NEWTON S PRINCIPIA. 143 



= Vb 2 + 2 (a + b)x = ^b* + fx I and D N = y = (by 

 construction) ^- nTT~~^ *^ e attract i ve f rce f 



the particles being supposed as the -th power of the dis- 



n 



tance, or inversely as (b 2 +fx)*. This equation to the 

 curve makes it always of the order - . If then the 



force is inversely as the distance, A N B is a conic hyper 

 bola ; if inversely as the square, it is a curve of the fifth 

 order; and if directly as the distance, it is a conic 

 parabola ; if inversely as the cube, the curve is a cubic 

 hyperbola. 



The area may next be determined. For this pur- 



i ro-x xx 



pose we have fydx = I -~5+r~ 



J 2(b*+fx)~ 

 2 (af + Z&amp;gt; 2 ) = h, this integral will be found to be 



4 (a 



- 



x (2 a + b^ (b* + fx)-iT - ^ 



I / tf-n 



+ C ; and the constant C is -y-r x ( 



4 (a + 6) 2 \6-Ji 



(2 a + bYb*- n h \ . 



_^_ ^ i 3-n 1 . This in every case gives 



i n o ?i / 



an easy and a finite 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, 



