a, we have . - rr- q X ( 7; - (2 a + &) 3 ~" 

 4 (a + b) a \3 n ^ 



x (2 a 



l -n 5 n ' 5-* ' 1 -n 



(2 a + Z>) 8 | for the whole area A N B, or the whole 



3 nj 



attraction. If P is at the surface, or AP = b = 0, and n = 2, 

 then the expression becomes as a, that is, as the distance from 

 the centre directly. We may also perceive from the form of 

 the expression, that if n is any number greater than 3, so that 

 n 3 = m, the terms & 3 ~" become inverted, and b is in 



their denominator thus : ~ r-*j. Hence if n > 3 and A P 



( JL ^"* lit j 



= 5=0, or the particle is in contact with the sphere, the 

 expression involves an infinite quantity, and becomes infinite. 

 The construction of Sir Isaac Newton by hyperbolic areas 

 leads to the same result for the case of n = 3, being one of 

 those three where the above formula fails. At the origin of 

 the abscissae we obtain, by that construction, an infinite area ; 

 and this law of attraction, where the force decreases in any 

 higher ratio than the square of the distance, is applicable to 

 the contact of all bodies of whatever form, the addition of any 



