NEWTON S PRINCIPIA. 197 



Then since 



d v _ x 2 

 d t~ m 



dv x , 



v 2 m 



integrating throughout the motion 



s-y-s - - (1 -&amp;gt; 



Again 



.-. = L-a-^ 



V 771 



. * . integrating throughout the motion 

 , t; x 



I K v = * 

 .-..-y. .--*. - - (2.) 



Since u = , this equation is the same as 



e *&amp;lt;/* = Vrf 

 integrating throughout the motion 



5 l-_i = * . v &amp;lt; 



m 



- - (3.) 



From these equations we may gather every circumstance 

 of the motion. From (1) we learn that if the times are in 

 Arithmetical Progression the velocities are in Harmonica! 

 Progression ; and that the velocity varies inversely as the 



time when counted from an era -^.units of time before 



x V 



o 3 



