NEWTON S PRINCIPIA. 189 



never be zero, it is always positive, and therefore 



m TT 

 x &amp;lt; - V. 



X 



The resisted particle can never reach a point distant 

 - V from the origin, and it takes an infinite time to de- 



X 



scribe this space. 



We may also represent the motion of the particle by the 

 several parts of an hyperbola. Construct an hyperbola 

 T P A, whose asymptotes are the per 



pendicular straight lines, O X, O Y. 

 Then P being any point, and 

 P N parallel to O Y, we know 

 that O. N . P N is constant, and 

 equal to one quarter the sum of 



the squares of the axes. Let the hyperbola be such that 

 this is equal to c 2 . 



Then take O B = V, and N = v. By (5), we 

 have, 



B N = - s. 

 m 



Hence the velocity being represented by O N, the 

 space described will be proportional to B N. 

 Also, P N = y 



m 

 = y d v 



X C 



m /&quot; 

 * xP / ^ 



or the time is proportional to the area P N B A. If the 

 hyperbola be so drawn that the number of units of area 



