NEWTON S PRINCIPIA. 205 



.. . 2 _ 2 . p = - ( 

 dx m ^ \ 



/* , N 

 a . (e - 1) 



+ MgMc M _*9(~*-V 



2 A V 2 / 4 A 2 V 2 cos 2 a 



which is the equation to the path.* 



4. PROBLEM. To determine the motion of a particle 

 moving in a straight line in a medium resisting partly in 

 the ratio of the velocity, and partly in the ratio of the square 

 of the velocity, and acted on by a uniform force. 



Let the symbols V, v, x, m, t, f have the same meaning 

 that they had in the corresponding problem in which the 

 resistance varied as the velocity. Let the whole resist 

 ance R be represented by the formula 



K = KV + -v 2 



a 



then the whole moving force will then be 



f X 2 



mf xv v 



a 



and our equations of motion are 

 d v 



m 



dt 



dv 



mv -j 

 d x 



&amp;gt; = mf xv v 1 . 



c*. 



First. Let f = o or the particle move by its innate 

 force only. 



Then 



dv x K v 2 



d t m in* ex. 



* Earnshaw s Dynamics. 



