NEWTON S PRINCIPIA. 



209 



If the resistance vary as v n and the particle move by its 

 own ft vis insita &quot; only. Then since 



d v __ d v 

 J~t = V d~x 





. . integrating both equations throughout the motion, 



- D * 



v n-i y 7 &quot;&quot;&quot; 1 m 



equations which never become nugatory except when 

 n=l or n 2) both which cases have been already con 

 sidered. 



From these equations we may learn several remarkable 

 facts. 



First. Suppose n greater than 2. Then both the right 

 hand members of the above equations are positive ; hence v 

 can never vanish, and the body will continue moving for 

 ever, with an ever diminishing velocity, and will pass over 

 an infinite space. 



Secondly. Suppose n greater than unity but less than 2. 

 Then v vanishes only when t is infinite, but then 



- 

 = x 



2 - n V&quot;- 2 



So that the particle t continues to move always with an 

 ever diminishing velocity, and will pass over a finite space. 

 Thirdly. Suppose n less than unity, then when v 

 vanishes, we have 



- - 

 = 



m 



= 



that is, the particle moves on with a diminished velocity, 



