144 NEWTON S PRINCIPIA. 



I s ) rt 



(2 a + ft} -- (2 



5 n o n 1 



for the whole area A N B, or the whole attrac- 

 n 



tion. If P is at the surface, or A P = b = 0, and 

 ft = 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 b 3 ~ n 

 become inverted, and b is in their denominator thus : 



fT~ n^&quot; Hence &amp;gt; if n &amp;gt; 3 and A P = b = 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 abscissa? 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 other matter to the 

 spherical bodies having manifestly no effect in lessening 

 the attraction. 



By similar methods we find the attraction of any por 

 tion or segment of a sphere upon a particle placed in the 

 centre, or upon a particle placed in any other part of the 

 axis. Thus in the case of the particle being in the centre 

 S, and the particles of the segment K B G attracting with 



forces as the - power of the distance S O or SI, the 



curve A N B will by its area express the attraction of 



I O 2 



the spherical segment, if D N or y be taken = 



