NEWTON S PRINCIPIA. 187 



Then the whole moving force on the body will clearly 

 be 



m f K v. 

 Also we know that for any particle the accelerating force 



is -=-, and therefore the moving force is m -j-. Equating 



these two 



dv 

 m j = mf K v 



dv K 



This equation contains the whole of the motion. 



dx 



First. We know that v = ~, substituting, we have 



dv _ K^dx 



d~t ~J &quot; m~Tt 



. . integrating throughout the time t, 



If x were nothing, or the medium did not resist the 

 particle, we should have, 



v-V=ft - (3.) 



Hence the motion lost by resistance is 



x x 



that is, it is proportional to the space gone over. This is 

 Newton s first proposition. A similar proof will apply if 

 f be not constant. 



Secondly. Suppose/ =0; or that the particle moves 

 by its &quot; vis insita &quot; only. Then, 

 dv _ K 

 d t~ m 



m 



