196 NEWTON S PRINCIPIA. 



portional to the tangent P T. This is Newton s sixth 

 corollary. 



If / be the latus rectum of the parabola that would be 

 described under the same circumstances of projection, if 

 the medium offered no resistance, then 

 l _ 2 Y 2 cos 2 a 



OCf 

 9 



3. PROBLEM. To determine the motion of a particle moving 

 in a straight line in a medium resisting in the ratio of the 

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



Let the symbols V, v, x, m, t, * 3 have the same meaning 

 that they had in the corresponding problem in which the 

 resistance varied as the velocity. Then the whole moving 

 force upon the particle will clearly be 



mf x v 2 

 also we have the accelerating force on a particle moving 



-, . dv , . . . dx , 



m any manner equal to ; which, since v = , may also 



u t u t 



be put under the form v . Taking both these forms, 

 we have 



dv 



mv-j- = mf K 

 dx 



which are identical equations. The first equation gives v 

 in terms of t } the second v in terms of x. 



First. Suppose / = 0, or that the body moves by its 

 &quot; vis insita &quot; only. 



