NEWTON S PRINCIPIA. 211 



locity&quot; when the body moves in a straight line, under the 

 action of a uniform force. This may be defined to be that 

 velocity which makes the resistance equal to the moving 

 force acting on the particle. Let &amp;lt;p (v) be the law of re 

 sistance, and mg the moving force. Then if u be the ter 

 minal velocity, we have 



&amp;lt;p (u)=mg 



an equation to find u. Suppose the resistance to vary as 

 the square of the velocity and $ (v) = x v 2 , then 



&quot;A/ 



m g 

 x 



It is manifest that if the body were projected with this 

 velocity, it would continue to move uniformly in the me 

 dium. This is a consequence of the first law of motion. 

 Also if the particle began to fall from rest, its velocity will 

 continually increase under the action of the force g, and 

 approach equality with the &quot; terminal &quot; velocity, and only 

 become equal to it when the time is infinite. All this is 

 quite manifest and needs no analytical investigation. 



Though the velocity of the particle never becomes equal 

 to the terminal velocity, yet it soon becomes so little dif 

 ferent from it that, for all practical purposes, we may con 

 sider the particle as moving with an uniform velocity equal 

 to the terminal velocity. In considering the motion of a 

 falling body we arrive at the equation, 



. s m 



Now if -- be a very small quantity, it does not require a 



very large value of x to render the second factor so small 

 that we may without much error consider 



X 



p 2 



