192 NEWTON S PRINCIPIA. 



As time goes on, the second factor on the right hand of 

 the first equation continually decreases. Hence v decreases, 

 until a time comes when 



= y 4- 



X 



or, 



m/ V v\ 



1 If T /C \ 



= ; lo s( 1 +?) 



/m, 



then v = or the body comes to rest. 



After this time, v becomes negative, or the body begins 

 to move in the opposite direction. This case has been 

 already considered. 



2. PROBLEM. Supposing the force of gravity to be uni 

 form and to tend perpendicularly to the plane of the 

 horizon, to determine the motion of a projectile in a 

 medium which resists in the ratio of the velocity. 



Suppose the body projected with a velocity V and in a 

 direction making an angle with the plane of the horizon. 

 Take this also as the epoch from which we measure the 

 time, and let v be the velocity and the angle the direction 

 of motion makes with the horizon and s the arc described 

 at any other time t. Let the position of the particle be 

 defined by two co-ordinates x and y, giving respectively its 

 distance from the point of projection measured along a ho 

 rizontal line, and its altitude at the time t. Our first ob 

 ject is manifestly to find x and y in terms of t. 



By the second law of motion each force produces its own 

 effect as if the particle were at rest and it the only acting 

 force. In considering then the motions parallel to any di- 



