NEWTON S PRINCIPIA. 195 



If P be any point in the curve, and N P M vertical 



x cos a 



MN= 



m cosa/ 

 put N A = f , N P = *j 



CM 



^co- 



-,log| - - (6.) 



m cos a TT 

 where a = - -5 V. 

 x cos p 



This is a very simple form of the equation to the curve 

 and enables us to investigate many of its properties with 

 ease. We learn that if the successive values of N A are 

 in geometric progression those of N P will be in arithme 

 tical progression. This is Newton s second corollary. 



It is also manifest from the manner in which we eli 

 minated t, that we have 



,-( 



tan a + . ^_ ) x - Z^L # - 

 x V COS a&amp;gt; 



or the particle moves in such a manner that its distance 

 from O A, measured parallel to any fixed straight line, 

 varies as the time. This is Newton s first corollary. 



Since O B . cos a = O C, and O C = V cos a 

 .-.OB-5V5 



X 



and since any point may be considered as the origin of 

 projection, we learn that the velocity at P is always pro- 



o 2 



